Arabic Origins Of Cryptology Vol. 3

Series on Arabic Origins of Cryptology Volume Three

ibn ad-Durayhim's Treatise on Cryptanalysis

Series editors MOHAMMED MRAYATI, Ph. D. YAHYA MEER ALAM, Ph. D.

M. HASSAN at-TAYYAN, Ph. D.

Published by KFCRIS & KACST

Acknowledgment The editors of this series greatly appreciate the encouragement they had from Dr. Yahya Mahmoud Ben Jonayd, Secretary General of King Faisal Center for Research and Islamic Studies, to publish this Series. We are also in the debt to Dr. Saleh Athel, the president of King Abdulaziz City for Science and Technology (KACST), for supporting the project of translating this series to English. Many thanks to Dr. Daham Ismail Alani, the Secretary General of the Scientific Council of KACST, for all his efforts to make this publication possible. The typing and set up, of this bilingual version of the series, was realized with skill and dedication by Mr. Ousama Rajab, we offer hearty thanks to him. Finally, we would like to re-mention our recognition to the many who had previously contributed to the Arabic version of this series, and particularly to Dr. Wathek Shaheed, Dr. Chaker al-Faham, the late Prof. Rateb an-Naffakh, and Dr. Fouad Seskeen.

Series on Arabic Origins of Cryptology Volume 3

Translated by Said M. al-Asaad

Revised by Mohammed I. AL-Suwaiyel, Ph. D. Ibrahim A. Kadi, Ph. D. Marwan al-Bawab

Contents List of Figures .…........................................................................... vii List of Tables ......…….................................................................... viii Transliterating Arabic words ....................................................... ix Preface ............................................................................................ xi

Chapter 1: Analytical study of ibn ad-Durayhim’s treatise: Mift al-Kun z f ' al-Marm z ...................... 1 1.1 Biography of ibn ad-Durayhim ......................…................... 3 1.2 Study and analysis of ibn ad-Durayhim’s treatise ................ 5 1.3 Structure of the treatise .......................................................... 6 1.3.1 Essentials for those practicing cryptanalysis .................. 6 1.3.2 Types of encipherment ................................................... 7 1.3.2.1 Transposition ………............................................... 8 1.3.2.2 Substitution ..........................................…............... 10 1.3.2.3 The augmentation or reduction of the number of letters ...................................…................................ 21 1.3.2.4 The utilization of cipher devices ............................. 21 1.3.2.5 The replacement of letters by numbers, using the decimally-weighted numerical alphabet ................. 22 1.3.2.6 The encipherment of letters by using words ........... 25 1.3.2.7 Replacing letters by generic names ......................... 27 1.3.2.8 Using invented symbols or signs to represent letters ...................................…............................... 29 1.3.3 Morphological introduction ........................................... 30 1.3.4 Algorithm for cryptanalysis ........................................... 33 1.3.5 Two practical examples of cryptanalysis ....................... 33 1.4 Originality of ibn ad-Durayhim .............................................. 34

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Chapter 2: ibn ad-Durayhim's edited treatise: Mift al-Kun z f al-Marm z ........................ 35 2.1 Editing methodology ............................................................. 37 2.2 Description of the manuscript ............................................... 38 2.3 Al ibn ad-Durayhim Treatise on Cryptanalysis ................... 47  Introduction ....................................................................... 50  Essentials for those practicing cryptanalysis ..................... 52  Types of encipherment ...................................................... 56 1. On transposition ............................................................ 56 2. On substitution .............................................................. 60 3. On the augmentation or reduction of the number of letters ............................................................................ 66 4. On the utilization of cipher devices .............................. 66 5. On the replacement of letters using the decimallyweighted numerical alphabet ...................................... 68 6. On the encipherment of letters by using words ............ 70 7. On enciphering by relationship and diffusion method 76 8. A return to the type on the utilization of cipher devices 80 9. On using invented symbols or signs to represent letters 82  Morphological introduction .............................................. 84  Algorithm for cryptanalysis .............................................. 98  Example 1 ....................................................................... 102  Example 2 ....................................................................... 116

vi

List of Figures 1.1 ibn ad-Durayhim's circle of letters ............................................. 13 1.2 ibn ad-Durayhim's method of encipherment by substitution of numbers for letters using the decimally-weighted numerical alphabet .................................................................................... 23 2.1 A photocopy of the index of the assemblage incorporating ibn ad-Durayhim's treatise ....................................................... 40 2.2 A photocopy of the first page of ibn ad-Durayhim's treatise. .. 41 2.3 A photocopy of a page of ibn ad-Durayhim's treatise illustrating encipherment using the "branched" calligraphy ..................... 42 2.4 A photocopy of a page of ibn ad-Durayhim's treatise demonstrating the encipherment of the first of two examples 43 2.5 A photocopy of a page of ibn ad-Durayhim's treatise demonstrating the encipherment of the second of two examples ................................................................................. 44 2.6 A photocopy of the encipherment of ibn ad-Durayhim's second example as set out in ub al-'A ........................................ 45 2.7 A photocopy of the last page of ibn ad-Durayhim's treatise ...… 46

vii

List of Tables 1.1 Calligraphs and alphabet size (number of letters) in different languages, as given by ibn ad-Durayhim ................................ 7 1.2 The alphabetical and numerical-alphabet letters, with their corresponding cipher alphabets .............................................. 11 1.3 Dual order of letters for some cipher alphabets ....................... 12 1.4 Table of encipherment, following the first method, using the numerical-alphabet order ......................................................... 14 1.5 Table of encipherment, following the second method, using the alphabetical order .............................................................. 15 1.6 Table of encipherment as given by ibn ad-Durayhim, following the third method using the numerical-alphabet order .....….... 17 1.7 Table of encipherment as given by ibn ad-Durayhim, following the fourth method using the numerical-alphabet order ....….... 18 1.8 Table of the order of letters of both the alphabet and numerical alphabet in eastern and western Arab worlds, together with an Indian numerical alphabet .....…............................................. 20 1.9 Table of the letters of the numerical alphabet with corresponding decimal numerical values ................................ 24 1.10 A list of generic names which match the Arabic alphabet, used by ibn ad-Durayhim ........................................................ 28 1.11 Table of non-combinable letters as observed by ibn ad-Durayhim ........................................................................... 32

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Transliterating Arabic words For transliterating Arabic words (names, titles, etc.) we have adopted the International System for the Transliteration of Arabic characters, devised by the International Standards Organization (ISO). The system constitutes ISO Recommendation R233 (December 1961). Given below is this system, with some additional explanations found to be necessary.

Vowels: Arabic characters

Short Vowels

Transliteration

Examples

(fat a)

a

as u in cup.

( amma)

u

(kasra)

i

as o in rock, and u in put. as e in red, and i in big. As a in last.

Long Vowels

‫( ڇ‬preceded by

)

as oo in moon.

‫( ي‬preceded by

)

as ee in sheet.

Consonants: Arabic characters

Transliteration

'

Examples (e.g. 'amr, 'ibr h m, fu' d, kis ' , t '). as a in add (e.g. ' dam, qur' n).

‫ة‬ ‫د‬ ‫س‬ ‫ط‬ ‫ػ‬ ‫ؿ‬ ‫ك‬ ‫م‬ ‫ه‬

b t

as b in back. as t in tea. as th in thin. as g in logic.

d

(e.g.

tim).

(e.g.

lid).

as d in day. as th in then.

r

as r in red.

ix

‫ى‬ ً ُ ٓ ٗ ٛ ٟ ٣ ٧ ٫ ‫ٯ‬ ‫ٳ‬ ‫ٷ‬ ‫ٻ‬ ‫ٿ‬ ‫څـ‬ ‫ڇ‬ ‫ي‬

z s

as z in zoo. as s in soon. as sh in show. (e.g. mi r). (e.g. ir r). (e.g. riq). (e.g.

fir).

(e.g. Abb s). (e.g.

f q k l m n h w Y

lib).

as f in few. (e.g. qur' n). as k in key. as l in led. as m in sum. as n in sun. as h in hot. as w in wet (e.g. wahab, nawfal). as ie in orient (e.g. y q t, dunayn r).

Notes: (t ' marb a): In the absolute state, ignored in transliteration (e.g. mad na); in the construct state, rendered by (t) (mad nat annab ). (suk n): Ignored in transliteration. ( adda): rendered by doubling the consonant.

x

Preface This is the third book of the The Arabic Origins of Cryptology series, which addresses the cryptological contributions of the Arabs, and translates the treatises of Arab cryptologists. An individual book is dedicated to each treatise. The first book was devoted to the oldest ever found treatise in cryptanalysis, which was written by the well-known Arab philosopher al-Kind about 1200 years ago. The second book of the series tackles the treatise of ibn Adl n, while the third book (this one) deals with the treatise of ibn ad-Durayhim. For the time being, nine books are envisaged, unless more manuscripts are discovered. As a matter of fact the first three books of the series are the translated copy of Volume One of our Arabic book entitled " ilm at-ta miya wasti r al-mu amm inda al- Arab" (Origins of Arab Cryptography and Cryptanalysis). This volume has been published in Damascus in 1987. In Book One we have allotted a full chapter to study and analyse cryptology among the Arabs. We hope this will prove useful for understanding the whole series.

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*

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We have divided this book into two chapters. The first chapter presents an analytical study of the edited treatise of ibn ad-Durayhim. It aims at elucidating difficult or vague points, spotting particular features and, more importantly, highlighting aspects of originality and innovation in the treatie. It is divided into four sections, the first of which gives a brief biography of ibn ad-Durayhim. The second section is a full study and analysis of ibn adDurayhim's Treatise, while the third section delineates its structure. This section contains a preface, rules in cryptanalysis, and a conclusion of practical example of live ciphered message, explaining the steps ibn ad-Durayhim follows in cryptanalysing it.

xi

The fourth section concludes the analysis of the treatise by a summary exposing the aspects of ibn ad-Durayhim's scientific originality.

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The second chapter comprises a translation of the original text of the edited treatise of ibn ad-Durayhim. It opens with a preamble to the editing methodology adopted (Section 1), which basically conforms to that commonly used by the scientific community. The treatise is preceded with a brief description of the manuscript, followed by sample photocopies of pages from the original (Section 2). The treatise itself (in Arabic) together with its English translation represent Section 3, with the English translation on the left-hand pages, and the original Arabic text on the right-hand pages. The task of editing the manuscript text was a challenge indeed. No effort has been spared correcting the mistakes and clearing the ambiguous. Wherever appropriate, lead-in headings have been added to designate the different divisions of the treatise itself.

Damascus, December 2003

Dr. Mohammed Mrayati Dr. Y. Meer Alam

Dr. M. H. Tayyan

xii

Chapter 1

Analytical Study of ibn ad-Durayhim’s treatise: Miftah al-Kunuz fi 'Idah al-Marmuz

1.1 Biography of ibn ad-Durayhim He is Al ibn Mu ammad ibn Abd al- Az z, T ad-D n, alias ibn ad-Durayhim. He was born in Mosul in the month of a b n of the year AH 712/ AD 1312. Raised there as a wealthy orphan, ibn adDurayhim received his education at the hands of many prominent scholars of that time. He travelled frequently as a merchant between Cairo and Damascus, and was appointed as a teacher at the 'Umayyad Mosque in Damascus. He moved to Egypt in AH 760/ AD 1359 and was sent by Sult n an-N ir as an emissary to the king of Abyssinia (now Ethiopia). Going against his will, he reached Qaw , and died there in the month of afar of the year AH 762/ AD 1361. He was well-versed in many sciences such as the fiqh (Islamic jurisprudence), the ad (Prophetic tradition), the modes of reading the Holy Koran (different phonetic phenomena of the Koranic language) and interpreting its meanings. In addition, he was famous for his ingenuity in arithmetic, solving riddles and rebuses, and in cryptanalysis. He was also knowledgeable in al-'awf q (an old science dealing with numbers: their special combinations, values and secret characteristics), and in the letters of the alphabet and their statistical and phonetical properties. He wrote many works in these fields which testify to his distinction.

His works ibn ad-Durayhim was a very prolific writer, despite his short life of less than fifty years. His works were as diversified as was his encyclopedic knowledge. We found a - afad to be the most thorough biographer in listing his works, as he mentioned approximately eighty of his compilations, most of which were not mentioned in other published biographical sources. What makes a - afad 's biography of ibn ad-Durayhim more valuable is the fact that the biographer explicity stated that the works he listed were handwritten by ibn adDurayhim himself. The following are the titles of some of his books that are likely to be related to esoteric sciences, and to cryptology in particular:

3

1. 'iqtin 2. '

al- u

q f 'anw al-'awf q. (On types of al-awf q)

al-mubham f

all al-mutar am.

(On cryptanalysing cipher texts)

3. ' q

al-mu b f a - a ran wal-man

4. bas al-faw 'id f ar

b. (On chess games)

is b al-qaw id. (On languages)

5. baw dir al- ul m f naw dir al- ul m. (On knowledge and science)

6. ta r f ad-dahr f ta r f az-za r. (On languages) 7. tan ' al-man ir f al-mar ' wal-man ir. (On physics) 8. sabr a - arf f sirr al- arf. (On spirituals) 9. sullam al- ir sa f

ilm al-fir sa. (On physiognomy)

10. ar al-as ardiyya f al- is b. (On arithmetic and computation) 11.

yat al-'i

z f al-'a

wal-'al

z. (On riddles and enigmas)

12.

yat al-ma nam f al-'Ism al-'A am.

(On the supreme name of God)

13. A poem in all rum z al-'aql m al-makt ba al al-bar b . (A poem on cryptanalysis)

14. kanz ad-durar f

ur f 'awa'il as-suwar.

(On the individual letters introducing some Koranic chapters)

15. mu ta ar al-mubham f

all al-mutar am.

(A résumé in cryptanalysis)

16. mift

al-kun z f '

al-marm z. [This book]

(Key to treasures in clarifying ciphers)

17. al-mun sab t al- adadiyya f al-'asm ' al-mu ammadiyya. (Numerical proportions in the names of Prophet Mu ammad)

18. mun sabat al- is b f 'asm ' al-'anbiy ' al-ma k r n f al-Kit b. (Numerical relations in the names of prophets mentioned in the Koran)

19. na m liqaw id fann al-mutar am wa aw bi ih. (On the rules and regulations of cryptology)

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1.2 Study and analysis of ibn ad-Durayhim’s treatise ibn ad-Durayhim's treatise Mift al-Kun z f ' al-Marm z is a masterpiece in its precision and coverage of the bulk of information known of this science at the time, insomuch that it is the most comprehensive and far-ranging of all the manuscripts that have come under our hands1. It also marks a fully-fledged past master who practiced cryptology since he worked in the employ of such kings as Sultan al-malik an-N ir. ibn ad-Durayhim wrote several books on cryptology, some of which he touched upon in the introduction of this treatise. He had first written his book al-Mubham all al-mutar am, then abridged it in another book, which was lost. Some time later, he committed to paper a sufficient amount of what had remained in his memory of the rules of this art, in compliance with the request of a notability "Who must be obeyed, and whose request cannot be refused"2. Mift alKun z was really a by-product of that work.

1

Some historians of science and cryptology numbered ibn ad-Durayhim’s manuscript among the lost books. David Kahn, for instance, says of Mift alKun z f ' al-marm z: "Though this must be included among the Lost Books of cryptology, most of its information was probably preserved in Qalqashandi". See The Codebreakers, p. 95. 2 See his treatise, p. 51.

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1.3 Structure of the treatise ibn ad-Durayhim's treatise is divided into five sections, each of which is composed of several related topics. Following the introduction these are: 1. Essentials for those practicing cryptanalysis. 2. Types of encipherment. 3. Morphological introduction. 4. Algorithm for cryptanalysis. 5. Two practical examples of cryptanalysis.

1.3.1 Essentials for those practicing cryptanalysis In this section ibn ad-Durayhim sums up what the cryptanalyst needs to know before plunging into cryptanalysis. A cryptanalyst should be well-acquainted with: 1. The cryptogram language he is seeking to break. 2. Language grammar. 3. Frequency of occurrence of letters, and their order. 4. Letters detachable and conjoint with both preceding and following letters. 5. Number of letters of each language. 6. Alphabets and numerical alphabets. 7. Types and methods of cryptography. ibn ad-Durayhim accommodates his treatise with significant information about the various languages known at the time ‫ـــ‬an accomplishment that bears out his familiarity with them. He also confirms what al-Kind had already mentioned concerning the fact that long vowels have the highest frequency of occurrence in all languages. Yet, he does not adopt al-Kind 's designation; nor is there any mention of the short vowels along the lines of al-Kind . In this bearing, no doubt, ibn ad-Durayhim comes short of al-Kind 's standing and depth of comprehension. He moves on to discuss the letters of frequent occurrence in certain languages, drawing the conclusion that it is (A) "alif" in Arabic, (S) in Latin and Armenian, and (N) in Mongol. He has been so perceptive as

6

to invite attention to calligraphs with detachable letters, and those with conjoint letters, concluding that all calligraphs have detached letters, short of Mongol, Syriac and Arabic, some letters of which are detached, while others can be both detachable or conjoint. It is worthy of remark that in cryptanalysis, acquaintance with the number of letters of many languages is a matter of primary importance. ibn ad-Durayhim's erudition and vast knowledge of a good few languages of his time seem to have enabled him to grasp those languages with the largest number of characters (e.g. Armenian and Hindi), and those with the fewest number (e.g. Mongol and Sumerian). The following table (Table 1.1) demonstrates those languages and number of letters in their alphabets according to ibn ad-Durayhim's citation: Calligraphs

Number of letters

Mongol Sumerian Persian Turkish Hebrew Syriac Astankily Greek

17 18 20 20 22 22 22 24

Old Latin

24

French Latin One of the Hindi languages Coptic Armenian Another of the Hindi languages

27 27 28 32 36 52

Remarks

They have another calligraphy (30) for the uncertain of their letters.

