We will now consider the class of substitution ciphers where a number of alphabets are used, the number and choice of alphabets depending on a key word or equivalent and being used periodically throughout the message.In this class belong the methods of Vigenere, Porta, Beaufort, St. Cyr, and many others. These methods date back several hundred years, but variations of them are constantly appearing as new ciphers. The Larrabee cipher, used for communication between government departments, is the Vigenere cipher of the 17th Century. The cipher disk method is practically the Vigenere cipher with reversed alphabets.In using these ciphers, there is provided a number of different cipher alphabets, usually twenty-six, and each cipher alphabet is identified by a different letter or number. A key word or phrase (or key number) is agreed upon by the correspondents. The message to be enciphered is written in lines containing a number of letters which is a multiple of the number of letters of the key. The key is written as the first line. Then each column under a letter of the key is enciphered by the cipher alphabet pertaining to that letter of the key. For example, let us take the message, “All radio messages must hereafter be put in cipher,” with the keyGrant, using the Vigenere cipher alphabets given below. Each of these alphabets isidentifiedby the first or left hand letter which representsAof the text. We thus willuse in turn the alphabets beginning withG, withR, withA, withN, and withT.GRANTGRANTALLRADIOMESSAGESMUSTHEREAFTERBEPUTINCIPHERUsing the alphabet indicated byG, we getGJYYNLKTKContinuing for the other alphabets, we getGCLETJZOZXYJATXYDUFMNVRRTLKEEUKGUGBTTICAKIThis method of arranging the message into lines and columns and then enciphering whole columns with each cipher alphabet is much shorter than the method of handling each letter of the message separately. The chance of error is also greatly reduced.All these cipher methods can be operated by means of squares containing the various alphabets, cipher disks or arrangements of fixed and sliding alphabets. For example, this was the original cipher of Vigenere:ABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZACDEFGHIJKLMNOPQRSTUVWXYZABDEFGHIJKLMNOPQRSTUVWXYZABCEFGHIJKLMNOPQRSTUVWXYZABCDFGHIJKLMNOPQRSTUVWXYZABCDEGHIJKLMNOPQRSTUVWXYZABCDEFHIJKLMNOPQRSTUVWXYZABCDEFGIJKLMNOPQRSTUVWXYZABCDEFGHJKLMNOPQRSTUVWXYZABCDEFGHIKLMNOPQRSTUVWXYZABCDEFGHIJLMNOPQRSTUVWXYZABCDEFGHIJKMNOPQRSTUVWXYZABCDEFGHIJKLNOPQRSTUVWXYZABCDEFGHIJKLMOPQRSTUVWXYZABCDEFGHIJKLMNPQRSTUVWXYZABCDEFGHIJKLMNOQRSTUVWXYZABCDEFGHIJKLMNOPRSTUVWXYZABCDEFGHIJKLMNOPQSTUVWXYZABCDEFGHIJKLMNOPQRTUVWXYZABCDEFGHIJKLMNOPQRSUVWXYZABCDEFGHIJKLMNOPQRSTVWXYZABCDEFGHIJKLMNOPQRSTUWXYZABCDEFGHIJKLMNOPQRSTUVXYZABCDEFGHIJKLMNOPQRSTUVWYZABCDEFGHIJKLMNOPQRSTUVWXZABCDEFGHIJKLMNOPQRSTUVWXYThe first horizontal alphabet is the alphabet of the plain text. Each substitution alphabet is designated by the letter at the left of a horizontal line. For example, if the key word isBAD, the second, first and fourth alphabets are used in turn and the wordWILLis encipheredXIOM.The Larrabee cipher is merely a slightly different arrangement of the Vigenere cipher and is printed on a card in this form:AABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyzBABCDEFGHIJKLMNOPQRSTUVWXYZbcdefghijklmnopqrstuvwxyzaCABCDEFGHIJKLMNOPQRSTUVWXYZcdefghijklmnopqrstuvwxyzabetc.YABCDEFGHIJKLMNOPQRSTUVWXYZyzabcdefghijklmnopqrstuvwxZABCDEFGHIJKLMNOPQRSTUVWXYZzabcdefghijklmnopqrstuvwxyThe large letters at the left are the letters of the key word. It will be noted that these letters correspond to the first letters of the cipher alphabets (in small letters) as in the Vigenere cipher.A much simpler arrangement of the Vigenere cipher is the use of a fixed and sliding alphabet. Either the fixed or sliding alphabet must be double in order to get coincidence for every letter whenAis set to the letter of the key word.Fixed Alphabet of TextABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZMovable Alphabet of CipherAs shown here,Aof the fixed or text alphabet coincides withTof the movable cipher alphabet. This is the setting whereTis the letter of the key word in use. The lower movable alphabet is moved for each letter of the message and theAof the fixed alphabet is made to coincide in turn with each letter of the key before the corresponding letter of the text is enciphered. It is obviously only a step from this arrangement to that of a cipher disk, where thefixed alphabet, (a single one will now serve) is printed in a circle and the movable alphabet, also in a circle, is on a separate rotatable disk. Coincidence of any letter on the disk withAof the fixed alphabet is obtained by rotating the disk.The well known U. S. Army Cipher Disk has just such an arrangement of the fixed alphabet but the alphabet of the disk is reversed. This has several advantages in simplicity of operation but none in increasing the indecipherability of the cipher prepared with it. The arrangement of fixed and sliding alphabets