Period Determination

There is a tendency for us to shy away from polyalphabetic cipher types which require a determination of its Period length, believing that complex mathematical formulas or a difficult factoring approach is necessary to arrive at the proper number of columns to satisfy Period determination.  I hope this chapter will persuade you that the pencil and paper solvers have no need to exclude these Period determination cipher types from their solving prowess.

There are two common methods applied in the determination of the periodicity of a periodic cipher type.

Index of Coincidence

William Friedman and his wife, Elizebeth (not a typographical error, the unusual spelling chosen by her mother to avoid the nickname of Eliza) , were American cryptanalysts who made substantial contributions to cryptology and played a significant role in the decryption of enemy ciphers during World War I and II.

The Index of Coincidence developed, by William, in 1920 is based on the likelihood that any pair of letters in a message are equal to each other. The Index of Coincidence for the twenty-six letters of the English alphabet as used in typical conversation or messaging is 0.0667. This can be arrived at by visualizing the assimilation of some hundreds of letters placed in a common container in the ratio of their frequency appearance in normal text or conversation. (See ACA and You Handbook, General Properties of Letters, Page 14). The calculation of drawing meaningful text dialogue from a random collection of these letters is 0.0667.

Computer analysis of polyalphabetic ciphertext can expeditiously determine the probability of proper Period determination by comparing the Index of Coincidence between possible Period lengths, discerning which length most closely corresponds to the 0.0667 average Index of Coincidence of common text. Needless to point out, this approach does not lend itself to pencil and paper computation. Let’s examine a more friendly paper and pencil approach to Period determination.

Kasiski Factoring System

Friedrich Kasiski was a Prussian Army Officer who rose from the ranks of an enlisted Private to that of Major in the 1860’s, serving as a cryptanalyst.

In 1863, Kasiski published the book, Die Geheimschriften und die Dechiffrirkunst (Secret Writing and the Art of Deciphering), that devoted most of its pages to the solution of periodic ciphers. It was in this book that he introduced the recognition of letter patterns as a way to break periodic ciphers. He cited the length of the interval between letter repetitions as the clue to the length of a cipher’s keyword or period length.

The Kasiski Factoring System is very user friendly to the pencil and paper solver and worthy of our study and attention. Those interested in the recreation of mathematics and/or the dynamics of letter frequencies upon the flow of the printed or linguistic word will take great delight in the examination of reoccurring digraph and trigraph patterns.

Factoring Process

  1. Identify Repetitions
  2. Indicate Positions
  3. Find Differences
  4. Factor the Differences
  5. Tabulate Factors

Kasiski Period Determination – Short Method – Identify Digraph & Trigraph Repetitions

Digraph and trigraph (two and three letter groupings) repetitions are identified, located, assessed and computed in the same manner. Locate the position in the cipher text of each repetition, subtract the first position location from the second, factor the difference and tabulate the factors. Factoring simply refers to the process of determining what whole numbers could have been multiplied to produce a specific product – in this case that the differences of repeated digraph/trigraph position numbers.

Let’s go through the process with the following cipher construction, a 2002 Period Five Gronsfeld. We will indicate the cipher’s letter positions above the ciphertext to facilitate computing the number of spaces which generate (factor) the differences.

 SO Cm, 2002 E-1 Gronsfeld.   Impaired Decision Making                LIFER

 2      8     13
FKIOK TYMSM XQIUY WHAXE DRNSI
   30       37&38       47        5 5
RLVEM SJGPY YKIUM YVBJQ JIWSE  SRBII WGZJR P.

Cipher Construction Positions

KI  2 37    MS 8 30
IU 13 38    IW 47 55

Find Differences

KI 37 - 2 = 35       MS 30 - 8 = 22
IU 38 - 13 = 25      IW 55 - 47 = 8

Factors

Difference 2 4 5 7 8 11
KI 35     5 7    
MS 22 2         11
IU 25     5(2)*      
IW 8 2 4     8  
Total 4 4 15 7 8 11

* Squared numbers carry a weight of two.

The factor columns represent the potential Period of the cipher. Each factor column entry is a whole divisor number for the difference entry. (KI-35 is divisible by 5 and 7.) The factor column with the highest total will indicate the proper (factor) Period for the cipher (5).

More Digraph Repetitions

JF Cm, 2002 E-4 Variant.     Face Cards,       RIG R. MORTIS

1  4   6              17                     33       40
ISHLU  VJFCW  FPPNW  UISYL  ERSRK  DVFKU   KOVJY      ZCYIP   VNDKC
                  59   61 (6   5)68           76 80      84
SIJVF  YLTJE   QNSMH   PKGTV   JGLUX   YQZFN     HTLRM   HWYHT   NMZZW
        97       104
NAJLV  VPVDA  JHRPK  SGSCZ.

Cipher Construction Positions:

IS  1 17   LU   4 68   VJ 6 33
VJ  6 65   VJ  33 65   PV 40 97
MH 59 80   PK  61 104  HT 76 84

Factors

Difference 2 3 4 7 8 9
IS 16 2   4(2)*   8  
LY 64 2   4   8(2)*  
VJ 27   3       9
VJ 59 Prime Number – No factors in range.
VJ 32 2   4   8  
PV 57 Prime Number – No factors in range.
MH 21   3   7    
PK 43 Prime Number – No factors in range.
HT 8 2   4   8  
Total 8 6 20 7 40 9

* Squared numbers carry a weight of two.

The factor column with the highest total (8 = 40) will indicate the proper Period.