They also have a numerical alphabet. For some Indians; called by ibn ad-Durayhim "the triangular Hindi"

Table 1.1: Calligraphs and alphabet size (number of letters) in different languages, as given by ibn ad-Durayhim

1.3.2 Types of encipherment According to ibn ad-Durayhim, encipherment types may be itemized under the following eight types:

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1. Transposition. 2. Substitution. 3. The augmentation or reduction of the number of letters. 4. The utilization of cipher devices. 5. The replacement of letters by numbers, using the decimallyweighted numerical alphabet. 6. The encipherment of letters by using words. 7. Replacing letters by generic names (i.e. applying the relationship and diffusion method). 8. Using invented symbols or signs to represent letters. ibn ad-Durayhim preeminently excelled in explaining methods of encipherment, analyzing different potentialities, invoking illustrative examples, and propounding qualifications for each method. This made ibn ad-Durayhim rank literally first in this regard (i.e. in his extensive explanation and analysis of methods) among all authors of the manuscripts we have so far edited. Besides, he represented, for the first time ever, a number of topical completely new methods. By comparison, al-Kind , for instance, had thoroughly covered the various methods, but without touching upon the possibilities and conditions that govern them; furthermore, ibn Adl n had not at all mentioned in his al-Mu'allaf lil Malik al-'A raf any method of encipherment, although he might probably have done so in his other work al-Mu lam. Notwithstanding the fact that many of ibn adDurayhim's methods were already approached by ibn Dunayn r, the latter did not rise up to the ibn ad-Durayhim's status in terms of elaboration, analysis and clarity. Hence we believe that the importance of ibn ad-Durayhim's treatise stems more from his citation and analysis of encipherment methods than from his practice of cryptanalysis. The following paragraphs give a brief account of each of the eight types:

1.3.2.1 Transposition Opening right from the beginning with this type, and following it by the substitution type, evinces ibn ad-Durayhim's awareness and realization of the significance of these two methods, as the core of the

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whole science of cryptography in all times. Transposition, according to him, can be divided into three types, depending on the extent of the transposition process. These are transposition within:  a single word.  two words.  the whole message. He then gets down to details, making ramifications for every type. It suffices here to give two examples of his transposition encipherment, which are: Arrangement of plaintext 1234567 Ascending alternate horizontal transposition 1726354 Descending alternate horizontal transposition 7 1 6 2 5 3 4 It should be noteworthy here to exhibit an important method based on the idea of taking up the first letter and leaving out a set number of letters, and so forth till the end of the text, then taking up the second letter, leaving out the same number of letters till the end of the text. For example, Arrangement of plaintext letters: 1

2

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

t

h e m o

3

4

s

t

i

m

p

o

r

t

a

n

t

f

o

r

m

Arrangement of ciphertext letters, by taking every fifth letter: 1

6

11

16

2

7

12

17

3

8

13

18

4

9

14

19

5

10

15

20

t

s

o

t

h

t

r

f

e

i

t

o

m m

a

r

o

p

n

m

This method is in effect equivalent to the transposition method commonly used, whereby the plaintext is written in lines of five letters each, and the cryptogram is formed by transposition, reading the plaintext vertically. Thus, the above example would run:

Ciphertext

Plaintext 1 t 6 s 11 o 16 t

2 h 7 t 12 r 17 f

3 e 8 i 13 t 18 o

9

4 m 9 m 14 a 19 r

5 o 10 p 15 n 20 m

1.3.2.2 Substitution The analysis by ibn ad-Durayhim of the common substitution methods is truly impressive. He has stated his intention to explain the norms governing them. He says: "Encipherment methods are of various types, too many to enumerate. I mean to mention the basic principles and rules that govern their laws" 3. To him, encipherment by substitution falls into two kinds: 1. unregulated. 2. regulated. In the unregulated encipherment, the substitution is determined by a set key, such as a line of verse. He declares that this kind of encipherment "is generative of innumerable cipher alphabets"4. It is an accomplished fact that the possible number of cipher alphabets for a 29-letter alphabet is the permutation of 29 elements. This is expressed by the equation: n! = 29! , which is a large number indeed  5 x 1028 ibn ad-Durayhim exemplifies the unregulated simple substitution by three cipher alphabets, namely al-Qumm , al-Fahlaw , (This had already been mentioned in ibn Adlan's al-Mu'allaf lil Malik al'A raf), and a third cipher alphabet not credited to anybody. All three cipher alphabets, in addition to a forth one given by ibn Adlan, are illustrated in the two tables to follow. Order of letters for those cipher alphabets are specified through mnemonic verses, serving as keys to ciphering. These verses are used in two ways: The one is by substituting for each letter of the verse the respective letter of the alphabet or numerical alphabet, as demonstrated in the following table (Table 1.2):

3 4

See his treatise, p. 57. Ibid., p. 63.

10

The alphabet The numerical alphabet (ab ad ) Al-Qummi cipher alphabet ibn ad-Durayhim's cipher alphabet

Al-Fahlawi cipher alphabet ibn Adl n's cipher alphabet

‫أ‬ ٛ ‫أ‬ ٣ ‫ٳ‬ ‫ي‬ ٛ ٣ ‫ٯ‬ ‫م‬ ٓ ‫أ‬

‫ة‬ ٟ ‫ة‬ ٫ ‫ٻ‬ ‫ة‬ ‫ه‬ ‫ؿ‬ ‫ك‬ ً ‫ػ‬ ‫ؿ‬

‫د‬ ٣ ‫ط‬ ٓ ‫أ‬ ‫ى‬ ‫ٯ‬ ‫ة‬ ٗ ‫ؿ‬ ٣ ‫م‬

‫س‬ ٧ ‫ك‬ ‫ٯ‬ ‫ڇ‬ ‫ؿ‬ ‫د‬ ‫ال‬ ‫ط‬ ٛ ‫ٿ‬ ٫

‫ط‬ ٫ ‫څـ‬ ‫ه‬ ‫ػ‬ ُ ُ ‫ػ‬ ‫ى‬ ‫د‬ ‫ك‬ ٟ

‫ػ‬ ‫ٯ‬ ‫ڇ‬ ُ ٛ ٧ ‫ٻ‬ ‫ك‬ ‫ػ‬ ٧ ‫ي‬ ‫ٳ‬

‫ؿ‬ ‫ٳ‬ ‫ى‬ ‫د‬ ٓ ٗ ‫ڇ‬ ‫ي‬ ‫ه‬ ٓ ‫ڇ‬ ‫س‬

‫ك‬ ‫ٷ‬ ‫ػ‬ ‫س‬ ‫ال‬ ‫س‬ ً ‫س‬ ‫ڇ‬ ‫ٿ‬ ‫ٯ‬ ‫ى‬

‫م‬ ‫ٻ‬ ٛ ‫ؿ‬ ‫ٷ‬ ‫ط‬ ٫ ‫ٳ‬ ُ ٣ ‫د‬ ٛ

‫ه‬ ‫ٿ‬ ‫ي‬ ‫م‬ ‫څـ‬ ‫د‬ ٟ ‫ٿ‬ ‫ٳ‬ ‫ٷ‬ ُ ٗ

‫ى‬ ‫څـ‬ ‫ٳ‬ ٗ ‫ك‬ ‫م‬ ‫ٷ‬ ٓ ‫أ‬ ‫ي‬ ٧ ‫ه‬

ً ‫ڇ‬ ‫ٷ‬ ٟ ‫ه‬ ‫ٿ‬ ‫م‬ ‫څـ‬ ‫ة‬ ‫ال‬ ‫ٷ‬ ‫ط‬

ُ ‫ال‬ ‫ٻ‬ ٧ ً ‫ٯ‬ ‫أ‬ ٧ ‫س‬ ٫ ‫ة‬ ً

ٓ ‫ي‬ ‫ٿ‬

ٗ

٣

٫

‫ط‬ ٗ ‫څـ‬ ٟ ‫څـ‬

‫ى‬

Al-Qummi cipher alphabet: ibn ad-Durayhim's cipher alphabet: Al-Fahlawi cipher alphabet: ibn Adl n's cipher alphabet:

Table 1.2: The alphabetical and numerical-alphabet letters, with their corresponding cipher alphabets.

11

ً

‫ٻ‬ ‫ٻ‬

The other is by considering the verse as composed of pairs of letters; the substitution is done reciprocally between the letters within each pair, as shown in the following table (Table 1.3): al-Qumm cipher alphabet ibn adDurayhim's cipher alphabet al-Fahlaw cipher alphabet ibn Adl n's cipher alphabet

‫ٳ‬ ‫ٻ‬ ٛ

‫أ‬ ‫ڇ‬ ‫ٯ‬

‫ػ‬ ٛ ُ

ٓ ‫ال‬ ‫ڇ‬

‫ٷ‬ ‫څـ‬ ٫

‫ك‬ ‫ه‬ ‫ٷ‬

ً ٣ ‫أ‬

٫ ‫ي‬ ‫ى‬

‫ة‬ ‫ى‬ ‫ؿ‬

‫ؿ‬ ُ ‫ال‬

٧ ٗ ‫ك‬

‫س‬ ‫ط‬ ‫س‬

‫د‬ ‫م‬ ‫ٿ‬

‫ٿ‬ ‫ٯ‬ ‫څـ‬

ٗ

‫ه‬

‫د‬

‫ٻ‬

ً

ٟ

‫م‬

‫ط‬

٣

‫ة‬

‫ػ‬

‫ي‬

‫ٳ‬

ٓ

٧

ٗ

‫ٯ‬ ‫ك‬ ٓ ‫ػ‬

ٗ ‫ط‬ ٣ ‫ٿ‬

‫ى‬ ‫ػ‬ ‫ك‬ ‫ي‬

‫ه‬ ‫ڇ‬ ‫ڇ‬ ‫ٯ‬

ُ ‫ٳ‬ ‫د‬ ُ

‫أ‬ ‫ة‬ ٧ ‫ٷ‬

‫س‬ ‫څـ‬ ‫ة‬ ‫څـ‬

‫ٻ‬ ‫م‬ ‫ٻ‬ ‫أ‬

ً ‫ؿ‬ ‫ؿ‬ ‫م‬

ٛ ‫د‬ ٫ ٟ

٧ ٓ ‫ٳ‬ ‫س‬

‫ٿ‬ ٣ ‫ى‬ ٛ

‫ٷ‬ ‫ي‬ ٗ ‫ه‬

‫ال‬ ٫ ‫ط‬ ً

ٟ ٟ

Table 1.3: Dual order of letters for some cipher alphabets In the regulated simple substitution, on the other hand, substitution is applied in accordance with a fixed rule. Of this kind ibn adDurayhim proposes four methods that are grouped under two classes: Class I: - The first method, in which the substitute for each letter of an alphabet is the one immediately following, or every third or fourth letter next to it …(Table 1.4) - The second method, in which the substitute for each letter is the one immediately preceding, or every third or fourth letter before it …(Table 1.5) Class II: - The third method lends itself to considering the alphabet as composed of pairs of letters; the substitution is done reciprocally between the letters within each pair. The pairs are formed by systematically taking every letter with the following one, or with every third, fourth, etc. letter next to it. (Table 1.6) - The fourth method is similar to the third, except that pairs are formed by taking every letter with the one immediately preceding, or with every second, third, fourth, etc. letter before it. (Table 1.7) ibn ad-Durayhim viewed the letters of the alphabet -when applying the regulated substitution encipherment- as located on a circle circumference or a disk, "because letters," he says, "are like a circle in

12

that the last letter is replaceable by the first letter as if to follow or precede it" 5. This notion was certainly the rudimentary basis of the concept of the cipher disk which became in common use during later centuries. The following figure (Figure 1.1) exhibits ibn adDurayhim's circle of letters:

Figure 1.1: ibn ad-Durayhim's circle of letters We shall try in what follows to elucidate the four above-mentioned methods of ibn ad-Durayhim's regulated substitution: 5

See his treatise, p. 63.

13

The rule for processing the first method was to substitute for each letter the one immediately following, or every third or fourth letter, and so forth… This resulted in (28) cipher alphabets when using the numerical alphabet, and (29) cipher alphabets using the alphabet (since the latter has one more character, i.e., ‫)ال‬. In so doing, ibn adDurayhim counted the natural order of letters as one of these cipher alphabets. The following tables (1.4 & 1.5) represent the cipher alphabets resulting from adopting the numerical-alphabet order, and those resulting from the alphabetical order respectively. 27

1

26

2

25

3

24

4

23

5

22

6

21

7

20

8

19

9

18

10

17

11

16

12

15

13

14

14

13

15

12

16

11

17

10

18

9

19

8

20

7

21

6

22

5

23

4

24

3

25

2

26

1

27

14

Table 1.4: Table of encipherment, following the first method, using the numerical-alphabet order. 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Table 1.5: Table of encipherment, following the second method, using the alphabetical order.

As for the second method of the regulated simple substitution, ibn ad-Durayhim states that, similar to the first method where we substitute for a letter the one following it, we can substitute for a letter the one preceding it. As he puts it, "… or by substituting for a letter the one preceding it. This would bring about a number of cipher alphabets amounting to 58". In fact substituting for a letter the one before generates quite the same cipher alphabets as when substituting

15

for a letter the one after. The only difference lies in the key number. Notice for example in the first of the previous couple of tables that substituting for (‫ )أ‬the next letter would effect key No.1 in the first method; whereas substituting for (‫ )أ‬the preceding letter would effect key No.27 in the first method, which is key No.1 in the second method. It is important to note that the previous tables (1.4 & 1.5) are reminiscent of what is commonly known in the West as the Vigenère6 Table. It would have rightly been more appropriate to have been termed the ibn ad-Durayhim Table, distant though the two eras are from each other. The rule for processing the third and fourth methods (Class II) was to look at the letters of the alphabet as pairs formed by taking every letter with the following/preceding one, or every second, third, etc. next /prior to it. This would produce 58 cipher alphabets as stated by ibn ad-Durayhim above. The following two tables (1.6 & 1.7) display respectively the cipher alphabets engendered by substituting the two methods of Class II. Note that the lower half of the tables (i.e., Nos. 15-28) is no different from one of the tables attributed to Porta7, the well-known cryptologist of the West, called "the double substitution system".

6 7

Vigenère (1523-1596). Porta (born 1535).

16

‫‪ٟ‬‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ؿ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ُ‬ ‫‪٧‬‬ ‫د‬ ‫‪٧‬‬ ‫س‬ ‫‪٧‬‬ ‫ؿ‬ ‫‪٧‬‬ ‫م‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫‪ٟ‬‬ ‫‪٧‬‬ ‫ٿ‬ ‫‪٧‬‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬ ‫ٿ‬ ‫ً‬

‫م‬ ‫ٗ‬ ‫م‬ ‫‪ٟ‬‬ ‫م‬ ‫‪ٟ‬‬ ‫م‬ ‫‪ٟ‬‬ ‫س‬ ‫‪ٟ‬‬ ‫م‬ ‫‪ٟ‬‬ ‫ه‬ ‫‪ٟ‬‬ ‫ُ‬ ‫‪ٟ‬‬ ‫د‬ ‫‪ٟ‬‬ ‫س‬ ‫‪ٟ‬‬ ‫ؿ‬ ‫‪ٟ‬‬ ‫م‬ ‫‪ٟ‬‬ ‫ٻ‬ ‫ٗ‬ ‫ٻ‬ ‫‪ٟ‬‬ ‫ٻ‬ ‫‪٧‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٻ‬ ‫‪٣‬‬

‫س‬ ‫ؿ‬ ‫د‬ ‫ؿ‬ ‫ُ‬ ‫ؿ‬ ‫ه‬ ‫ؿ‬ ‫د‬ ‫ٗ‬ ‫ٓ‬ ‫ؿ‬ ‫ٯ‬ ‫ٗ‬ ‫ه‬ ‫ٗ‬ ‫ُ‬ ‫ٗ‬ ‫د‬ ‫ٗ‬ ‫س‬ ‫ٗ‬ ‫ٷ‬ ‫ؿ‬ ‫ٷ‬ ‫م‬ ‫ٷ‬ ‫ٗ‬ ‫ٷ‬ ‫‪ٟ‬‬ ‫ٷ‬ ‫‪٧‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٫‬‬

‫ُ‬ ‫د‬ ‫ُ‬ ‫س‬ ‫ه‬ ‫س‬ ‫ٯ‬ ‫س‬ ‫ُ‬ ‫م‬ ‫‪٫‬‬ ‫س‬ ‫ٓ‬ ‫م‬ ‫ٯ‬ ‫م‬ ‫ه‬ ‫م‬ ‫ُ‬ ‫م‬ ‫ٳ‬ ‫د‬ ‫ٳ‬ ‫س‬ ‫ٳ‬ ‫ؿ‬ ‫ٳ‬ ‫م‬ ‫ٳ‬ ‫ٗ‬ ‫ٳ‬ ‫‪ٟ‬‬ ‫ٳ‬ ‫‪٧‬‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬ ‫ٳ‬ ‫ٓ‬

‫ٯ‬ ‫ه‬ ‫ٓ‬ ‫ه‬ ‫ٯ‬ ‫د‬ ‫ٓ‬ ‫د‬ ‫ً‬ ‫ه‬ ‫‪٣‬‬ ‫د‬ ‫‪٫‬‬ ‫ؿ‬ ‫ٓ‬ ‫ؿ‬ ‫ٯ‬ ‫ؿ‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ُ‬ ‫ي‬ ‫د‬ ‫ي‬ ‫س‬ ‫ي‬ ‫ؿ‬ ‫ي‬ ‫م‬ ‫ي‬ ‫ٗ‬ ‫ي‬ ‫‪ٟ‬‬ ‫ي‬ ‫‪٧‬‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬ ‫ي‬ ‫ٯ‬

‫‪٫‬‬ ‫ٓ‬ ‫‪٫‬‬ ‫ٯ‬ ‫ً‬ ‫ٓ‬ ‫‪٫‬‬ ‫ُ‬ ‫ٿ‬ ‫ٯ‬ ‫ً‬ ‫ُ‬ ‫‪٣‬‬ ‫س‬ ‫‪٫‬‬ ‫س‬ ‫‪ٛ‬‬ ‫ٓ‬ ‫‪ٛ‬‬ ‫ٯ‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫د‬ ‫‪ٛ‬‬ ‫س‬ ‫‪ٛ‬‬ ‫ؿ‬ ‫‪ٛ‬‬ ‫م‬ ‫‪ٛ‬‬ ‫ٗ‬ ‫‪ٛ‬‬ ‫‪ٟ‬‬ ‫‪ٛ‬‬ ‫‪٧‬‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ه‬

‫ً‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٫‬‬ ‫ٷ‬ ‫‪٣‬‬ ‫ٻ‬ ‫ٓ‬ ‫ٿ‬ ‫ه‬ ‫ً‬ ‫د‬ ‫ػ‬ ‫‪٣‬‬ ‫ػ‬ ‫‪٫‬‬ ‫ػ‬ ‫ٓ‬ ‫ػ‬ ‫ٯ‬ ‫ػ‬ ‫ه‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫س‬ ‫ػ‬ ‫ؿ‬ ‫ػ‬ ‫م‬ ‫ػ‬ ‫ٗ‬ ‫ػ‬ ‫‪ٟ‬‬ ‫ػ‬ ‫‪٧‬‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫ُ‬

‫ٻ‬ ‫ٿ‬ ‫ٻ‬ ‫ً‬ ‫ٻ‬ ‫‪٣‬‬ ‫ٳ‬ ‫ً‬ ‫ٷ‬ ‫‪٫‬‬ ‫ٻ‬ ‫ٯ‬ ‫ى‬ ‫ٿ‬ ‫ى‬ ‫ً‬ ‫ى‬ ‫‪٣‬‬ ‫ى‬ ‫‪٫‬‬ ‫ى‬ ‫ٓ‬ ‫ى‬ ‫ٯ‬ ‫ى‬ ‫ه‬ ‫ى‬ ‫ُ‬ ‫ى‬ ‫د‬ ‫ى‬ ‫س‬ ‫ى‬ ‫ؿ‬ ‫ى‬ ‫م‬ ‫ى‬ ‫ٗ‬ ‫ى‬ ‫‪ٟ‬‬ ‫ى‬ ‫‪٧‬‬ ‫ى‬ ‫د‬ ‫ى‬ ‫د‬ ‫ى‬ ‫د‬ ‫ى‬ ‫د‬ ‫ى‬ ‫د‬ ‫ى‬ ‫د‬

‫ٳ‬ ‫ٷ‬ ‫ي‬ ‫ٷ‬ ‫‪ٛ‬‬ ‫ٷ‬ ‫ي‬ ‫ٿ‬ ‫ٳ‬ ‫‪٣‬‬ ‫ڇ‬ ‫ٷ‬ ‫ڇ‬ ‫ٻ‬ ‫ڇ‬ ‫ٿ‬ ‫ڇ‬ ‫ً‬ ‫ڇ‬ ‫‪٣‬‬ ‫ڇ‬ ‫‪٫‬‬ ‫ڇ‬ ‫ٓ‬ ‫ڇ‬ ‫ٯ‬ ‫ڇ‬ ‫ه‬ ‫ڇ‬ ‫ُ‬ ‫ڇ‬ ‫د‬ ‫ڇ‬ ‫س‬ ‫ڇ‬ ‫ؿ‬ ‫ڇ‬ ‫م‬ ‫ڇ‬ ‫ٗ‬ ‫ڇ‬ ‫‪ٟ‬‬ ‫ڇ‬ ‫‪٧‬‬ ‫ڇ‬ ‫س‬ ‫ڇ‬ ‫س‬ ‫ڇ‬ ‫س‬ ‫ڇ‬ ‫س‬ ‫ڇ‬ ‫س‬

‫‪ٛ‬‬ ‫ي‬ ‫‪ٛ‬‬ ‫ٳ‬ ‫ػ‬ ‫ٳ‬ ‫‪ٛ‬‬ ‫ٻ‬ ‫څـ‬ ‫ي‬ ‫څـ‬ ‫ٳ‬ ‫څـ‬ ‫ٷ‬ ‫څـ‬ ‫ٻ‬ ‫څـ‬ ‫ٿ‬ ‫څـ‬ ‫ً‬ ‫څـ‬ ‫‪٣‬‬ ‫څـ‬ ‫‪٫‬‬ ‫څـ‬ ‫ٓ‬ ‫څـ‬ ‫ٯ‬ ‫څـ‬ ‫ه‬ ‫څـ‬ ‫ُ‬ ‫څـ‬ ‫د‬ ‫څـ‬ ‫س‬ ‫څـ‬ ‫ؿ‬ ‫څـ‬ ‫م‬ ‫څـ‬ ‫ٗ‬ ‫څـ‬ ‫‪ٟ‬‬ ‫څـ‬ ‫‪٧‬‬ ‫څـ‬ ‫ؿ‬ ‫څـ‬ ‫ؿ‬ ‫څـ‬ ‫ؿ‬ ‫څـ‬ ‫ؿ‬

‫ى‬ ‫ػ‬ ‫ڇ‬ ‫ػ‬ ‫ى‬ ‫ي‬ ‫ك‬ ‫ػ‬ ‫ك‬ ‫‪ٛ‬‬ ‫ك‬ ‫ي‬ ‫ك‬ ‫ٳ‬ ‫ك‬ ‫ٷ‬ ‫ك‬ ‫ٻ‬ ‫ك‬ ‫ٿ‬ ‫ك‬ ‫ً‬ ‫ك‬ ‫‪٣‬‬ ‫ك‬ ‫‪٫‬‬ ‫ك‬ ‫ٓ‬ ‫ك‬ ‫ٯ‬ ‫ك‬ ‫ه‬ ‫ك‬ ‫ُ‬ ‫ك‬ ‫د‬ ‫ك‬ ‫س‬ ‫ك‬ ‫ؿ‬ ‫ك‬ ‫م‬ ‫ك‬ ‫ٗ‬ ‫ك‬ ‫‪ٟ‬‬ ‫ك‬ ‫‪٧‬‬ ‫ك‬ ‫م‬ ‫ك‬ ‫م‬ ‫ك‬ ‫م‬

‫څـ‬ ‫ڇ‬ ‫څـ‬ ‫ى‬ ‫ط‬ ‫ڇ‬ ‫ط‬ ‫ى‬ ‫ط‬ ‫ػ‬ ‫ط‬ ‫‪ٛ‬‬ ‫ط‬ ‫ي‬ ‫ط‬ ‫ٳ‬ ‫ط‬ ‫ٷ‬ ‫ط‬ ‫ٻ‬ ‫ط‬ ‫ٿ‬ ‫ط‬ ‫ً‬ ‫ط‬ ‫‪٣‬‬ ‫ط‬ ‫‪٫‬‬ ‫ط‬ ‫ٓ‬ ‫ط‬ ‫ٯ‬ ‫ط‬ ‫ه‬ ‫ط‬ ‫ُ‬ ‫ط‬ ‫د‬ ‫ط‬ ‫س‬ ‫ط‬ ‫ؿ‬ ‫ط‬ ‫م‬ ‫ط‬ ‫ٗ‬ ‫ط‬ ‫‪ٟ‬‬ ‫ط‬ ‫‪٧‬‬ ‫ط‬ ‫ٗ‬ ‫ط‬ ‫ٗ‬

‫ط‬ ‫ك‬ ‫ة‬ ‫ك‬ ‫ة‬ ‫څـ‬ ‫ة‬ ‫ڇ‬ ‫ة‬ ‫ى‬ ‫ة‬ ‫ػ‬ ‫ة‬ ‫‪ٛ‬‬ ‫ة‬ ‫ي‬ ‫ة‬ ‫ٳ‬ ‫ة‬ ‫ٷ‬ ‫ة‬ ‫ٻ‬ ‫ة‬ ‫ٿ‬ ‫ة‬ ‫ً‬ ‫ة‬ ‫‪٣‬‬ ‫ة‬ ‫‪٫‬‬ ‫ة‬ ‫ٓ‬ ‫ة‬ ‫ٯ‬ ‫ة‬ ‫ه‬ ‫ة‬ ‫ُ‬ ‫ة‬ ‫د‬ ‫ة‬ ‫س‬ ‫ة‬ ‫ؿ‬ ‫ة‬ ‫م‬ ‫ة‬ ‫ٗ‬ ‫ة‬ ‫‪ٟ‬‬ ‫ة‬ ‫‪٧‬‬ ‫ة‬ ‫‪ٟ‬‬

‫أ‬ ‫ة‬ ‫أ‬ ‫ط‬ ‫أ‬ ‫ك‬ ‫أ‬ ‫څـ‬ ‫أ‬ ‫ڇ‬ ‫أ‬ ‫ى‬ ‫أ‬ ‫ػ‬ ‫أ‬ ‫‪ٛ‬‬ ‫أ‬ ‫ي‬ ‫أ‬ ‫ٳ‬ ‫أ‬ ‫ٷ‬ ‫أ‬ ‫ٻ‬ ‫أ‬ ‫ٿ‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫‪٣‬‬ ‫أ‬ ‫‪٫‬‬ ‫أ‬ ‫ٓ‬ ‫أ‬ ‫ٯ‬ ‫أ‬ ‫ه‬ ‫أ‬ ‫ُ‬ ‫أ‬ ‫د‬ ‫أ‬ ‫س‬ ‫أ‬ ‫ؿ‬ ‫أ‬ ‫م‬ ‫أ‬ ‫ٗ‬ ‫أ‬ ‫‪ٟ‬‬ ‫أ‬ ‫‪٧‬‬

‫‪+1‬‬ ‫‪+2‬‬ ‫‪+3‬‬ ‫‪+4‬‬ ‫‪+5‬‬ ‫‪+6‬‬ ‫‪+7‬‬ ‫‪+8‬‬ ‫‪+9‬‬ ‫‪+10‬‬ ‫‪+11‬‬ ‫‪+12‬‬ ‫‪+13‬‬ ‫‪+14‬‬ ‫‪+15‬‬ ‫‪+16‬‬ ‫‪+17‬‬ ‫‪+18‬‬ ‫‪+19‬‬ ‫‪+20‬‬ ‫‪+21‬‬ ‫‪+22‬‬ ‫‪+23‬‬ ‫‪+24‬‬ ‫‪+25‬‬ ‫‪+26‬‬ ‫‪+27‬‬

‫‪Table 1.6: Table of encipherment as given by ibn ad-Durayhim, following the‬‬ ‫‪third method using the numerical-alphabet order.‬‬ ‫‪Note: Porta's double substitution system is part of this table.‬‬

‫‪17‬‬

‫ط‬ ‫ة‬ ‫ك‬ ‫ة‬ ‫څـ‬ ‫ة‬ ‫ڇ‬ ‫ة‬ ‫ى‬ ‫ة‬ ‫ػ‬ ‫ة‬ ‫‪ٛ‬‬ ‫ة‬ ‫ي‬ ‫ة‬ ‫ٳ‬ ‫ة‬ ‫ٷ‬ ‫ة‬ ‫ٻ‬ ‫ة‬ ‫ٿ‬ ‫ة‬ ‫ً‬ ‫ة‬ ‫‪٣‬‬ ‫ة‬ ‫‪٫‬‬ ‫ة‬ ‫ٓ‬ ‫ة‬ ‫ٯ‬ ‫ة‬ ‫ه‬ ‫ة‬ ‫ُ‬ ‫ة‬ ‫د‬ ‫ة‬ ‫س‬ ‫ة‬ ‫ؿ‬ ‫ة‬ ‫م‬ ‫ة‬ ‫ٗ‬ ‫ة‬ ‫‪ٟ‬‬ ‫ة‬ ‫‪٧‬‬ ‫ة‬

‫څـ‬ ‫ك‬ ‫څـ‬ ‫ط‬ ‫ڇ‬ ‫ط‬ ‫ى‬ ‫ط‬ ‫ػ‬ ‫ط‬ ‫‪ٛ‬‬ ‫ط‬ ‫ي‬ ‫ط‬ ‫ٳ‬ ‫ط‬ ‫ٷ‬ ‫ط‬ ‫ٻ‬ ‫ط‬ ‫ٿ‬ ‫ط‬ ‫ً‬ ‫ط‬ ‫‪٣‬‬ ‫ط‬ ‫‪٫‬‬ ‫ط‬ ‫ٓ‬ ‫ط‬ ‫ٯ‬ ‫ط‬ ‫ه‬ ‫ط‬ ‫ُ‬ ‫ط‬ ‫د‬ ‫ط‬ ‫س‬ ‫ط‬ ‫ؿ‬ ‫ط‬ ‫م‬ ‫ط‬ ‫ٗ‬ ‫ط‬ ‫‪ٟ‬‬ ‫ط‬ ‫‪٧‬‬ ‫ط‬ ‫‪ٟ‬‬ ‫ط‬

‫ى‬ ‫ڇ‬ ‫ػ‬ ‫ڇ‬ ‫ى‬ ‫ك‬ ‫ػ‬ ‫ك‬ ‫‪ٛ‬‬ ‫ك‬ ‫ي‬ ‫ك‬ ‫ٳ‬ ‫ك‬ ‫ٷ‬ ‫ك‬ ‫ٻ‬ ‫ك‬ ‫ٿ‬ ‫ك‬ ‫ً‬ ‫ك‬ ‫‪٣‬‬ ‫ك‬ ‫‪٫‬‬ ‫ك‬ ‫ٓ‬ ‫ك‬ ‫ٯ‬ ‫ك‬ ‫ه‬ ‫ك‬ ‫ُ‬ ‫ك‬ ‫د‬ ‫ك‬ ‫س‬ ‫ك‬ ‫ؿ‬ ‫ك‬ ‫م‬ ‫ك‬ ‫ٗ‬ ‫ك‬ ‫‪ٟ‬‬ ‫ك‬ ‫‪٧‬‬ ‫ك‬ ‫ٗ‬ ‫ك‬ ‫ٗ‬ ‫ك‬

‫‪ٛ‬‬ ‫ػ‬ ‫‪ٛ‬‬ ‫ى‬ ‫ٳ‬ ‫ػ‬ ‫‪ٛ‬‬ ‫څـ‬ ‫ي‬ ‫څـ‬ ‫ٳ‬ ‫څـ‬ ‫ٷ‬ ‫څـ‬ ‫ٻ‬ ‫څـ‬ ‫ٿ‬ ‫څـ‬ ‫ً‬ ‫څـ‬ ‫‪٣‬‬ ‫څـ‬ ‫‪٫‬‬ ‫څـ‬ ‫ٓ‬ ‫څـ‬ ‫ٯ‬ ‫څـ‬ ‫ه‬ ‫څـ‬ ‫ُ‬ ‫څـ‬ ‫د‬ ‫څـ‬ ‫س‬ ‫څـ‬ ‫ؿ‬ ‫څـ‬ ‫م‬ ‫څـ‬ ‫ٗ‬ ‫څـ‬ ‫‪ٟ‬‬ ‫څـ‬ ‫‪٧‬‬ ‫څـ‬ ‫م‬ ‫څـ‬ ‫م‬ ‫څـ‬ ‫م‬ ‫څـ‬

‫ٳ‬ ‫ي‬ ‫ٷ‬ ‫ي‬ ‫ٷ‬ ‫‪ٛ‬‬ ‫ٿ‬ ‫ي‬ ‫ٳ‬ ‫ڇ‬ ‫ٷ‬ ‫ڇ‬ ‫ٻ‬ ‫ڇ‬ ‫ٿ‬ ‫ڇ‬ ‫ً‬ ‫ڇ‬ ‫‪٣‬‬ ‫ڇ‬ ‫‪٫‬‬ ‫ڇ‬ ‫ٓ‬ ‫ڇ‬ ‫ٯ‬ ‫ڇ‬ ‫ه‬ ‫ڇ‬ ‫ُ‬ ‫ڇ‬ ‫د‬ ‫ڇ‬ ‫س‬ ‫ڇ‬ ‫ؿ‬ ‫ڇ‬ ‫م‬ ‫ڇ‬ ‫ٗ‬ ‫ڇ‬ ‫‪ٟ‬‬ ‫ڇ‬ ‫‪٧‬‬ ‫ڇ‬ ‫ؿ‬ ‫ڇ‬ ‫ؿ‬ ‫ڇ‬ ‫ؿ‬ ‫ڇ‬ ‫ؿ‬ ‫ڇ‬

‫ٻ‬ ‫ٷ‬ ‫ٻ‬ ‫ٳ‬ ‫ٻ‬ ‫ي‬ ‫ً‬ ‫ٳ‬ ‫‪٫‬‬ ‫ٷ‬ ‫ٻ‬ ‫ى‬ ‫ٿ‬ ‫ى‬ ‫ً‬ ‫ى‬ ‫‪٣‬‬ ‫ى‬ ‫‪٫‬‬ ‫ى‬ ‫ٓ‬ ‫ى‬ ‫ٯ‬ ‫ى‬ ‫ه‬ ‫ى‬ ‫ُ‬ ‫ى‬ ‫د‬ ‫ى‬ ‫س‬ ‫ى‬ ‫ؿ‬ ‫ى‬ ‫م‬ ‫ى‬ ‫ٗ‬ ‫ى‬ ‫‪ٟ‬‬ ‫ى‬ ‫‪٧‬‬ ‫ى‬ ‫س‬ ‫ى‬ ‫س‬ ‫ى‬ ‫س‬ ‫ى‬ ‫س‬ ‫ى‬ ‫س‬ ‫ى‬

‫ً‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٫‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٷ‬ ‫ٓ‬ ‫ٻ‬ ‫ه‬ ‫ٿ‬ ‫ً‬ ‫ػ‬ ‫‪٣‬‬ ‫ػ‬ ‫‪٫‬‬ ‫ػ‬ ‫ٓ‬ ‫ػ‬ ‫ٯ‬ ‫ػ‬ ‫ه‬ ‫ػ‬ ‫ُ‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫س‬ ‫ػ‬ ‫ؿ‬ ‫ػ‬ ‫م‬ ‫ػ‬ ‫ٗ‬ ‫ػ‬ ‫‪ٟ‬‬ ‫ػ‬ ‫‪٧‬‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫د‬ ‫ػ‬ ‫د‬ ‫ػ‬

‫‪٫‬‬ ‫‪٣‬‬ ‫‪٫‬‬ ‫ً‬ ‫ٓ‬ ‫ً‬ ‫‪٫‬‬ ‫ٻ‬ ‫ٯ‬ ‫ٿ‬ ‫ُ‬ ‫ً‬ ‫س‬ ‫‪٣‬‬ ‫‪٫‬‬ ‫‪ٛ‬‬ ‫ٓ‬ ‫‪ٛ‬‬ ‫ٯ‬ ‫‪ٛ‬‬ ‫ه‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫د‬ ‫‪ٛ‬‬ ‫س‬ ‫‪ٛ‬‬ ‫ؿ‬ ‫‪ٛ‬‬ ‫م‬ ‫‪ٛ‬‬ ‫ٗ‬ ‫‪ٛ‬‬ ‫‪ٟ‬‬ ‫‪ٛ‬‬ ‫‪٧‬‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬ ‫ُ‬ ‫‪ٛ‬‬

‫ٯ‬ ‫ٓ‬ ‫ه‬ ‫ٓ‬ ‫ٯ‬ ‫‪٣‬‬ ‫د‬ ‫ٓ‬ ‫ه‬ ‫ً‬ ‫د‬ ‫‪٣‬‬ ‫ؿ‬ ‫‪٫‬‬ ‫ؿ‬ ‫ٓ‬ ‫ٯ‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ُ‬ ‫ي‬ ‫د‬ ‫ي‬ ‫س‬ ‫ي‬ ‫ؿ‬ ‫ي‬ ‫م‬ ‫ي‬ ‫ٗ‬ ‫ي‬ ‫‪ٟ‬‬ ‫ي‬ ‫‪٧‬‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬ ‫ه‬ ‫ي‬

‫ُ‬ ‫ه‬ ‫ُ‬ ‫ٯ‬ ‫س‬ ‫ه‬ ‫س‬ ‫ٯ‬ ‫ُ‬ ‫‪٣‬‬ ‫س‬ ‫‪٫‬‬ ‫م‬ ‫ٓ‬ ‫م‬ ‫ٯ‬ ‫م‬ ‫ه‬ ‫ُ‬ ‫ٳ‬ ‫د‬ ‫ٳ‬ ‫س‬ ‫ٳ‬ ‫ؿ‬ ‫ٳ‬ ‫م‬ ‫ٳ‬ ‫ٗ‬ ‫ٳ‬ ‫‪ٟ‬‬ ‫ٳ‬ ‫‪٧‬‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬ ‫ٯ‬ ‫ٳ‬

‫س‬ ‫د‬ ‫ؿ‬ ‫د‬ ‫ؿ‬ ‫ُ‬ ‫ؿ‬ ‫ه‬ ‫ٗ‬ ‫د‬ ‫ؿ‬ ‫ٓ‬ ‫ٗ‬ ‫ٯ‬ ‫ٗ‬ ‫ه‬ ‫ٗ‬ ‫ُ‬ ‫ٗ‬ ‫د‬ ‫س‬ ‫ٷ‬ ‫ؿ‬ ‫ٷ‬ ‫م‬ ‫ٷ‬ ‫ٗ‬ ‫ٷ‬ ‫‪ٟ‬‬ ‫ٷ‬ ‫‪٧‬‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬ ‫ٓ‬ ‫ٷ‬

‫م‬ ‫ؿ‬ ‫م‬ ‫س‬ ‫م‬ ‫د‬ ‫م‬ ‫ُ‬ ‫‪ٟ‬‬ ‫س‬ ‫م‬ ‫ٯ‬ ‫‪ٟ‬‬ ‫ه‬ ‫‪ٟ‬‬ ‫ُ‬ ‫‪ٟ‬‬ ‫د‬ ‫‪ٟ‬‬ ‫س‬ ‫‪ٟ‬‬ ‫ؿ‬ ‫م‬ ‫ٻ‬ ‫ٗ‬ ‫ٻ‬ ‫‪ٟ‬‬ ‫ٻ‬ ‫‪٧‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬ ‫‪٫‬‬ ‫ٻ‬

‫‪ٟ‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ؿ‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪٧‬‬ ‫ُ‬ ‫‪٧‬‬ ‫د‬ ‫‪٧‬‬ ‫س‬ ‫‪٧‬‬ ‫ؿ‬ ‫‪٧‬‬ ‫م‬ ‫‪٧‬‬ ‫ٗ‬ ‫‪ٟ‬‬ ‫ٿ‬ ‫‪٧‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬ ‫‪٣‬‬ ‫ٿ‬

‫أ‬ ‫‪٧‬‬ ‫أ‬ ‫‪ٟ‬‬ ‫أ‬ ‫‪ٟ‬‬ ‫أ‬ ‫‪ٟ‬‬ ‫أ‬ ‫م‬ ‫أ‬ ‫‪ٟ‬‬ ‫أ‬ ‫د‬ ‫أ‬ ‫س‬ ‫أ‬ ‫ؿ‬ ‫أ‬ ‫م‬ ‫أ‬ ‫ٗ‬ ‫أ‬ ‫‪ٟ‬‬ ‫أ‬ ‫‪٧‬‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬ ‫أ‬ ‫ً‬

‫‪+1‬‬ ‫‪+2‬‬ ‫‪+3‬‬ ‫‪+4‬‬ ‫‪+5‬‬ ‫‪+6‬‬ ‫‪+7‬‬ ‫‪+8‬‬ ‫‪+9‬‬ ‫‪+10‬‬ ‫‪+11‬‬ ‫‪+12‬‬ ‫‪+13‬‬ ‫‪+14‬‬ ‫‪+15‬‬ ‫‪+16‬‬ ‫‪+17‬‬ ‫‪+18‬‬ ‫‪+19‬‬ ‫‪+20‬‬ ‫‪+21‬‬ ‫‪+22‬‬ ‫‪+23‬‬ ‫‪+24‬‬ ‫‪+25‬‬ ‫‪+26‬‬

‫‪Table 1.7: Table of encipherment as given by ibn ad-Durayhim, following the‬‬ ‫‪fourth method using the numerical-alphabet order.‬‬ ‫‪Note: Porta's double substitution system is part of this table.‬‬

‫‪18‬‬

ibn ad-Durayhim directed attention to an important issue related to the already mentioned cipher alphabets; namely, when the encipherer is "Maghrebi" -from North Africa or Arab Spain-, because the order of letters in their numerical alphabet differs from that of the one used in the Muslim East. He set out to mention the Maghrebi numerical alphabet as it was actually used. This indicates that correspondence was active between North-Africa/Arab Spain in the west, and Egypt, Syria and Iraq in the east. Still, he never failed to point out that, in encipherment by regulated substitution, the (‫ )ڇ‬was made by some to occur before the ( ‫)څـ‬, contrary to the usual order of the ( ‫ )څـ‬coming first. Moreover, he provided one of the numerical alphabets of the Indian calligraphs. All this promotes the belief of ibn ad-Durayhim's all-round acquaintance with, and in-depth knowledge of, the possibilities of encipherment by substitution. The following table (Table 1.8) manifests the order of Arabic letters of both the alphabet and numerical alphabet in eastern and western Arab worlds, together with a numerical alphabet of an Indian calligraph.

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Order of Arabic Alphabet Eastern Western ‫أ‬ ‫أ‬ ‫ب‬ ‫ب‬ ‫ت‬ ‫ت‬ ‫ث‬ ‫ث‬ ‫ج‬ ‫ج‬ ‫ح‬ ‫ح‬ ‫خ‬ ‫خ‬ ‫د‬ ‫د‬ ‫ر‬ ‫ر‬ ‫س‬ ‫س‬ ‫ص‬ ‫ص‬ ‫س‬ ‫س‬ ‫ش‬ ‫ش‬ ‫ص‬ ‫ص‬ ‫ض‬ ‫ض‬ ‫ط‬ ‫ط‬ ‫ظ‬ ‫ظ‬ ‫ؼ‬ ‫ؼ‬ ‫غ‬ ‫غ‬ ‫ف‬ ‫ف‬ ‫ق‬ ‫ق‬ ‫ك‬ ‫ك‬ ‫ل‬ ‫ل‬ ‫م‬ ‫م‬ ‫ى‬ ‫ى‬ 8 ‫هـ‬ ‫هـ‬ 8 ‫و‬ ‫و‬ ‫ال‬ ‫ال‬ ‫ي‬ ‫ي‬

Order of Arabic Numerical alphabet Eastern Western ‫أ‬ ‫أ‬ ‫ب‬ ‫ب‬ ‫أبجذ‬ ‫أبجذ‬ ‫ج‬ ‫ج‬ ‫د‬ ‫د‬ ‫هـ‬ ‫هـ‬ ‫و‬ ‫هوص‬ ‫و‬ ‫هوص‬ ‫ص‬ ‫ص‬ ‫ح‬ ‫ح‬ ‫ط‬ ‫حطي‬ ‫ط‬ ‫حطي‬ ‫ي‬ ‫ي‬ ‫ك‬ ‫ك‬ ‫ل‬ ‫ل‬ ‫كلوي‬ ‫كلوي‬ ‫م‬ ‫م‬ ‫ى‬ ‫ى‬ ‫س‬ ‫ص‬ ‫ؼ‬ ‫ؼ‬ ‫سعفص‬ ‫صعفط‬ ‫ف‬ ‫ف‬ ‫ص‬ ‫ض‬ ‫ق‬ ‫ق‬ ‫س‬ ‫س‬ ‫قشضت‬ ‫قشست‬ ‫ش‬ ‫س‬ ‫ت‬ ‫ت‬ ‫ث‬ ‫ث‬ ‫خ‬ ‫ثخز‬ ‫خ‬ ‫ثخز‬ ‫ر‬ ‫ر‬ ‫ض‬ ‫ظ‬ ‫ظ‬ ‫ظػػ‬ ‫غ‬ ‫غغص‬ ‫غ‬ ‫ش‬

Order of an Indian Numerical alphabet ‫أ‬ ‫ي‬ ‫ق‬ ‫غ‬ ‫ب‬ ‫ك‬ ‫س‬ ‫ج‬ ‫ل‬ ‫ش‬ ‫د‬ ‫م‬ ‫ت‬ ‫هـ‬ ‫ى‬ ‫ث‬ ‫و‬ ‫س‬ ‫خ‬ ‫ص‬ ‫ؼ‬ ‫ر‬ ‫ح‬ ‫غ‬ ‫ض‬ ‫ط‬ ‫ص‬ ‫ظ‬

‫أيقػ‬

‫بكش‬ ‫جلص‬

‫دهت‬

‫هنث‬

‫وسخ‬ ‫صعز‬

‫حغط‬

‫طصع‬

Table 1.8: Table of the order of letters of both the alphabet and numerical alphabet in eastern and western Arab worlds, together with an Indian numerical alphabet.

8

Order reversed by some.

20

1.3.2.3 The augmentation or reduction of the number of letters Under this type, ibn ad-Durayhim reported three methods, the counterparts of which we had earlier met with al-Kind 's "simple encipherment where letters retain their forms". ibn ad-Durayhim contributed in enriching and enhancing those methods through advancing several variations within each. In his third method, he enunciated the important practice based on adding one or more letters to each word, following a fixed key; for instance, adding (‫ )أ‬in the first word, (‫ )ة‬in the second, and so forth… This bespeaks his awareness of changing the rule from one word to another. We do not really know for certain what precluded him from mentioning the polyalphabetic substitution, though many of the basic concepts behind it he seemed to be fully aware of.

1.3.2.4 The utilization of cipher devices Four simple devices have been mentioned by ibn ad-Durayhim in two different places, the latter of which we have entitled: "A return to the type on the utilization of cipher devices"9. These cipher devices are: a. Chessboard, assigning each square to a letter. b. Punched board, with a number of holes equal to the language letters; the cryptogram is represented by a thread marking a route which defines the letters of the message successively. c. Coloured beads threaded on a string as a rosary. Each letter of the alphabet is coded by a certain number of coloured beads. d. Paper folded in pleats. The message is written on a folded paper and concealed by unfolding it and adding other superfluous figures to it for further complication. ibn ad-Durayhim comments on using such devices like that of the folded paper, for example: "This device, however, is not really an 9

See his treatise, p. 81.

21

encipherment; therefore we say that such matters need sound common sense lest the decryptor should deviate from the right solution"10.

1.3.2.5 The replacement of letters by numbers, using the decimally-weighted numerical alphabet This type was overlooked by al-Kind in his treatise: F isti r almu amm (On cryptanalysing ciphered messages) and ibn Adl n in his al-mu'allaf lil malik al-'A raf (The book written for King al-'A raf), but had already been mentioned by ibn Dunayn r in his treatise Maq id al-fu l al-mutar ima an all at-tar ama11 (Expositional chapters on solving ciphers). The method belongs under substitution, with the possibility of substituting for a letter one or more letters or words pursuant to a set rule, as this "involves more sophistication", ibn ad-Durayhim proclaims12. This process may be represented by the following model (Figure 1.2), showing ibn adDurayhim's method of encipherment, followed by a table (Table 1.9) of the numerical alphabet with the corresponding numbers in this type of encipherment; namely, the arithmetic using decimally-weighted numerical alphabet or " is b al- ummal".

10

See his treatise, p. 83. "Collected Papers on Cryptology", 66/B and 67/A. 12 See his treatise, p. 69. 11

22

Figure 1.2: ibn ad-Durayhim's method of encipherment by substitution of numbers for letters using the decimally-weighted numerical alphabet

The letter (CLEARTEXT)

Substitution of numbers for letters

Arithmetic operation according to a set rule

Re-substitution: by substituting for a number one or several letters

The ciphertext (letters or words) (CRYPTOGRAM)

A numerical cryptogram expressed in one of the following forms: (a) numbers written in words; e.g. thirty, seven, forty, …… (b) numbers written in Arabic numerals; e.g. 30, 7, 40, …… (c) numbers written to look like a page of financial register. (d) numbers communicated to a recipient (it is not a written form) through the well-known Arab method of signalling by finger - bending (manual alphabet).

23

‫ا‬

‫ة‬

‫ط‬

‫ك‬

‫څـ‬

‫ڇ‬

‫ى‬

‫ػ‬

1

2

3

4

5

6

7

8

ٛ 9

‫ي‬

‫ٳ‬

‫ٷ‬

‫ٻ‬

‫ٿ‬

ً

٣

٫

ٓ

10

20

30

40

50

60

70

80

90

‫ٯ‬

‫ه‬

ُ

‫د‬

‫س‬

‫ؿ‬

‫م‬

ٗ

ٟ

100

200

300

400

500

600

700

800

900

x

1

2

3

4

5

6

7

8

9

1 10 100 1000

‫ا‬ ‫ي‬ ‫ٯ‬ ٧

‫ة‬ ‫ٳ‬ ‫ه‬

‫ط‬ ‫ٷ‬ ُ

‫ك‬ ‫ٻ‬ ‫د‬

‫څـ‬ ‫ٿ‬ ‫س‬

‫ڇ‬ ً ‫ؿ‬

‫ى‬ ٣ ‫م‬

‫ػ‬ ٫ ٗ

ٛ ٓ ٟ

٧ 1000

Table 1.9: Table of the letters of the numerical alphabet with corresponding decimal numerical values.

To illustrate this method, let us now give a case in point by enciphering the proper name Muhammad (‫)ٽؾپل‬: The numerical cryptogram of this name can be expressed in one of the following forms: - In words; i.e., forty, eight, forty, four. - In Arabic numbers; i.e., 40, 8, 40, 4, and the cryptogram look like a list of figures. - By giving the cryptogram a semblance of a financial register. This method is too obvious to require illustration. - Through manual signalling by finger-bending. This way is used, for instance, in communication by deaf-mutes. When the cryptographer used his fingers to convey the numbers: 40, 8, 40 and 4, the recipient would understand the message to have meant the name ‫ٽؾپل‬, which is a manual alphabet. - Resubstituting letters for the number representative of the intended letter. Thus, our example (‫ )ٽؾپل‬becomes:

24

‫ اط‬،ً‫ ٹ‬،‫ ثڈ‬،ً‫ ٹ‬By breaking up the number into a sum of two numbers, so long as their numerical values add up to the numerical value of the original letter. (40=30+10, 8=2+6, 40=30+10, 4=1+3) ‫ ثت‬،‫ ٵٴ‬،‫ اى‬،‫ ٵٴ‬By breaking up the number into a sum of two numbers of another choice. (40=20+20, 8=1+7, 40=20+20, 4=2+2)

‫ ػ‬،٫ ،‫ ٌڈ‬،٫ By doubling the number. (40x2=80, 8x2=10+6, 40x2=80, 4x2=8)

‫ ٌت‬،‫ ٱٴ‬،‫ ٵل‬،‫ ٱٴ‬By tripling the number. (40x3=120, 8x3=24, 40x3=120, 4x3=12)

This method is based on the principle of substituting letters to stand for the number representing the intended clear letter. This may be done by means of analyzing the number into its immediate constituents (which, of course, adds to the process of analysis for cryptanalysing); or through making it two, three, four, or five times as greater in value; or by employing any other arithmetic rule. It is to be noted that this method is extremely important, as it is the first in the history of cryptography to represent a marked departure from previous practice, in which more than one symbol or letter are substituted for a single letter, and numbers for letters.

1.3.2.6 The encipherment of letters by using words Reverting to al-Kind , we may conveniently term this method encipherment by substitution of letters without "relationship" but with "diffusion" -i.e., without a key but with expansion of the number of characters-, whereby a word is substituted for each letter, with the intended letter embedded in the word, in keeping with a set rule. ibn ad-Durayhim cites four methods under this type, pioneering in some, and taking up the others at the point where his predecessors had left off. These methods are: A) Substituting for a letter its spelling, or the spelling reversed, or enciphering by synthesizing both elements in a certain manner (e.g. by alternately writing the straight spelling of a letter and the reversed spelling of the next letter).

25

The name ( ‫)ٽؾپل‬, by way of example, is enciphered by substituting for each letter its spelling thus: ‫ ٽٍپؾب ٽٍپلاٷ‬:‫ٽؾپل‬ )‫ ك = كاٷ‬،‫ ٻ = ٽٍټ‬،‫ ػ = ؽب‬،‫(ٻ = ٽٍټ‬ And by using this spelling reversed alternately thus: ‫ ٽٍپبؽپٍپالك‬:‫ٽؾپل‬ The above rules give rise to "many ramifications"13. B) Encipherment by letters embedded in the words conformably with a fixed rule; for example by taking the first letter of each word, so that the name ( ً‫ٺ‬٥) can be enciphered: ( ‫ٺپذ ٹپٌٍ ٌؾٍى‬٥); or by taking the last letter of each word where (ً‫ٺ‬٥) becomes: (ً‫ ٽبٷ أث‬٤ٍٙ). It can also be done by taking the odd letters only, or even letters only, or by leaving out a specific number of letters throughout. ibn adDurayhim reports quite a few of these methods and their offshoots. Those methods later came to be called the "Grille systems". Amongst his examples of what may be brought under a regular Grille system is "taking the first letter and then every fourth letter throughout, so that in enciphering the words: ( ً‫ٺ‬٥ ‫ټ‬٥ ‫ )ٽؾپل اثڀ‬you may write: "‫لڃ اٹزجغٍٸ ٹڄ‬٦ٍ ‫لٿ أٽبٿ‬٦‫"ٽڀ اٹؾَڀ ٹپڀ ٌزلٌڀ ثبٹٲوثى ٹغڂبة ٽ‬14. ‫ٷ‬

‫ٿ‬

ً

‫ػ‬

‫ٷ‬

‫ا‬

‫ٿ‬

‫ٻ‬

‫ة‬

‫ٿ‬

‫ي‬

‫ك‬

‫د‬

‫ي‬

‫ٿ‬

‫ٻ‬

‫ط‬

‫ٷ‬

‫ډ‬

‫ة‬

‫ه‬

‫ٯ‬

‫ٷ‬

‫ا‬

‫أ‬

‫ٿ‬

‫ك‬

٣

‫ٻ‬

‫ة‬

‫ا‬

‫ٿ‬

‫ا‬

‫څـ‬

‫ك‬

٣

ً

‫ٿ‬

‫ا‬

‫ٻ‬

‫څـ‬

‫ٷ‬

‫ٷ‬

‫ي‬

‫ط‬

‫ة‬

‫د‬

‫ٷ‬

ibn ad-Durayhim further indicates another feasibility when merging the cleartext in the ciphertext, so that it may be read inversely (i.e., backwards: contrary to the writing direction)15. The use of this method has become familiar in the Grille systems later. 13

See his treatise, p. 71. Ibid., p. 73. 15 Ibid., p. 73. 14

26

C) Substituting a word for a letter: There are many possibilities for bringing this method to pass: letters may assume names of people, stars, mansions of the moon, months (lunar, Latin, Coptic), number of days in a month, hours of the day, days of the week, book names, suras of the Koran, region names, ointments, drugs, fruits, trees, etc. It is well worth mentioning here that some of these systems had been tackled by ibn Dunayn r16 in more detail. D) Substituting for a letter a picture or representation of all that can be symbolized -suggestive of rebus-, such as birds, animals, plants, etc. What is really peculiar to ibn ad-Durayhim in this connection is his indication of the special branched calligraph which looks like fancy flourishes and tails, and is based on the words of the numerical alphabet. Such calligraphy, in fact, stands to his credit as the first of its kind to be mentioned; none of ibn adDurayhim's predecessors whose treatises are covered in this study had mentioned it. However, it has recently been revealed to us that ibn Wa iyya, in his awq al-Mustah m f Ma rifat Rum z alAql m (Seekers’ joy in identifying other languages’ written symbols), had already mentioned it.

1.3.2.7 Replacing letters by generic names (i.e. applying the relationship and diffusion method) Encipherment in this type takes as a basis changing letter forms, using conceptual relationship and diffusion. ibn ad-Durayhim's utilization of al-Kind 's term of "conceptual relationship and diffusion" highlights the importance of the latter's treatise and its farreaching impact on his successors. "This," ibn ad-Durayhim notes, "relates to what has been denominated 'relationship and diffusion', where a genus or species is representative of a letter."17

16

See his treatise: Maq id al-Fu l al-Mutar ima an "Collected Papers on Cryptology", 64/A. 17 See his treatise, p. 79.

27

all at-tar ama; from

The following (Table 1.10) is a table of the generic names which stand for each letter of the Arabic alphabet, as given by ibn adDurayhim. It is interesting to notice that the first letter of the name of genus in Arabic is the letter to be ciphered: ‫أ‬ ‫ة‬ ‫د‬ ‫س‬ ‫ط‬ ‫ػ‬ ‫ؿ‬ d ‫ك‬ ‫م‬ r ‫ه‬ z ‫ى‬ s ً ُ ٓ ٗ ٛ ٟ ٣ ٧ f ٫ q ‫ٯ‬ k ‫ٳ‬ l ‫ٷ‬ m ‫ٻ‬ n ‫ٿ‬ w ‫ڇ‬ h ‫څـ‬ ‫ال‬ l y ‫ي‬ a b t

People Vegetable Dates, soil, or spices Clothing Leather Cereals or iron Wood Animals or ointments Gold Aromatic plants Glass Weaponry or fishes Months, hair, or chess Dyes, brass, gum, or wool Light or regions Birds Dark or deer Perfume, eyes (or springs), or number (or tools)

Sheep Fruits Villages or reed Books or planets Milk Towns Stars or cooper Wild animal, currency (coin), or paper

Vermin, pests, etc. Scissors Jewellery

‫أځبٻ‬ ‫ثٲڈٷ‬ ‫رپڈه أڇ رواة أڇ رڈاثٸ‬ ‫صٍبة‬ ‫عٺڈك‬ ‫ؽجڈة أڇ ؽلٌل‬ ‫فْت‬ ‫كڇاة أڇ كڅبٿ‬ ‫مڅت‬ ‫هٌبؽٍڀ‬ ‫ىعبط‬ ‫ٍالػ أڇ ٍپٴ‬ ‫وځظ‬ّٞ ‫ڈه أڇ‬٦ّ ‫ّچڈه أڇ‬ ٫‫ أڇ ٕڈ‬٧‫ أڇ ٕٮو أڇ ٕپڈ‬٧‫ٕجڈ‬

٣‫ٍب‬ٙ ‫ڈء أڇ‬ٙ ‫ٍڈه‬ٝ ‫جبء‬١ ‫الٻ أڇ‬١ ‫لك‬٥ ‫ٍڈٿ أڇ‬٥ ‫و أڇ‬ٞ٥ ‫ڂى‬٩ ‫ڂټ أڇ‬٩ ‫٭ڈاٵڄ‬ ‫ٱوډ أڇ ٱٖت‬ ‫ٵزت أڇ ٵڈاٵت‬ ‫ٹجڀ‬ ‫ٽلٿ‬ ً‫ځغڈٻ أڇ ځؾب‬ ‫ڇؽڈُ أڇ ڇُهْٯ أڇ ڇَهَٯ‬ ‫څڈاٻ‬ ّٔ‫ٽٲ‬ ‫ٌڈاٱٍذ‬

Table 1.10: A list of generic names which match the Arabic alphabet, used by ibn ad-Durayhim.

28

To be noted is the clear uniformity in designation between ibn adDurayhim's statements here and al-Kind 's exemplifications in his treatise, as well as ibn Dunayn r's tables18. ibn ad-Durayhim's new contribution manifests itself in indicating various possibilities of methods and their number. Thus, he handles encipherment in terms of genus/species relationship and the resultant cipher alphabets, some of which are "restricted", or "committed", others are not. He says: "From this emerge thirty-two cipher alphabets, the first of which is unrestricted, the second is restricted to the letter (‫)ا‬, the third to the letter (‫)ة‬, and so on till the end of the alphabet"19. The cipher alphabets he mentions are the following arrangement: 1 unrestricted. 29 each restricted to one letter of the alphabet. 1 by changing the restriction according to the numericalalphabet order. 1 by changing the restriction according to alphabetical order. ----32 cipher alphabets

1.3.2.8 Using invented symbols or signs to represent letters With this type ibn ad-Durayhim concludes his discussion of encipherment methods. This is a variety of simple substitution he would utilize in his examples on cryptanalysing. The keynote of the method is substituting for each letter of the clear alphabet a distinctive symbol. Among its conveniencies are the susceptibility to fill up spaces with hyphens, dots, blanks, circles; or by a symbol similar to that devised for the letters. Applicable to this method also is the addition of extra symbols; namely, nulls which render cryptanalysis more complicated.

18

Maq id al-Fu l al-Mutar ima an Cryptology", 64/A. 19 See his treatise, p. 79.

all at-tar ama; from "Collected Papers on

29

However, ibn ad-Durayhim could be criticized for bypassing the possibility of sparing the space; what ibn Adl n termed "the no-wordspacer cryptogram", or "al-mudma ". This spells ibn Adl n's superiority as concerns the presentation of cryptanalysis methods.

1.3.3 Morphological introduction In this significant introduction, ibn ad-Durayhim sums up some linguistic information on the Arabic language (linguistic, morphological and phonetic rules). He deems that it is an indispensable precondition for working out cipher. He says: "Cryptanalysing the afore-stated and all kindred ciphers needs a 'genial' introduction that serves as a guide"20. The contents of ibn adDurayhim's introduction may be summarized as follows: A. On word-length - Definition of the "word" according to "writers" and grammarians, and the fact that cryptology is more concerned with the "writers" definition. - The shortest word in Arabic is one letter, and the longest word is fourteen letters, depending on the word being a noun, a verb or an article. - The maximum length of a noun prior to affixation is five letters. - The maximum length of a verb prior to affixation is four letters. - No word of four- or five- letter root is devoid of at least one of the "liquid letters"; i.e. the letters: (‫ ٻ‬،‫ ة‬،٫ ،‫ ٿ‬،‫ ه‬،‫)ٷ‬. B. The maximum repetition of a letter in one word - The same letter can be repeated in one word five consecutive times at the very most.

20

See his treatise, p. 85.

30

C. Combinable letters These are of various kinds: - Non-combinable letters (neither in anterior nor in posterior positions). - Combinable in anterior position only. - Combinable in posterior position only. - Letter repetition at the beginning of words. ibn ad-Durayhim elaborates these kinds with highly admirable competence. We have opted for devoting an independent study to the related linguistic information, abundant in books on cryptology, particularly those reported by al-Kind , ibn Dunayn r, ibn Adl n and ibn ad-Durayhim. The following table (Table 1.11) covers the cases of the noncombinable letters, those non-combinable in anterior position only, or in posterior position only, as given by ibn ad-Durayhim in his treatise Mift al-Kun z.

31

Letter symbol

Letters non-combinable with it

Resultant non-combinable bigrams

Table 1.11: Table of non-combinable letters as observed by ibn ad-Durayhim Key:

post-positively (posteriorly) pre-positively (anteriorly) neither post- nor pre-positively.

32

1.3.4 Algorithm for cryptanalysis ibn ad-Durayhim's algorithm for breaking a message enciphered by substitution may be summed up by the following stages: a) Counting the letters or symbols. b) Checking the times of recurrence of each letter or symbol. c) Cryptanalysing the space, so as to properly separate the words. d) Matching the symbol or letter frequency of occurrence in the cryptogram against the frequency of the language letters. ibn adDurayhim stresses the decisive importance of the message being long enough to allow for acceptable matching. e) Utilizing word lengths (two-character words, three- character words,…) and the probable word principle. f) Making use of the fact that the letters preceding (‫ )اٷ‬at the beginning of a word may, all too often, be: (‫)ة‬, (٫), (‫)ٳ‬, or (‫)ڇ‬. It merits consideration that ibn ad-Durayhim, unlike al-Kind , ibn Dunayn r and ibn Adl n, departs from observing the same letter order; instead, he depends on the statistics of letters of the Holy Koran. Again, unlike his predecessors, he considers (‫ )ال‬a letter of the alphabet.

1.3.5 Two practical examples of cryptanalysis ibn ad-Durayhim concludes with an interesting minute analysis of two examples ciphered by substitution, utilizing devised symbols or shapes as ciphertext replacements. The researcher would find that, for the most part, al-Qalqa and 's quotations21 have been derived from these very examples. David Kahn22 believes that ibn ad-Durayhim's work was "the first exposition on cryptanalysis in history". In fact, and as a result of our subsequent investigation, it was brought home to us that it was al-Kind , ibn Dunayn r and ibn Adl n who actually took 21

See ub al-'A , 9/240 and 245. (Published by al-Mu'assasa al-Mi riyya almma, 1963) 22 See his book: The Codebreakers, p. 96.

33

the lead, since the earliest of those mentioned, al-Kind , lived five centuries ahead of ibn ad-Durayhim! Still, there is no gainsaying the merit of ibn ad-Durayhim's treatment of cryptanalysis as the most detailed of all the past cryptographic legacy handed down to us through generations.

1.4

Originality of ibn ad-Durayhim

From our analysis of ibn ad-Durayhim's treatise we arrive at the following conclusions: 1. ibn ad-Durayhim's originality manifested itself first and foremost in his explanation and analysis of ciphering methods, their individual capabilities and qualifications, especially the substitution cipher. His originality was more evident in cryptography than cryptanalysis. 2. We believe that he was familiar with ibn Dunayn r's treatise Maq id al-fu l al-mutar ima an all at-tar ama. This is evident from the uniformity in using some encipherment devices and methods, such as the chessboard, thread, beads, and the decimal numerical alphabet (i.e. the arithmetic using decimallyweighted letters: is b al- ummal); unlike al-Kind and ibn Adl n who did not make any reference to them. 3. ibn ad-Durayhim did not refer to composite encipherment, neither did he mention the no-word-spacer encipherment as ibn Adl n had done a century earlier. Similarly, he only made a passing reference to ciphering poetry. 4. As noted before, al-Kind and ibn Adl n made no attempt to deal with the decimal numerical alphabet, contrary to ibn Dunayn r who paved the way, and ibn ad-Durayhim who continued along his lines.

34

Chapter 2

ibn ad-Durayhim's edited treatise: Mift

al-Kun z f '

al-Marm z

2.1 Editing methodology The main purpose of editing is the reproduction of a text as close to the author's original as possible. In line with this objective we have opted for conserving the statement of the original whenever possible.  The very nature of the original manuscripts required the addition -where appropriate- of explicatory titles in the interest of marking out divisions or classifications. This would prove useful for easy understanding and clarity of ideas.  No effort has been spared in the interpretation of citations (Koranic verses, Prophetic traditions, lines of poetry, sayings, etc.) contained in the treatises. We have given brief biographical identification of personalities, relegating the interested reader to such authorities as al-A l m by ayr alD n al-Zirkily or Mu am al-mu'allif n by Omar Ri Ka ala, for further and more detailed biographical reference. Those citations and personalities that our efforts fell short of their interpretation or identification have also been properly recorded.  In explaining the linguistic terms included in the treatise we have made use of various dictionaries, old and modern, foremost of which are: Lis n al- Arab and Matn al-lu a. Unless otherwise helpful, no reference has been made to any dictionary.  We have adopted the same symbols and signs commonly employed by editors of Arabic manuscripts. We conformed to the modern spelling norms, and we enclosed requisite contextual additions -i.e. explanatory insertions and comments other than the writer's own words- within square brackets [ ]; examples illustrating rules of encipherment have been set off by round brackets (parentheses) ( ); book titles in italics, quoted material and Prophetic traditions have appeared within quotation marks “ ” , while floral brackets  have been used to enclose Koranic verses.23 23

Translator's explanatory additions are placed between pairs of hyphens: -…-.

37

2.2 Description of the manuscript The original manuscript is part of an assemblage of small-sized sheets, comprising several treatises on such occult sciences as numerology (z yir a), divination ( afr), al-awf q, geomancy, talismans, and others. The assemblage is handwritten in fine penmanship, and housed in 'As ad 'Afand 's Library of asSulaymaniyya Ottoman Archives in Istanbul, under the number 3558. The first sheet is an index, written by the scribe, of the titles of treatises included. Each title is written in two lines, with the number of the first sheet of each treatise affixed thereunder. The index reads as follows: "What is contained in this unique paper:  ar ka f ar-r n f al-z yir a ................................................. 2 (Exposition of " " unveiling in "z yir a").  ar bayt minh , by al- amr ............................................... 11 (al- amr 's exposition of a line of the above).  'Isti r al-'a wiba min al- afr al- mi .............................. 14 (Drawing the answers out of the extensive afr).  F naw m s al- aw riq lil- d t ............................................ 27 (On the laws of the supernatural).  Man mat al-'Im m al-Sabt ................................................. 41 (al-'Im m al-Sabt 's poems).  Mift al-Kun z f ' al-Marm z ...................................... 47 (Key to treasures on clarifying ciphers).  all at- illasm f al-z yir a .................................................... 60 (Solution of talisman in z yir a).  Ad-durra al-munta aba f al-'a wiba .................................... 63 (The gem: a collection of answers).  F al-'awf q al-mu awwaqa .................................................... 67 (On confined 'awf q).  'Istin q al-'a ruf min al-' y t ................................................ 74 (Elicitation of letters from Koranic verses).  Ras 'il f al-raml, by Na r a - s ......................................... 77 (Na r a - s 's treatises on geomancy).  Kit b al-'akt f ......................................................................... 80 (The book of al-'akt f).

38

     

F al-mu ammas al- l al-wasa ........................................ 85 (On poetical quintets). Da aw t as-s t, by al-Ba n ............................................... 89 (al-Ba n 's invocations of times). F ilm al-'awf q, by al-qabb n ........................................... 102 (On al-'awf q science by: al-Qabb n ). ar sim al-hindiyya f al-wafq ............................................. 105 (On letters and their secrets) Kalim t ibn al a f al-waq 'i ............................................ 109 (ibn al a 's words on events). Bay n as-s a, by as-Suy ............................................... 121" (On Doomsday)

Beside the last title the scribe has made this note: "

"(i.e. copied by the poor scribe),

immediately appended by a seal on which has been inscribed in Persian-style Arabic script: "My lord, I ask Thee a creditable end". ibn ad-Durayhim's treatise, whole and complete, occupies the pages 47/B to 59/A, and closes with a colophon of the scribe's name and date of copying.

39

Figure 2.1: A photocopy of the index of the assemblage incorporating ibn ad-Durayhim's treatise (Document No. 3558, as-Sulaym niyya Ottoman Archives, Turkey)

40

Figure 2.2: A photocopy of the first page of treatise

ibn ad-Durayhim's

(Document No. 3558, as-Sulaym niyya Ottoman Archives, Turkey)

41

Figure 2.3: A photocopy of a page of ibn ad-Durayhim's treatise illustrating encipherment using the "branched" calligraphy (Document No. 3558, as-Sulaym niyya Ottoman Archives, Turkey)

42

Figure 2.4: A photocopy of a page of ibn ad-Durayhim's treatise demonstrating the encipherment of the first of two examples. (Document No. 3558, as-Sulaym niyya Ottoman Archives, Turkey)

43

Figure 2.5: A photocopy of a page of ibn ad-Durayhim's treatise demonstrating the encipherment of the second of two examples. (Document No. 3558, as-Sulaym niyya Ottoman Archives, Turkey)

44

Figure 2.6: A photocopy of the encipherment of ibn ad-Durayhim's second example as set out in ub al-'A 9/245. (Published by al-Mu'assasa al-Mi riyya al- mma, 1963)

45

Figure 2.7: A photocopy of the last page of ibn ad-Durayhim's treatise. (Document No. 3558, as-Sulaym niyya Ottoman Archives, Turkey)

46

2.3 Ali ibn ad-Durayhim Treatise on Cryptanalysis (Original Arabic Text and English Translation)

47

Mift

al-kun z f ' al-marm z

by Al ibn ad-Durayhim

48

In the name of God the Most Gracious, the Most Merciful. [Introduction] Glory be to God, who set out with the creation of the pen, disposeth it on the guarded tablet, and dealt out tongues amongst nations. He is the Omniscient, Knower of all secrets. Praise be to Him who made known something of His knowledge. We profess that there is no god but Allah, the One without partner, and that Mu ammad is His prophet to all people and His favourite whom He brought so nigh to Him, and made the seal of prophets. May God's blessing and peace without end be upon him and his noble household and companions. I had earlier written a book on the formulation of ciphers and their cryptanalysis, which I called: '

al-mubham f

all al-mutar am

(The clarification of ambiguities in cryptanalysing cipher texts). A period of time had elapsed since I abridged it. I had no other copy in my possession. Who must be obeyed, and whose request cannot be refused, asked me to write this book. I have thus put down what came to my mind of the rules and regulations of this art, and I have written this preface in order to explain the sructure of the book and facilitate its comprehension, God willing. I have called my book Mift Kun z f '

al-

al-Marm z (Key to treasures on clarifying ciphers). I

pray to God for help and success, most sufficient unto us is He in whom we trust.

50

Know that cryptanalysing cipher texts and cryptograms is a very worthy cause. It is indispensable in times of need, and useful in understanding the symbols of the ancients in their sciences and books and other material they have bequeathed.

*

*

*

[Essentials for those practicing cryptanalysis] -The cryptanalyst's toolsIt is necessary for one experiencing cryptanalysis to develop a thorough knowledge of the cryptogram language he seeks to cryptanalyse, as well as the language grammar. He should also know the frequency of occurrence of letters and their order, such as long vowels which have the highest frequency of occurrence in all languages. Letters of highest frequency in certain languages are (a) "alif" in Arabic, (s) in Latin and Armenian, and (n) in Mongol. According to the numerical alphabet, all calligraphs have detachable letters short of

Mongol, Syriac and Arabic, of which

letters can be both detachable and conjoint. Syriac letters are detached and conjoined as in Arabic.

52

The shortest of all calligraphs [alphabets] is Mongol, consisting of 17 letters; the longest Armenian: 36 letters; the Turkish 20 letters, and as many for the Persian calligraph, with three letters in it not in the Turkish, namely, h ( ‫)څـ‬, f (٫), and

(‫)م‬. It follows that there are three

letters in the Turkish calligraph not there in the Persian, namely, (ٓ), (ٛ) and q (‫)ٯ‬, with the letters:

(‫)ػ‬,

(٣),

(‫)س‬,

(ٗ) and

(ٟ)

absent. The Hebrew, Syriac and Astank ly calligraphs are made up of 22 letters each, i.e. from the letter (‫ )ا‬to the letter (‫ )د‬of the numerical alphabet. The French and Latin are 27 letters; the old Latin and Greek 24 letters (these two have another calligraph of 30 letters for the uncertain of their letters); the Coptic 32 letters (it also has a numerical alphabet). One Hindi calligraph differs in their language from that of their numerical alphabet which comprises 28 letters in nine forms with the following orders: (٨‫اٌٲ‬, ‫ثٶو‬, ِ‫عٺ‬, ‫كٽذ‬, ‫څڂذ‬, ‫ڇٍـ‬, ‫ن‬٥‫ى‬, ٘‫ ؽٮ‬and ٠ٖٝ). For some Indians there is another calligraph of 52 letters called the triangular Hindi. The Sumerian calligraph is four letters less than the Hebrew. The letters wanting, being from the Old Testament, are: hamza (‫)ء‬,

(‫)ػ‬, (٣) and h (‫ ;)څـ‬so for (‫إثواڅٍټ‬,, ‫ٍٸ‬٥‫إٍپب‬, ‫ إٍؾبٯ‬and ‫ٲڈة‬٦ٌ)

they say: ( ‫ثٍوٻ‬, ‫ّپڈٌٸ‬, ‫ ٌٖبٯ‬and ‫ )ٌبٱت‬respectively, with ( ‫)ٌبٱت‬ pronounced halfway between ‫ ي‬and ‫ ڇ‬and all in the same enunciation. They have no such letter as

(‫)ؿ‬, and the letters (٣) and hamza (‫)ء‬

may on occasion be pronounced as such, thus ( ‫ )إثواڅٍټ‬may be pronounced (‫)إثوٻ‬, and would probably be enunciated (‫ اٷ‬٤‫)ٌْپ‬.

54

Encipherment methods are of various types, too many to enumerate. I mean to mention the basic principles and rules that govern their laws. *

*

*

[Types of encipherment] Some people opt for changing places of the letters within a cryptogram, conformably with some criteria. This is called the transposition type. [1. On transposition]  By writing a word in reverse, e.g. ( ‫ )ٽؾپل‬is enciphered: (‫;)كٽؾټ‬ (ً‫ٺ‬٥: ٤‫)ٌٺ‬.  By writing the last letter first thus: ( ‫ ٽؾپل‬: ‫) كٽؾټ‬, ( ً‫ٺ‬٥: ‫ٸ‬٦ٌ).  By transposing the first letter of a word with the last, e.g. (‫ ٽؾپل‬: ‫) كؽپټ‬, (ً‫ٺ‬٥: ٤‫)ٌٺ‬.  By changing positions of even letters with odd letters, e.g. ( ‫ ٽؾپل‬: ‫) ؽپلٻ‬, ( ً‫ٺ‬٥: ً٦‫)ٹ‬. This rule may be extended for application in multi-word texts; thus: ً‫ٺ‬٥ ‫ٽؾپل أفڈ‬ (plaintext) becames: ‫ڈٌٸ‬٥ ‫( ؽپلٻ فب‬ciphertext).  By transposing the first letter of a word with the third letter, e.g. ( ‫ڈك‬٦َ‫ٽ‬: ‫َپڈك‬٥), ( ً‫ٺ‬٥: ٤‫)ٌٺ‬, ( ‫أؽپل‬: ‫ ;)ٽؾأك‬or by bringing every two consecutive letters in front of the preceding two. This can be done throughout a multi-word message, looking at it as an integral whole, e.g. ‫ڈك أفڈ‬٦َ‫ٽ‬ ‫( أؽپل‬plaintext): ‫ڈٽَقڈك أٽلأػ‬٥ (ciphertext), or within individual words, e.g. ‫ڈك أفڈ أؽپل‬٦َ‫ ٽ‬: ‫ڈٽَل ڇفأ ٽلأػ‬٥.  By transposing the first letter of a word with the fourth, e.g. ( ‫ڈك‬٦َ‫ٽ‬: ‫پل‬٦ٍ‫)ڇ‬, ( ‫ٽؾپل‬: ‫)كؽپټ‬. This, again, may be observed for a multi-word text, tackled as one integral unit.

56

 By alternating one initial letter of a word with the corresponding terminal one until the word is exhausted, in terms of the ascending alternate horizontal transposition, e.g. ( ‫ڈك‬٦َ‫ )ٽ‬is enciphered: ( ٣‫ )ٽلٍڈ‬, ( ‫ ٽؾپل‬: ‫ )ٽلؽټ‬and ( ً‫ٺ‬٥ : ‫ٍٸ‬٥). This process also holds in multiword texts; that is by taking alternately one letter from the beginning followed by the corresponding letter from the end of the message until all are used. This method may be performed the other way round, namely in term of the descending alternate horizontal transposition, starting from the end of the message by taking alternately one letter from the end followed by the corresponding letter from the beginning until all are used. Thus: ( ‫ڈك‬٦َ‫ ٽ‬is enciphered: ٤ٍ‫)كٽڈ‬, ( ‫ ٽؾپل‬: ‫ )كٽپؼ‬and ( ً‫ٺ‬٥ : ‫ٸ‬٦ٌ). This process, again, also holds in multi-word texts.  By taking the words of the text in pairs (1), transposing the first letter of one word with the first letter of the other word, e.g. ‫ټ ؽَڀ‬٥ ‫( ٽؾپل اثڀ‬clear): ‫َڀ‬٥ ‫( اؽپل ٽجڀ ؽټ‬cipher), (2) or transposing the last letter of one word with the last letter of the other, e.g. ‫ټ ؽَڀ‬٥ ‫( ٽؾپل اثڀ‬clear): ‫ڀ‬٥ ‫ٽؾپڀ اثل‬ ‫( ؽَټ‬cipher), (3) or transposing the first letter of one with the last letter of the other; so that the above example is enciphered: ٤َ‫ځؾپل اثټ ځټ ؽ‬, (4) or transposing the last letter of one with the first of the other, so that the above example is enciphered: ‫ؼ ٽَڀ‬٥ ‫ٽؾپب كثڀ‬, (5) or, as an extra measure, transposing the first letter of one word with the first letter of the other, while simultaneously transposing the last letter of one word with the last letter of the other. Thus, the example: ‫ټ ؽَڀ‬٥ ‫ ٽؾپل اثڀ‬would be written in cipher: ‫َټ‬٥ ‫اؽپڀ ٽجل ؽڀ‬, (6) or transposing the first letter of one word with the last letter of the other, and the last letter of one with the first letter of the other. In so doing, the above example is enciphered: ٤َ‫ځؾپب كثټ ځؼ ٽ‬.

58

 By taking every other letter right through, dropping the others for the time being, the intended plaintext is obtained. Letters dropped are now considered in the same way to complete the text, e.g. ‫ټ ؽَڀ‬٥ ‫( ٽؾپل اثڀ‬clear): ‫پپلؽبٍجڀ‬٦‫( ٽڂؾ‬cipher). You can also take one letter and drop two till the end. You do the same with the second letter, then with the third. So you write the above example: ‫ڀ‬٥ ‫ ٽبٽؾجؾپڂَل‬. You may take one letter and leave out the following one, or four, five, etc.. letters as you please. Anyhow, it is advisable to separate words by as many spaces as the cipher alphabets according to a fixed rule. The above example would be written, using the last mentioned method: ‫ڀ‬٥‫ٽبٻ ؽجؼ ٽڂٌ ك‬. From this a good many configurations arise, all of which are the very letters constituting a message, no more, no less, but transposed. [2. On substitution] -Encipherment of this type can be exercised-:  By always substituting a specific letter for another according to a set key, as in the Qummi cipher alphabet represented by this line of verse:

in which the letter m (‫ )ٻ‬is substituted for the letter k (‫)ٳ‬, and vice versa, the letter o (‫ )ڇ‬for

(‫ )ا‬and the other way round, and

so forth. Accordingly, the word ( ‫ )ٽؾپل‬, for instance, is enciphered: (‫ٶو‬ٞ‫)ٵ‬, (ً‫ٺ‬٥: ٬‫)ٍچ‬, and (‫ڈك‬٦َ‫ ٽ‬: ‫َبه‬٦‫)ٵ‬.

60

There is another cipher alphabet represented by this verse: so that the word (‫پو‬٥) is written in cipher: (ّٜ‫)ى‬. And also the Fahlawi cipher alphabet: e.g. ( ‫ٽؾپل‬: ‫)مىمٯ‬, ( ً‫ٺ‬٥ : ‫)ځٍٸ‬, ( ‫پو‬٥ : ‫)ځنڇ‬. This relates to the unregulated encipherment by substitution, which is generative of innumerable cipher alphabets.  Using the numerical alphabet order, by substituting for each letter the one immediately following; thus the letter (‫ )ة‬is substituted for (‫)ا‬, and (‫ )ط‬for (‫)ة‬, and so on until the end. The letter (‫ )ا‬is substituted for (٧). This is because letters are like a circle, [i.e. they are viewed as located on a circle circumference or a disc] in that the last letter is replaceable by the first letter as if to follow or preceded it. Example: ‫( ٽؾپل‬clear text): ‫ڂڄ‬ٞ‫( ځ‬cipher text). You may substitute for each letter every third letter next to it, so that ‫( ٽؾپل‬clear) becomes ‫( ٍٍَڈ‬cipher), and ( ً‫ٺ‬٥ : ‫ ;)ٕڂٸ‬or every fourth letter, so that ( ‫ ٽؾپل‬becomes: ‫ي‬٦‫ٶ‬٥) and ( ً‫ٺ‬٥ : ‫ )ٱَټ‬and so on and so forth till the end of letters. This results in 28 cipher alphabets.  By considering the numerical alphabet as composed of pairs of letters; the substitution is performed between the letters within each pair. So the word (‫ ٽؾپل‬is enciphered: ‫ )ځيځظ‬and ( ً‫ٺ‬٥ : ٜ‫)ٍٶ‬. The pairs are formed by systematically taking every letter with the one immediately following it; or with every third letter next to it, e.g. ( ‫ٽؾپل‬: ‫)ٍڈٍت‬, ( ً‫ٺ‬٥ : ‫ ;)ځٍٸ‬or with every fourth, fifth, etc.. letter next to it.  Alternatively, -similar to this last-mentioned method- by substituting for a letter the one preceding it. This would bring about a number of cipher alphabets amounting to 58.

62

But care should be taken, while considering the 112 cipher alphabets, that the encipherer is not Maghrebi, since the order of letters in our numerical alphabet is different from the Maghrebi's, which runs as follows: ‫أثغل‬, ‫څڈى‬, ًٞ‫ؽ‬, ‫ٵٺپڀ‬, ٘‫ٮ‬٦ٕ, ‫ٱوٍذ‬, ‫صقن‬, ِ٪١. And this is the order which a - ib , peace be on his soul, adopted for assigning symbols to reciters [in a wellknown poem attributed to him on the Koranic modes of reading].  Adopting the alphabetical order, by substituting for each letter the one immediately following, thus substituting the letter (‫)ة‬ for (‫ )ا‬, (‫ )د‬for (‫)ة‬, (‫ )س‬for (‫)د‬,etc. until the end. The letter (‫ )ا‬is substituted for the (‫)ي‬. Examples: ‫( ٽؾپل‬clear): ‫( ځقڂن‬cipher), ( ً‫ٺ‬٥: ‫پب‬٩). You may substitute for each letter every third letter next to it, so that the word ‫ ٽؾپل‬is enciphered either ‫ ڇكڇه‬in terms of the key in which the letter (‫ )ڇ‬goes before the letter ( ‫)څـ‬, or ‫ څلڅو‬in terms of the key in which (‫ )ڇ‬follows ( ‫)څـ‬. Also, ً‫ٺ‬٥ (plain): ‫( ٭ڂت‬cipher). This brings forth a number of cipher alphabets amounting to 29. You can, in the same way, substitute for each letter the one immediately preceding; thus the letter (‫ )ي‬is substituted for (‫)ا‬, the letter (‫ )ا‬for (‫)ة‬, (‫ )ة‬for (‫)د‬, etc. Examples: ( ‫ٽؾپل‬: ‫)ٹغٺـ‬, (ً‫ٺ‬٥: ‫ٶال‬١). This engenders 29 cipher alphabets, too.  By considering the alphabet as composed of pairs of letters; the substitution is done between the letters within each pair. The pairs are formed by systematically taking every letter with the one immediately preceding, e. g.: ( ‫ٽؾپل‬: ‫)ٹغٺـ‬, ( ً‫ٺ‬٥: ‫ٶڈ‬١), adopting the cipher alphabet in which the letter ( ‫ )څـ‬precedes the letter (‫)ٿ‬. The (‫ )ال‬may be spared, or substituted leaving the (‫)ي‬ out. In this manner substitution can be conducted by taking every letter with every second, third, fourth, etc. preceding letter, as already mentioned. This would produce 58 cipher alphabets, too.

64

Analogous to these four divisions in the Maghrebi system are 116 cipher alphabets. Their alphabet runs: (‫ا‬, ‫ة‬, ‫د‬, ‫س‬, ‫ط‬, ‫ػ‬, ‫ؿ‬, ‫ك‬, ‫م‬, ‫ه‬, ‫ى‬, ٛ, ٟ, ‫ٳ‬, ‫ٷ‬, ‫ٻ‬, ‫ٿ‬, ٓ, ٗ, ٣, ٧, ٫, ‫ٯ‬, ً, ُ, ‫څـ‬, ‫ڇ‬, ‫ال‬, ‫)ي‬. In substituting for a letter the immediately preceding one, the word ( ً‫ٺ‬٥), for instance, becomes: (‫ٶال‬ٙ), and (‫ڈك‬٦َ‫ ٽ‬: ‫چـ‬ٚ‫)ٹٲ‬. All these cipher alphabets do not involve any addition (augmentation) of letters. [3. On the augmentation or reduction of the number of letters] -This can be performed-:  By repeating all letters, or only odd letters.  By dropping a certain letter wherever it occurs, or choosing pairs of letters with or without some charactaristic in common -such as (ٟ ٛ) and (٣ ‫ )ة‬respectively-, and regarding them as single letters throughout.  By inserting an extra letter somewhere within each word, or inserting a pair of similar or dissimilar letters, or adding a certain letter (e.g. (‫ ))ا‬to one word and another letter (e.g. (‫ ))ة‬to the next word, and so on until the end, using either the alphabet (‫ا‬, ‫ة‬, ‫د‬, ‫س‬, …) or the numerical alphabet (‫)أثغل‬.  By applying any of the above rules anywhere at will, thereby producing many cipher alphabets. [4. On the utilization of cipher devices]  The chessboard, which can be utilized by assigning each square to a letter. The message is sent by placing certain chessmen on intended squares, and the reply is likewise received. In either case the order of the alphabet (‫ا‬, ‫ة‬, ‫د‬, ‫س‬, …) or the numerical alphabet (‫ )أثغل‬is observed.

66

 The punched board, with 28 holes standing for the letters. The cryptogram is represented by a thread driven through the intended holes so as to make a route defining the letters of the message successively. To represent ‫أؽپل‬, for example, the thread is driven through the holes: 1, 8, 13 and 4 consecutively, using the numerical alphabet ( ‫)أثغل‬. The algorithm for decipherment, regardless of the length of the cryptogram, is by reading the letters through which goes the thread. For each hole you write a letter. The order of letters is then reversed so that the last one is made the first. By so reading to the first letter, you are correct. [5. On the replacement of letters using the decimally-weighted numerical alphabet]  By substituting decimal numerical alphabet for letters in four different ways: by writing the numbers in words as pronounced; or by finger-bending, using the fingers to communicate the message visually to a recipient; or by writing the numbers as numerals, such as writing ( ‫ٽؾپل‬: fourty, eight, fourty, four); or by giving the crytogram a semblance of a page of a financial register.  By reconverting the cryptogram numerals into a number of letters ‫ ـ‬a method of encipherment which involves more sophistication. There are many combinations that can be used in this method; for example in ( ‫ٽؾپل‬: ً‫ٹ‬. ‫ثڈ‬. ً‫ٹ‬. ‫ )اط‬or ( ‫ٵٴ‬. ‫اى‬. ‫ٵٴ‬. ‫)ثت‬. One can even form delusive words such as (‫ٌؾجٴ‬. ‫اثلا‬. ‫ڇٹل‬. ‫)عب‬, or substitute two words for a letter, e. g. (ً‫ٺ‬٥: ‫ٍجؼ‬. ً‫ڇڅبثبً عڈاكا‬. ‫)څلأ‬, in which case a line is to be drawn over the two words to denote that they represent one letter.

68

 By multiplying the number representing the letter by two, and so write ( ‫ٽؾپل‬: ٫ ‫ ٌڈ‬٫ ‫ )ػ‬and ( ً‫ٺ‬٥: ‫)ٳ ً ٱټ‬, etc.; or multiply it by three, thus writing ( ‫ٽؾپل‬:

‫ )ٌت ٱٴ ٵل ٱٴ‬and ( ً‫ٺ‬٥: ًٍ ٗ ‫)ٷ‬.

Numbers can also be multiplied by four or five. [6. On the encipherment of letters by using words]  By substituting for a letter its spelling, or by alternately writing the straight spelling of one letter and the reversed spelling of the next, e.g. ( ‫ٽؾپل‬: ‫ )ٽٍپبؽپٍپالك‬and ( ً‫ٺ‬٥: ‫ٍڂپبٹٍب‬٥). One may start with the reversed spelling followed by the straight, and so write (‫أؽپل‬: ‫ )٭الؽب ٽٍپلاٷ‬and (ً‫ٺ‬٥: ‫الٽبي‬٦ٍ‫)ځ‬. The above rules may be partly applied in various ways, giving rise to many ramifications.  By feigning words, - conformably with a set rule -, in which the intended letters are made to be the first letter of each word, so that the word ( ‫)ٽؾپل‬, for example, may be enciphered: ( ٬‫)ٽب ؽبٷ ٽَٶٍڀ كځ‬, and ( ً‫ٺ‬٥: ً‫و٭ذ األٽو ٌٍَوا‬٥); or the last letter of each word, where ( ‫ )ٽؾپل‬becomes: ( ‫جل‬٦‫ٺټ ٕوٌؼ ّټ اٹ‬١), and ( ً‫ٺ‬٥: ً‫ ٽبٷ أث‬٤ٍٙ); or the middle letter of each word, thus ( ‫)ٽؾپل‬ may be expressed: (‫)ٌپڀ ثؾت ّپٌ فله‬, and (ً‫ٺ‬٥: ‫ٺى فٍو‬٥ ‫ل‬٥‫)ك‬, and suchlike.  By taking the second letter of each feigned word, e.g. ( ‫ٽؾپل‬: ‫لڃ‬٩ ‫پٸ‬٥ ‫)ٹټ ٌؾَڀ‬, and ( ً‫ٺ‬٥: ‫ اٹٖجو فٍو‬٤‫ ;)ٽ‬or by taking the third letter of each word throughout, e.g. ( ‫ٽؾپل‬: ‫)أٹټ أهؽٴ ٌڈٻ ځغل‬, and (ً‫ٺ‬٥: ‫ذ ٱڈٹٴ ٭غٍذ‬٦‫)ٍپ‬, and so on.

70

 By taking from every three words the first letter of the first word, the second letter of the second word and the third of the third word, so the word ( ‫ل‬٦ٍ) would be enciphered: ( ‫ ٽٲلاهڃ‬٫‫و‬٦ٌ ‫ ;)ٍٍّل‬or by adopting odd letters only, i.e. the first, third, fifth, etc., e.g. ً‫ٺ‬٥ ‫ټ‬٥ ‫( ٽؾپل اثڀ‬clear): ‫ٍټ‬٦‫ٽب ؽزټ ٱل ٱبهة ٽڀ ځ‬ ‫پٸ فٍو‬٦‫( ث‬cipher); or even letters only, -i.e. the second, fourth, sixth, etc.-, writing the same example in cipher thus: (ً‫غٺڂ‬٦ٌ ‫ٍټ‬٦‫)ٵټ رؾڈٻ ٱلٽبي ثٍڀ ځ‬.  By taking up one letter and leaving out the next two letters, e.g. (ً‫ٺ‬٥ ‫ټ‬٥ ‫ٽؾپل اثڀ‬: ‫ٲڈٷ ٱبٌلح‬٥ ‫ٺڈٽچټ‬٥ ً‫)ٽب أؽَڀ ٽڂبكٽخ أٵبثو اٹڂب‬, and the like. Conversely, some may start by leaving out rather than taking up letters, so that, of the above example, the third, sixth, ninth, etc. letters are taken. The cipher may look like this: ( ‫پٸ ٹلځٍبڃ‬٦ٌ ‫چب ٽڀ‬٦ٌٚ ‫)أٹټ أهؽپٶټ ثجلهح أڇعجذ أٿ‬, and so on. Another method is by taking the first letter and then every fourth letter throughout, so that in enciphering the words: ( ً‫ٺ‬٥ ‫ټ‬٥ ‫)ٽؾپل اثڀ‬, you may write: (‫لڃ اٹزجغٍٸ ٹڄ‬٦ٍ ‫لٿ أٽبٿ‬٦‫)ٽڀ اٹؾَڀ ٹپڀ ٌزلٌڀ ثبٹٲوثى ٹغڂبة ٽ‬. You can of course start by leaving out letters rather than taking them, as already stated, with a feasibility of dropping four, five, etc. of the extraneous letters at a time while taking one throughout the cryptogram.  The encipherer may choose to make his key known to the recipient. One way of doing that is to agree that starting the cryptogram with the letter (‫ )ا‬suggests to the recipient that every second letter is to be taken, starting with the letter (‫ )ة‬means that every third letter should be taken, starting with (‫ )ط‬means that every fourth letter should be taken, and so on and so forth. Some start by enciphering the opening: ( ‫)ثَټ اهلل اٹوؽپڀ اٹوؽٍټ‬, from which the key is detected without toil and applied all through.  In so merging ciphertexts with plaintexts, the cryptogram may be made to read backwords, i.e. from left to right.

72

 The encipherer can substitute for a letter a proper name, so that every letter of the cryptogram is represented by the name of a person. He may also build on the names of stars, mansions of the moon, (either according to respective letters of the numerical alphabet or at random; thus the lunar mansions are in succession: ura n (for ‫)ا‬, Bu n (for ‫)ة‬, Pleiades (for ‫)ط‬, and so on until the last one, Ra (for the letter ٧)), months (lunar, Latin, Coptic, etc.), the number of days in a month, hours of the day, days of the week and its hours, book names, suras of the Koran, names of countries, ointments, drugs, an n t*, fruits, trees, etc., or any other word of his choice repeated every time the letter it represents occurs. The cryptographer may perform that verbally, in writing, or as a picture or symbol, such as birds, animals, plants or trees, whichever he pleases.  It is well worth mentioning here the branched calligraph, which is based on the words of the numerical alphabet , and practicable in writing only - i.e. not feasible verbally -. The first letter of the cryptogram is represented by a single branch on the right of the trunk if that letter is one of the constituent letters of the word (‫)أثغل‬, which is the first word of the numerical alphabet. Likewise, if the second letter of the cryptogram is part of the word ( ‫( )څڈى‬the second word of the numerical alphabet), it is represented by two branches on the right of the next trunk, and so on. Note that a maximum number of eight right branches is possible, which is the total sum of the numerical alphabet words. Now you look at the position of each letter of the cryptogram within the word of the numerical alphabet of which it is part; thus if the first letter of the cryptogram is , say, (‫ )ط‬you draw three branches on the left of the first trunk, because (‫ )ط‬is the third letter of the first word of the numerical alphabet. Similarly, the letter (‫)ٿ‬, for example, is represented by four left branches on the relevant trunk. Clrearly, a number of four left branches in the maximum possibility for representing a letter, as none of the numerical alphabet words comprises more than four letters.

*

-There is no reference whatsoever to " an n t" in Arabic dictionaries, hard as I have searched, though. It might be a scribe's error.** The numerical alphabet is: ‫أثغل‬, ‫څڈى‬, ًٞ‫ؽ‬, ‫ٵٺپڀ‬, ٔ‫ٮ‬٦ٍ, ‫ٱوّذ‬, ‫صقن‬, ٨٢ٙ; i.e. ‫أ‬, ‫ة‬, ‫ط‬, ‫ك‬, ‫څـ‬, ‫ڇ‬, ‫ى‬, ‫ػ‬, etc.

74

Example: ( ً‫ٺ‬٥ ‫ټ‬٥ ‫ )ٽؾپل اثڀ‬is enciphered using the branched calligraph as follows: (*)

[7. On enciphering by relationship and diffusion method]  Encipherment may be done by substituting for each letter of the Arabic alphabet a generic name as follows: The letter (‫ )ا‬is used for people; (‫ )ة‬for legume (vegetables); (‫)د‬ for dates, soil or spices; (‫ )س‬for clothing; (‫ )ط‬for leather; (‫ )ػ‬for cereals or iron; (‫ )ؿ‬for wood; (‫ )ك‬for animals or ointments; (‫ )م‬for gold; (‫ )ه‬for aromatic plants; (‫ )ى‬for glass; (ً) for weaponry or fish; (ُ) for months, hair (or feeling) or chess; (ٓ) for dyes, brass, gum or wool; (ٗ) for light or region; (ٛ) for birds; (ٟ) for dark or deer; (٣) for perfume, eyes (or springs), or tooling; (٧) for sheep; (٫) for fruits; (‫ )ٯ‬for village or reed; (‫ )ٳ‬for books or planets; (‫ )ٷ‬for milk; (‫ )ٻ‬for towns; (‫ )ٿ‬for stars or copper; (‫ )ڇ‬for wild animals, currency (coin), or paper; ( ‫ )څـ‬for vermin, pests, etc.; (‫ )ال‬for pair of scissors or a sum of (‫ )ٷ‬and (‫)ا‬ (which is still better); and (‫ )ي‬for jewellery.

(*)

It is so in the original, but there is an error in branching the two letters (‫ )ا‬and (‫)ك‬.

76

This relates to what has been denominated "relationship and diffusion", where a genus or species is representative of a letter. From this emerge thirty-two cipher alphabets, the first of which is unrestricted - or uncommitted -, the second is restricted - or committed - to the letter (‫)ا‬, the third to the letter (‫)ة‬, and so on till the end of the alphabet. The encipherer may opt for making the first cipher alphabet restricted to the letter (‫)ا‬, the second cipher alphabet to the letter (‫)ة‬, and so on, changing the restriction either according to the numerical alphabet (‫ )أثغل‬order or according to the alphabetical order (‫ا‬, ‫ة‬, ‫د‬, ‫)س‬. Examples: * of encipherment uncommitted to a certain cipher alphabet:

= * of encipherment committed to the letter ( ):

= * of encipherment committed to the letter ( ):

= * of encipherment committed to the letter ( ):

= * of encipherment committed to the letter ( ):

= * of encipherment committed to the letter (

= and so forth.

78

):

Some generic names may contain a few letters which are difficult to encipher. This requires the encipherer to be conversant enough with language, to cope well and make the intended choice. Encipherment committed to the numerical alphabet may be exemplified thus: ‫ٺى اٹلڇاة = ٽؾپل‬٥ ‫ إهثٸ ثُوّڅب ٌُؾپٸ ٽڀ اٹغيٌوح‬, while encipherment committed to the alphabet can produce something like this: ‫ٺى اٹضٍواٿ = ٽؾپل‬٥ ‫ إهثٸ ثُوّڅب ٌُؾپٸ ٽڀ رجوٌي‬.  Building on encipherment by substitution using generic names, the cryptographer can express further purport quite different from the apparent meaning, such as through commitment to initial, second, middle, or terminal letters of words, as already mentioned. He can apply the restriction to all the words of the cryptogram or only to the words of genera intended in it. Example: ‫ څبٹزڄ ثلهّح رجچو‬ٍٜ‫ ڇ‬٤ٕ‫چو ٽغڂً اٹپو‬١ ‫ڈٽبً ٭ٖلٽڄ‬ٞ‫هأٌذ ثؾٺت أٍلاً ؽ‬, of which ( َ‫ ٽب ڇڅجذ‬٠‫ )هَةِ اؽٮ‬is formed by taking the first letter of each word. [8. A return to the type on the utilization of cipher devices]  Some substitute for letters coloured beads threaded on a string as a rosary. One way for so enciphering is to devote white beads to serve as spaces between letters. The letter (‫ )ا‬is represented by a yellow bead, the letter (‫ )ة‬by a blue one, (‫ )ط‬by a red, (‫ )ك‬a green, ( ‫ )څـ‬a dark blue, (‫ )ڇ‬a black. Then beads are added in the same order of colours by twos, i.e. two yellows to represent the letter (‫)ى‬, two blues to represent (‫)ػ‬, etc. until you come to (‫)ٷ‬ with two balcks. Now you carry on with beads added by threes in the same order, viz. 3 yellows to represent (‫)ٻ‬, 3 blues to stand for (‫)ٿ‬, until you add 3 blacks to mark the letter (ٓ). Next, 4 yellows are made to represent (‫)ٯ‬, until 4 blacks are provided for (‫)ؿ‬. Then 5 yellows designate (‫ )م‬in this way until all the letters are exhausted. This type gives rise to scores of ramifications.

80

It is even better to have the thread made of silk, and assign a certain colour for each of the 28 letters represented by beads. The beads are shuffled and recognized by their colours.  Another device is based on writing the message on a folded paper in such a way as to place part of a word on one edge of the fold and the other part on the opposite edge, and so on till the end of the message. The paper then is unfolded, thereby concealing the writing, which looks like cipher. Once the paper is folded, the message will come out. This device, however, is not really an encipherment; therefore we say that such matters need sound common sense lest the decryptor should deviate from the right solution. [9. On using invented symbols or signs to represent letters]  In this type the encipherer devises symbols of his own, serving as substitutes for the letters of the plaintext. We shall give two examples on cryptanalysing this type of encipherment [at the end of the book]. The technique is to write beneath each letter of the alphabet a distinctive symbol representing that letter. The symbols are unmistakably substituted for the letters. Spaces between the words of the cryptogram can be filled up with hyphens, dots, blanks, circles, or, in the interest of complicating the cipher, with symbols similar to those devised for letters. This would add to the beginner's difficulty in cryptanalysis. Furthermore, extra symbols, namely nulls may also be added in order to make cryptanalysing yet more intricate.

82

Most of the earlier cryptographers represented a geminated letter by doubling it, -unlike later encipherers, who expressed gemination by one letter only-. Cryptanalysing the afore-stated and all kindred ciphers needs a 'genial' introduction that serves as a guide.

[Morphological introduction] The shortest length of an Arabic word is one letter, such as: ,

, ,

,

. These are verbs in the imperative, the past of which pertain to

the so-called "al-laf f al-mafrouq"; i.e. (‫ڇأډ‬, ‫ڇ٭ى‬, ‫ڇٱى‬, ‫ڇكډ‬, and ‫ى‬٥‫)ڇ‬ respectively. Some Arabic words consist of two letters, e.g. (of verbs): (

) and (

); (of articles): ( ‫ٽڀ‬, ً‫٭‬, ‫هة‬, ‫څٸ‬, ‫ثٸ‬,) etc.; (of indeclinable

nouns): (‫مي‬, ‫ما‬,

ْ‫ٽَڀ‬,

‫( ;)ٵټ‬of combinations of pronouns and

prepositions): ( ‫ثٴ‬, ‫)ٹڄ‬. Some other words (articles, verbs and nouns) are made up of three, four and five letters.

84

There are ten affixing letters; namely, ( ‫څـ‬, ‫ڇ‬, ‫ي‬, ‫د‬, ‫ا‬, ‫ٷ‬, ً, ‫ٻ‬, ‫ أ‬and ‫)ٿ‬, rounded up skillfully four times in the following line of verse by Sheik am l id-D n ibn M lik: In addition to three more letters; i.e., ٫, prepositional ‫ة‬, and ‫( ٳ‬of comparison as well as that used in addressing the second person). A word can be made longer by adding from these affixing letters. The possible length of a word so formed is fourteen letters at the very most. By "word" I do not mean the definition of grammarians, to whom even a pronoun is a word, but I mean the definition of writers and according to the way they counted the words of the Koran. For instance, you say addressing two owners of orchards or gardens: ( ) or ( ). These, if ciphered in terms of the afore-mentioned methods of encipherment, will produce such a number of letters amounting to 37 for the one, and 38 for the other respectively. Note that Arabic has no four- or five-letter root words devoid of at least one of the "liquid letters" - al-hur f a - alaqiyya -; i.e. the letters: ( ‫ٷ‬, ‫ ٿ‬and ‫ ;)ه‬and the labial letters like: ( ٫, ‫ ٻ‬and ‫)ة‬, with a few exceptions, e.g. ‫َغل‬٥ (= gold). The maximum length of an Arabic noun prior to affixation is five letters (with exceptions such as: ‫ڂلٹٍت‬٥); and the maximum length of a verb before affixation is four letters. Note as well that there is no word in the Holy Koran with a five-letter root except for those proper names of non-Arabic origins, as (‫)إثواڅٍټ‬.

86

The same letter can be repeated in one word no more than five ( , the first ‫ٳ‬

consecutive times. In the example, in the word (

) is for comparison, the last for addressing the

second person singular, and (

) is the plural of (

), which is a

vessel or a large boat. Other examples in point are: (

) (singular):

(

) (plural); (

) (singular): (

) (plural). The following words

contain the letter ( ) repeated four times in each: ( and

,

).

In several words, a letter can be successively repeated as many times as nine at most. Consider this verse:

In which the first (‫( )كك‬dad) means play or frivolity, the second is a name of a particular place, and the last is a proper noun of person in the vocative case. Note also that there are letters that are noncombinable with each other in anterior nor in posterior position, some others are combinable in either anterior or posterior position only. Those letters which are noncombinable at all are: ‫ س‬: does not combine with any of these letters: (‫م‬, ‫ى‬, ً, ٓ, and ٗ).

88

‫ط‬: does not combine with (ٛ, ٟ, ٧, ‫ ٯ‬or ‫)ٳ‬. The word ( ‫غڀ‬ٝ) is not Arabic (it is Nabatean), nor all the following words - most of

,

them being Arabized Persian -: (

, word (

,

,

,

,

,

,

,

,

,

, ). The

) is also Arabized Persian meaning partridge (a bird).

‫ ك‬: does not combine with ٟ. ‫ م‬: does not combine with (‫ى‬, ٓ, ٗ, ٛ, ٟ). The word ‫جوىم‬ٝ (= sugar) is also Arabized Persian, with two more possibilities in common parlance; i.e. (‫جوىٷ‬ٝ) and (‫جوىٿ‬ٝ). ‫ ى‬: does not combine with (ً, ٓ, ٗ, ٛ, ٟ). The word ( ‫وى‬ٝ) is Persian, the word ( ّٛ‫ )اٹ ّي‬is Nabatean, and the word ( ‫ )ٍي‬is the imperative form of the Persian verb ( ‫)ٍبى‬. Also the word ( ‫ )ٍي‬is Turkish meaning : you [plural].

90

ً : does not combine with (ٓ, ٗ, ٟ). ٓ : does not combine with (ٗ, ٟ). ٗ : does not combine with (ُ, ٟ). ٛ : does not combine with (ٟ). ‫ ٯ‬: does not combine with (٧) nor with ‫ ٳ‬in a word root, with the exception of (َٰ٪َ َ‫( )ځ‬of a crow, to caw; of a she-camel, to ‫پذ‬٪‫)ث‬. ‫ ٳ‬: does not combine with (‫ )ؿ‬in a word root. ‫ ٻ‬: does not combine with (‫ة‬, ٫) in a word root except in the word (‫)٭ټ‬. The word (‫=( )ثَټ‬the lowest string of the lute) is non-Arabic. The guttural letters - ‫ء‬, ‫څـ‬, ٣, ‫ػ‬, ٧, ‫ ؿ‬- are noncombinable with one another save the letter ( ‫)څـ‬, which follows other gutturals in a word at end position as an affixing letter denoting feminine form or a pronoun. It also goes after letter (٣) as a basic letter as in: ( ‫چل‬٥, ‫چڀ‬٥). Over and above this, no two gutturals are possible in one root word. However, the letter ( ‫ )څـ‬may occur after another gutturals, but with a third letter separating the two gutturals, such as in: ( ‫ٍچت‬٩, ‫جچو‬٥). The word ( ‫)ؽٍّچٸ‬ are compounds.

92

Accordingly, no two of these five gutturals ( ‫څـ‬, ‫ػ‬, ٣, ٧, ‫ )ؿ‬occur together anteriorly in a word besides that mentioned above; nor do they occur in the middle of a word, except the letter ( ‫ )څـ‬with the letter (٣) as in: ( ٤‫ ;)څٺ‬the letter ( ‫ )څـ‬with (٧) as in ( ٨ٍ‫ ;)أڅ‬and ( ‫ )څـ‬with (‫)ؿ‬ which produces only the word (

).

The basic ( ‫ )څـ‬does not combine at all with (‫)ػ‬, whereas ‫ ؿ‬does

,

combine with (٣); e.g. (

,

). The letter (‫ )ػ‬is not

combinable with (‫)ؿ‬, nor with (٣) except in a compound word such as:

,

(

).

The repetition of the same letter in one word is in common use.

,

Examples: (

,

,

,

,

,

, ,

,

,

,

double letters. Other examples are: (

,

,

). These belong under ,

,

,

) and the like -

geminated letters -. The repeated letter may be such an inherent part of the word, e.g. (

,

,

,

,

94

,

).

Letters combinable in anterior position only or in posterior position only: [The letter] ‫ س‬is noncombinable with (ُ) in anterior position. [The letter] ‫ ك‬is noncombinable with (‫ى‬, ٓ, ٛ) in anterior position. That is why [the letter] ‫ ى‬in the word ‫[ ٽچڂلى‬engineer] has been replaced by (ً) when Arabized; so we say: (ً‫ )ٽچڂل‬and (‫)ٽچڂلٍخ‬. [The letter] ‫م‬

is also noncombinable in anterior position with

(‫ط‬, ً, ُ, ٣). The word ( ‫ )اٹٮبٹڈمط‬is Persian; when Arabized it became (‫)٭بٹڈمٯ‬. Some people pronounce the word (‫ )اٷڅڈمط‬wrongly so; the right form should be ( ‫ )اٷڅڈكط‬with (‫ )ك‬rather than (‫)م‬. The same applies to the word )‫)ٍبمط‬. The word (‫ )اٹَڂجبمط‬is also Persian. However, [the letter] ‫ م‬can combine anteriorly only with the letter (‫ )ط‬providing that they are separated by one or two letters in between; e.g. (‫ثبمهڇط‬, ‫ثبمڅڂظ‬, ‫ثبمهځغجڈٌڄ‬, ‫اٍٮٍناط‬, ‫)ثبمځغبٿ‬. [The letter] ُ is not combinable in posterior position with any of these letters: (‫ى‬, ً, ٓ). [The letter] ٛ is noncombinable in anterior position with [the letter] ‫ ٳ‬in a root word. You ought to know the letters that do not occur initially in a word, such as the letter (‫ )ط‬which does not go initially before (‫د‬, ٓ, ٗ, ٧). However, the word: ( ّٔ‫ )اٹغ‬is Arabized. The word ( ‫)اٹٖڂغخ‬, where the third letter (‫ )ٿ‬separates (ٓ) and (‫)ط‬, is debated among scholars as to whether it is originally Arabic or Arabized; the fact is that it is Arabized.

96

You should also know the letters that are seldom combinable with each other, such as ( ) after ( ) in ( (

,

), the letter ( ) before ( ) in

), the letter ( ) with ( ) as in: (

the letter ( ) as in: (

,

), the letter ( ) with

), the letter ( ) before ( ) as in: ( )

(imperative), the letter ( ) after the inherent ( ) as in: (

,

).

Finally you should know that the letters that can be repeated at the beginning of words are necessarily any of the following ten letters: (‫ٳ‬, ‫ٷ‬, ‫ٻ‬, ‫ٿ‬, ‫د‬, ‫ا‬, ‫ة‬, ‫ڇ‬, ٫ and ‫)ي‬, of which the least frequently repeated is )‫)ي‬. [Algorithm for cryptanalysis] Setting out to cryptanalyse a cryptogram, you begin first of all by counting the symbols in it, and then count the times of recurrence of each symbol and set down the totals individually. In case the cryptographer has elaborated his encipherment, i.e. by concealing the space within letters, you engage in working out the space first. The algorithm is to take a letter and assume the next letter to be the space, according to the rules I have already decided for you, and having regard for the possible combinations of letters of which the words may be composed. You keep trying, turning your assumption to the next letter after the second one and the next after that, and so on, until all spaces are detected and the words of the cryptogram appropriately separated.

98

Subsequently you match the symbol or letter frequency of occurrence in the cryptogram against the pattern of letter frequency previously mentioned. Bear in mind the following frequency order of Arabic letters (in descending order): (‫ا‬, ‫ٷ‬, ‫ٻ‬, ‫ي‬, ‫ڇ‬, ‫ٿ‬, ‫څـ‬, ‫ه‬, ً, ‫ة‬, ‫ٳ‬, ‫د‬, ٣, ٫, ‫ٯ‬, ‫ك‬, ‫م‬, ‫ال‬, ‫ػ‬, ‫ط‬, ٓ, ‫ؿ‬, ُ, ٗ, ‫ى‬, ‫س‬, ٛ, ٧ and ٟ). This is the letter order of frequency in the Holy Koran, although this order may differ in other language usages. Some deliberately encipher poetry and prose, dispensing with the letter (‫)ا‬, or without letter-dotting or without idle particles [those that do not affect the parsing of what follows]. The normal order of letter frequency may particularly be different when the cipher has few letters, i.e. when it is too short to cover a whole rotation of the letter order, and consequently not allowing for proper matching. Hence the prime significance of the message being long enough. On cryptanalysing a cipher, the most frequently occurring letter is considered to be the letter (‫ ;)ا‬the next most frequent letter should in all likelihood be (‫)ٷ‬, and what should lend credit to your conjecture is the fact that in a majority of contexts, (‫ )ٷ‬follows (‫)ا‬. - to form the definite article -. You then look into the cipher to see if it contains a letter of single occurrence throughout. That you would think likely to be (‫)ال‬, on account of the scarce occurrence of the imperative single letters mentioned earlier. Then the first words you try to work out in the message are the bigrams - two-character words - through somehow trying to have access to the most feasible combinations of their letters, until you are sure you have discovered something correct in them. You then examine their forms and write down the equivalents by them - whenever they occur in the message -. You apply the same principle to trigrams - three-character words - until you are sure you have got something, and write out the equivalents - all through the message -. Tackle tetragrams and pentagrams likewise. Whenever in doubt, posit two or three or more probable conjectures and write each one down. Wait until one of the conjectures proves to be the good one from cryptanalysing other parts of the cipher. Once this is done, you proceed as such in the rest of the cipher.

100

Remember that the letter preceding the definite article (‫ )اٷ‬at the beginning of a word may, all often, be one of these letters: (‫ة‬, ٫, ‫ ٳ‬or ‫)ڇ‬. A starter cryptanalyst should have each word of the cipher written separately - to set him going -, and poetry should be written to him in such a way as to enable him by the aid of metre to solve some letters like: ( ‫( )څـ‬denoting feminine form), (‫( )د‬also denoting femininity), (‫)ي‬ (indicating the first person singular), vowels, and the like. [Example 1] As an example, let us consider the following lines written in cipher [by substitution using devised symbols for letters]:

For cryptanalysis, we start first of all on counting the frequency of occurrence of each symbol right through the cryptogram, affixing the frequency numbers to symbols as follows:

102

We notice the symbol ( ) has far and away higher frequency of occurrence than all others. We conclude, therefore, that it must be the letter (‫)ا‬, and mark it so on the cryptogram. The next frequently-occurring symbol is found to be ( ), and so we settle our choice on the letter (‫)ٷ‬, supporting our belief by its occurrence immediately after (‫ )ا‬in seven places of the text. We then trace a single symbol representing a word, which we assume to be (‫)ال‬. We also notice that the third word is a bigram, the other letter of which is ‫ال‬. That means it might be one of these possibilities: ( ‫ثال‬, ‫رال‬, ‫عال‬, ‫ؽال‬, ‫فال‬, ‫ٍال‬, ‫ال‬٥, ‫ال‬٩, ‫٭ال‬, ‫ٵال‬, ‫څال‬, ‫)ڇال‬. But yet we realize that the first symbol of that bigram, i.e.

, is

repeated elsewhere as initial letter of a word where it is supposed not to admit being any of these letters: (‫ط‬, ‫ػ‬, ‫ؿ‬, ً, ٣, ٧ or ‫)څـ‬. Thus we rule out the possibilities: ( ‫عال‬, ‫ؽال‬, ‫فال‬, ‫ٍال‬, ‫ال‬٥, ‫ال‬٩ and ‫ )څال‬and retain the rest, namely: (‫ثال‬, ‫رال‬, ‫٭ال‬, ‫ ٵال‬and ‫)ڇال‬. Then we notice that the fifth word is a bigram, too, with its second letter being (‫)ا‬. So it might be one of the following words: ( ‫ثب‬, ‫عب‬, ‫كا‬, ‫ما‬, ‫ٍب‬, ‫ّب‬, ‫ب‬ٙ, ‫٭ب‬, ‫ٽب‬, ‫ ځب‬or ‫)ٌب‬. Seeing that the frequency of the symbol ( ) is higher than that of all other symbols, we conclude that it would probably be one of these letters: (‫ٻ‬, ‫ ي‬or ‫)ٿ‬. Because ( ‫ )ځب‬is less common in use, the letter (‫ )ٿ‬is excluded as unlikely. Therefore the bigram has to be either ( ‫ )ٽب‬or ( ‫)ٌب‬. We realize, too, that that symbol ( ) succeed the symbol associated with (‫)ال‬, which we believe is (‫ة‬, ‫د‬, ٫, ‫ ٳ‬or ‫ )ڇ‬in the trigram (

). We match the letters with the

letter (‫ )ٻ‬and get the word ( ‫ )رزټ‬only; then try them again with the letter (‫ )ي‬and the word )ً‫ )٭ٮ‬comes out.

104

Subsequently, we find the symbol ( ) repeated no more than four times throughout. That gives us a strong impression that it is the letter (٫), and not (‫)ي‬, owing to the high frequency expected of the latter in cryptograms such as this length. Thus we fix on the third word being ( ‫)٭ال‬, the fifth ( ‫)ٌب‬, the fifteenth ( ً‫)٭ٮ‬, and the single letter (‫)ال‬. Our guess is further promoted by the repetition, in the eleventh word after (‫)اٷ‬, of two letters followed by (‫ )ا‬and another letter. In this case no letter but (‫ )ٻ‬could possibly be repeated when checked against the letters, so we say: (‫اٹپپبد‬, ‫اٹپپبػ‬, ‫اٹپپبه‬, ً‫اٹپپب‬, ٣‫)اٹپپب‬. We also observe that the frequency of the symbol ( ) comes next to that of (‫ا‬, ‫ ٷ‬and ‫)ي‬, so we conclude it should be any of these possibilities: (‫ه‬, ً, ‫ د‬or ٣), considering that the letter (‫ )ٻ‬is already made out, and that it cannot be (‫ )ٿ‬either. Thus we mark the symbol for (‫ )ٻ‬in its locations.(*) The symbol ( ) is found to be the initial letter of the fourth trigram of which the middle and terminal letters have already been cryptanalysed as (‫ )ٷ‬and (‫ )ٻ‬respectively. On identifying it with the above letters, the letter (‫ )ه‬is eliminated, and the word is quite sure to be one of these: (‫ٍٺټ‬, ‫ رٺټ‬or ‫ٺټ‬٥). In the word next to (‫اٹپپبد‬, ٣‫اٹپپب‬, ً‫ )اٹپپب‬there is a letter prior to (‫)اٷ‬ which might be (‫ة‬, ‫ ٷ‬or ‫)ڇ‬, the letter (٫) having already been designated.

(*)

There may be an omission or distortion in scribing this paragraph.

106

We also notice that the symbol (

) succeeds (‫ )اٷ‬just before (‫)ي‬. It

is found between two ( )'s in a three-character word that might be ( ‫اثب‬, ‫اما‬, ‫ اٍب‬or ‫)اځب‬. We match the word against the letters (‫ة‬, ‫م‬, ً and ‫ )ٿ‬on condition that (ً) be the terminal letter, but no intelligible vocable is obtained; thus the word (‫ )ٍٺټ‬is dropped as irrelevant. Then we try the word again as before, provided that (٣) this time be the terminal letter, from which get out, after the initial letter, the word ( ٣‫)اٹجٍب‬. We try once more with the letter (‫ )د‬and derive these words: ( ‫اٹجٍبد‬, ‫ اٹڂٍبد‬and ‫)اٹٍَبد‬. Hence the letter (‫ )م‬is excluded and the words: (‫اثب‬, ‫ اٍب‬and ‫ )اځب‬are retained. Looking into the seventh word, a trigram of which the initial letter is (‫)ٷ‬, the middle is this symbol (

), and the terminal is ( ) which

can be either (٣) or (‫)د‬, we derive the word ( ‫)ٹَذ‬, dropping the letters (‫ )ة‬and (‫)ٿ‬. With the (‫ )ة‬dropped, the (٣) is also dropped from the word ( ٣‫ )اٹجٍب‬ipso facto. That is why the word ( ٤َ‫ )ٹ‬is eliminated. It follows that the words: ( ‫ )اٹٍَئبد‬and its counterpart ( ‫)اٹپپبد‬, and also the trigram ( ‫ )رٺټ‬are now hit right, and that the word ( ‫ٺټ‬٥) turns out irrelevant. So we mark the letters (‫ )د‬and (ً) in their locations, thus forming the trigram ( ‫)اٍب‬. As yet the following words of the cryptogram have been made out: ( ً‫)٭ال رٺټ ٌب ٹَذ اٹپپبد ال اٍب ٭ٮ‬, with the letter just preceding (‫ )اٹٍَئبد‬still obscure.

108

We examine the tenth [word], also a trigram with the two letters (‫ )د‬and (‫ )ي‬already clear. We check it against the letters, and come away with the word ( ‫ )ؽزى‬only, marking the letter (‫ )ػ‬all through the cipher. We switch over to a five-character word, with all its characters already laid open except the middle letter; thereby we put forward these possibilities: ( ‫ؽَواد‬, ‫ؽَٶبد‬, ‫)ؽَڂبد‬. Seeing that the symbol ( ) occur at higher frequency than all letters other than (‫ا‬, ‫ٷ‬, ‫ ي‬and ‫)د‬, we fix upon the word (‫)ؽَڂبد‬, the letter (‫ )ٻ‬having already turned out well. We indicate the (‫ )ٿ‬in its positions. Then we consider the symbol ( ) as the initial letter in two threecharacter words; of the one we have already known the letters (‫ ٿ‬and ‫)ي‬, and of other the letters (‫ ٷ‬and ‫)ي‬. We attempt the letter and find that it could probably be either (٣ or ‫)ڇ‬, and so conclude these possibilities: (ً‫ڂ‬٥, ً‫ڇځ‬, ً‫ٺ‬٥, ً‫)ڇٹ‬. But our choice is settled on the letter (٣) because the frequency of this letter does not rise up to the order of (‫(ڇ‬. Our attention is also drawn to a heptagram (seven-character word) of which just one letter is still covert. Upon experimentation with letters, no other word than (‫ )اٹجٍزبٿ‬arises. [The letter ‫ ]ة‬is represented by this symbol ( ) prior to the word ‫اٹٍَئبد‬, and so it is labelled in its due positions. We try likewise at a hexagram (six-character word), the third letter of which is yet concealed. The word (‫ )اٹٶزبة‬shows up.

110

We now consider the five-character word preceding the current word by two. It has the middle letter still obscure. We experiment with the letters and get the words: ( ٬‫ٹپغڂ‬, ٬‫ ٹپلځ‬and ٬‫)ٹپٖڂ‬. Of these the context renders necessary the choice of ( ٬‫ )ٹپٖڂ‬as it is the fittest of all three to chime in with the word (‫)اٹٶزبة‬. So we designate the letter (ٓ). The same procedure is repeated with the last word, the fourth letter of which is yet close. In consequence of experimentation with letters, the word ( ً‫ )اٹپڈٕٺ‬turns out clear, and this gives rise to uncovering the word (‫ )أٍٺڈ‬following (‫)ٹَذ‬. Thus we label the letter (‫)ڇ‬. We now look at the first word, a bigram starting with the letter )ٓ), the manipulation of which we have purposely delayed on account of the infrequency of its letters. Identification unveils the word ( ّ‫)ٕل‬. Labelling the letter (‫ )ك‬in its locations, we run across another bigram ending with the letter (‫)ك‬, which we match against the rest of letters unrevealed as yet. These words get out: (‫عل‬, [‫]فل‬, ‫ ٱل‬and ‫)څل‬. Then we take up a trigram, the middle letter of which is represented by the symbol ( ), the initial and terminal letters being (‫ )د‬and (‫)ٷ‬ respectively. Consequent upon matching the word against the four letters: (‫ط‬, ‫ؿ‬, ‫ ٯ‬and ‫)څـ‬, the letter ( ‫ )څـ‬is dropped as unfeasible, and these words: (‫رغٸ‬, ‫ رقٸ‬and ‫ )رٲٸ‬remain as possibilities. It is brought home to us from the context that the word before ( ‫)أٍب‬ is ( ), and the trigram ( ); thus the statement goes: (‫)ال رٲٸ ٱل أٍب‬. We seek the sixth word, with all its component letters already unfolded except the second. On experimentation with the rest of letters, we come away with the word ( ً‫نڇٹ‬٥). The letter (‫ )م‬is thus indicated in its positions on the cryptogram.

112

We examine the trigram placed between the words: ( ٬‫ )ٹپٖڂ‬and ( ‫)اٹٶزبة‬. This trigram starts with the symbol ( ) followed by (‫)ما‬. The word is decided to be (‫)څنا‬, and the letter (‫ )څـ‬is pointed out throughout. The pentagram in between ( ً‫ )٭ٮ‬and ( ‫ )ٽڂڄ‬is as much treated to unfold the fourth letter yet covert. The result is the word (‫)اٹڈعڄ‬. The second to last word, a heptagram, is likewise tackled to unveil the fourth and only remaining letter in the cipher. The word ( ‫)اٹلهٌچټ‬ flows. Cryptanalysis thereupon is done, and the plaintext is now developed full and for good:

- Do not you blame, oh, my admonisher; Never in all my life shall I forget his love. Say not he wronged me; for each flaw in him He had myriads of virtues. This verse is by the author of this treatise Al ibn ad-Durayhim alMaw il .It is after this pattern that cryptanalysis is carried out. Notice, in this example, how no more than 21 letters of the 28-letter alphabet were used to make the cryptogram. A look at the order of letter frequency in the Glorious Koran demonstrates that the 8 remaining letters in fact come last in the order of letters. It so happened that they did not mix up with one another in anterior nor posterior position. But that was only a fortuity, because a letter may occur near to its normal order, as has previously been stated. Hence, in the above cryptogram, the letters: (‫ي‬, ‫ د‬and ‫ )څـ‬occurred at higher frequency than the letter (‫)ٻ‬. The fact remains that it is really essential to gain acquaintance with the approximate order of letter frequency by matching against the words and developing from context.

114

[Example 2] let us give another example to further illustrate algorithms towards cryptanalysis:

We start straight away to count the frequency of occurrence of symbols, and affix the frequency number to each, as in example 1 above, thus:

116

On scrutinizing the symbols we learn that the highest frequency is for the symbol , then the following, in descending order: , , , and (equally), and (equally), , and and (equally). That order makes an impression that the symbol represents the letter (‫ )ا‬and the symbol the letter [‫]ٷ‬, being higher in frequency than all others. Our notion is based on the fact that (‫ )ٷ‬all often goes after (‫)ا‬ - to form the definite article -, not so here. The situation gives evidence to the contrary. We so realize that the symbol is (‫ )ا‬and this is (‫)ٷ‬, marking them both in their positions on the cryptogram. The second word holds our attention on the spot: a trigram with the letter (‫ )ٷ‬repeated as initial and middle letters. We match the word against the letters and find the letter ( ‫ )څـ‬only, which we label throughout as the terminal letter, and thus come away with the word (‫)هلل‬. We then search the fifth word, a pentagram with the fourth letter still unknown. In consequence of experimentation the following possibilities turn up: ( ‫اٹچجب‬, ‫اٹچغب‬, ‫ اٹچپب‬and ‫)اٹچڂب‬. We note that the letter we are seeking have the highest frequency of all letters next to (‫ )ا‬and (‫ ;)ٷ‬so we attach credence to the letter (‫)ٻ‬, leaving room for (‫)ٿ‬, too, and dropping the letters (‫ )ة‬and (‫)ط‬. We resolve on (‫ )ٻ‬as the choice in the light of its occurrence before (‫ )ا‬in two bigrams. Thus the letter (‫)ٻ‬ is designated throughout. We also observe that (‫ )ٻ‬is followed by a letter that could, together with ‫ٻ‬, be one of these bigrams: ( ‫ٽل‬, ‫ٽن‬, ‫ٽو‬, ٌ‫ٽ‬, ٔ‫ٽ‬, ٜ‫ٽ‬, ٤‫ ٽ‬and ‫)ٽڀ‬. Seeing that that letter occurs at high frequency, with that bigram repeated three times in the cipher, we hold the bigram to be ( ‫)ٽڀ‬, and rule out the others as irrelevant possibilities, labelling the (‫ )ٿ‬in its locations.

118

Next we consider this frequent symbol

before the definite article

at the beginning of words. According to the letter order of occurrence we fix upon the letter (‫)ڇ‬. We survey the last word of the cryptogram, the terminal letter of which is its fourth letter unknown as yet. Giving it a trial, we obtain the words: (‫ڇاٹجچټ‬, ‫ڇاٹزچټ‬, ‫ڇاٹغچټ‬, ‫ڇاٹلڅټ‬, ‫ڇاٹَچټ‬, ‫ڇاٹْچټ‬, ‫ ڇاٹٮچټ‬and ‫)ڇاٹٍچټ‬. The symbol

in this last word - of the cryptogram - occurs in a

bigram before another letter of which frequency of occurrence comes next to (‫)ا‬, (‫ )ٷ‬and (‫ ;)ٻ‬hence we say it might be (‫)ي‬. Our belief is enhanced through another word, with this very symbol still covert. The word (‫ )اٹڂچى‬comes out, and thereby the preceding word ( ً‫)أڇٹ‬, too. Deciding on (‫)ي‬, we match the symbol against it and come away with (ً‫ )ث‬and (ً‫)٭‬. We move on to a pentagram, with this symbol

standing for the

fourth letter, followed by another letter which we experiment with )‫)ة‬ and (٫), and elicit these words: ( ‫اٹٺجش‬, ‫اٹٺجل‬, ٌ‫اٹٺج‬, ٜ‫اٹٺج‬, ‫اٹٺجٴ‬, ‫اٹٺٮذ‬, ‫اٹٺٮظ‬, ‫اٹٺٮؼ‬, ٠‫ اٹٺٮ‬and ٰ‫)اٹٺٮ‬. Next we find this symbol ( ) representing initial letter in a word, followed by a double (‫)ٷ‬. On trail, these words develop: ( ‫كٹّٺڄ‬, ‫ٵٺٺڄ‬, ‫ رٺٺڄ‬, ‫عٺٺڄ‬, ‫ؽٺٺڄ‬, ‫ٺٺڄ‬١ and ‫)ٱٺٺڄ‬. The letters (‫ )س‬and (ٛ) are dismissed.

120

The following word is a hexagram, with all its component letters already unfolded except the third. We give it a trial and develop the words: ( ‫اٹزپبٻ‬, ‫اٹؾپبٻ‬, ‫اٹنٽبٻ‬, ‫اٹْپبٻ‬, ‫پبٻ‬٪‫ اٹ‬and ‫)اٹٶپبٻ‬. From the context it is quite easy to educe ( ‫پبٻ‬٪‫ٺّٺڄ اٹ‬١), and, as a corollary, we determine on the pentagram (

) and the other word ( ‫)ڇاٹٮچټ‬, besides the bigram

(ً‫)٭‬. We score the letters: (٫, ٟ and ٧). The third word, a trigram with the middle letter being (‫ )ٷ‬and the terminal (‫)ي‬, is followed by ( ‫)ٽبأٹچپب‬. Guided by the context, we hold the trigram to be (‫ٺى‬٥), and so mark the letter (٣) in the cryptogram. We now shift to the tetragram succeeding the word ( ‫)ڇآٹڄ‬, of which the third letter only is yet to be disclosed. Experimentation demonstrates ( ‫غڀ‬٦‫ ٽ‬and ‫لٿ‬٦‫)ٽ‬. We settle our choice on ( ‫لٿ‬٦‫)ٽ‬, and the bigram that follows (prior to ‫ٺټ‬٥) shows itself to be ( ‫)ٵٸ‬. We thus label the letter (‫ )ك‬in its positions. Reverting to the first word of the cipher, of which the middle letter is yet to break, we draw up: ( ‫اٹضپل‬, ‫اٹغپل‬, ‫ اٹؾپل‬and ‫)اٹٖپل‬. The phrase: ( ‫ٺى ٽب أٹچپب‬٥ ‫ )هلل‬that follows immediately leads us to settle on the word (‫)اٹؾپل‬, and we point out the letter (‫)ػ‬. Then we attend to the third tetragram situated in between ( ‫ٺى‬٥) and (‫ٺّٺڄ‬١). On experimentation, the word (‫ )اٹني‬transpires. Likewise we treat the pentagram next to the word ( ‫)ٽؾپل‬, of which the fourth letter remaines covert as yet. Trying, we get the word ( ً‫)اٹڂج‬, and designate the letter (‫)ة‬.

122

Still unsolved is the third letter, represented by the character ( ), of the hexagram next to the word ( ْ‫)ٽِڀ‬. It is also the third letter of a tetragram the initial letter of which is (‫)ا‬, the second (٫) and the terminal (‫)ػ‬. In yet another word the same character symbolizes the second letter of a pentagram starting with (‫)ڇ‬, ending with ( ‫ )څـ‬and having (‫ )ػ‬and (‫ )ة‬as the third and fourth letters. Consequently the letter (ٓ) is designated, and the first of the above-described words thus turns out to be ( ‫)اٹٖڈاة‬, the second ( ‫)أ٭ٖؼ‬, and the last ( ‫)ڇٕؾجڄ‬. The first word of the second line, a bigram, is thereby developed to be ( ‫)صټ‬, and the following word ( ‫ ;)ٕالح‬the letter (ً) of the word ( ‫)اٹَالٻ‬ shows up, too, so that the hemistich gets all clear and reads: ( ‫صټ ٕالح اهلل‬ ‫)ڇاٹَالٻ‬. It goes without saying that the more one gets and keeps one's hand in, the faster cryptanalysis is accomplished. It ensues that the fourth letter of the hexagram following ( ‫)أ٭ٖؼ ٽڀ‬ is the letter (ٗ) and that the word is (‫بك‬ٚ‫)ثبٹ‬. The context suggests that the word after ( ٠‫ )٭ً اٹٺٮ‬is ( ٰٞ‫)ځ‬, and so we mark the letter (‫)ٯ‬. This last-mentioned immediately gives rise to the appearance of its counterpart (ٰ‫ )فٺ‬in the first hemistich of the same line - i.e. the third -. The letter (‫ )ؿ‬is so designated as well, and therefore the word ( ‫)فٍو‬, preceding ( ٰ‫)ٽڀ فٺ‬, manifests itself, and thereupon manifesting the whole cleartext that flows:

124

- My Lord! By Thy grace I seek guidance; By Thine inspiration I learn and be favoured. Bestow Thy divine peace on him, oh, God: On the Prophet whom a canopy of clouds Shaded; who was endowed with eloquence; The gem of humankind of all ages, And all his folk of reason and intellect. That is quite enough for the sapient to be in the fair way to succeed. God is the One to be sought for help, and unto whom souls are committed. He is sufficient for us! Most Excellent is He in Whom we trust. May His bounteous blessing and peace be upon our Prophet Mu ammad and his magnanimous folk till the Day of Doom. Praise belongs to God, the Lord of the World. *

*

*

Handwritten by idq Mu af ibn li , on the day of Friday, 10th Ramadan of the year 1149 of the Hegira(*) of Prophet Mu ammad, blessing and peace on him and all his folk.

(*)

AD 1736.

49

‫بسن اهلل الشحوي الشحين‬

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(4) (5) (6)

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269 134 154 1

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(8) (9) (10) (11)

163

(12)

310 309

(13) (14)

271 5

91

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236 9

(2) (3) (4) (5)

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(6) (7)

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(3) (4) (5) (6)

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