which is equivalent to the U. S. Army cipher disk is this:Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBAMovable AlphabetIt will be noticed that with this arrangement of running the alphabets in opposite directions, it becomes immaterial which alphabet is used for the text and which for the cipher for ifA=GthenG=A. This is not true of the Vigenere cipher.It is perfectly feasible to substitute a card for the U. S. Army cipher disk. It would have this form:ABCDEFGHIJKLMNOPQRSTUVWXYZ1AZYXWVUTSRQPONMLKJIHGFEDCB2BAZYXWVUTSRQPONMLKJIHGFEDC3CBAZYXWVUTSRQPONMLKJIHGFEDetc.25YXWVUTSRQPONMLKJIHGFEDCBAZ26ZYXWVUTSRQPONMLKJIHGFEDCBAThe first horizontal line is the alphabet of the text. The other twenty-six lines are the cipher alphabets each corresponding to the letter of the key word which is at the left of the line.One of the ciphers of Porta was prepared with a card of this kind:ABABCDEFGHIJKLMNOPQRSTUVWXYZCDABCDEFGHIJKLMZNOPQRSTUVWXYEFABCDEFGHIJKLMYZABCDEFGHIJKetc.WXABCDEFGHIJKLMPQRSTUVWXYZNOYZABCDEFGHIJKLMOPQRSTUVWXYZNIn this cipher the large letters at the left correspond to the letters of the key and, in each alphabet, the lower letter is substituted for the upper and vice versa. For example, with keyBADto encipherWILLwe would getJVXY. Note that with eitherBorAas the key letter, the first alphabet would be used.A combination of the Vigenere and Porta ciphers is this:AABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZBCABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZADEABCDEFGHIJKLMNOPQRSTUVWXYZCDEFGHIJKLMNOPQRSTUVWXYZABetc.VWABCDEFGHIJKLMNOPQRSTUVWXYZLMNOPQRSTUVWXYZABCDEFGHIJKXYABCDEFGHIJKLMNOPQRSTUVWXYZMNOPQRSTUVWXYZABCDEFGHIJKLZABCDEFGHIJKLMNOPQRSTUVWXYZNOPQRSTUVWXYZABCDEFGHIJKLMHere again the large letters at the left correspond to the letters of the key and, in each pair of alphabets, the upper one is that of the plain text and the lower is that of the cipher.This cipher can also be operated by a fixed and sliding alphabet.Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZSliding Alphabet of Cipher\/IndexBDFHJLNPRTVXAZLetters of Key.CEGIKMOQSUWYThe other ciphers mentioned are merely variations of these that have been discussed. It is immaterial, in the following analysis, which variety has been used. The analysis is really based on what can be done with a cipher made up with a mixed cipher alphabet which may be moved with reference to the fixed alphabet of the text, (See Case 7-b). Clearly this is a much more difficult proposition than dealing with a cipher in which the cipher alphabets run in their regular sequence, either backward or forward. In fact, in the analysis of Case 7, we may consider any cipher prepared by the method of Vigenere or any of its variations as a special and simple case.It was long ago discovered that, in any cipher of this class, (1) two like groups of letters in the cipher are most probably the result of two like groups of letters of the text enciphered by the same alphabets and (2) the number of letters in one group plus the number of letters to the beginning of the second group is a multiple of the number of alphabets used. It is evident, of course, that we may have similar groups in the cipher which are not the result of encipheringsimilar groups of the text by the same alphabets but if we take all recurring groups in a message and investigate the number of intervening letters, we will find that the majority of such cases will conform to these two principles.Changing the key word and message to illustrate more clearly the above points, the following is quoted from the Signal Book, 1914, with reference to the use of the cipher disk in preparing a message with a key word.1“—This simple disk can be used with a cipher word or, preferably, cipher words, known only to the correspondents.... Using the key word ‘disk’ to encipher the message ‘Artillery commander will order all guns withdrawn,’ we will proceed as follows: Write out the message to be enciphered and above it write the key word ... letter over letter, thus:DISKDISKDISKDISKDISKDISKDISKDISKDISKDISKDISARTILLERYCOMMANDERWILLORDERALLGUNSWITHDRAWNDRZCSXOTFGEYRIFHZRWCSXETAEBKSXMQQQWCXBPTDMP“Now bring the ‘a’ of the upper disk under the first letter of the key word on the lower disk, in this case ‘D’. The first letter of the message to be enciphered is ‘A’: ‘d’ is found to be the letter connected with ‘A’, and it is put down as the first cipher letter. The letter ‘a’ is then brought under ‘I’ which is the second letter of the key word. ‘R’ is to be enciphered and ‘r’ is found to be the second cipher letter.... Proceed in this manner until the last letter of the key word is used and beginning again with the letter ‘D’, so continue until all letters ofthe message have been enciphered. Divided into groups of five letters, it will be as follows:“DRZCS XOTFG EYRIF HZRWC SXETA EBKSX MQQQW CKBPT DMF.”So much for the Signal Book; now let us examine the above message for pairs or similar groups and count the intervening letters to demonstrate principles (1) and (2);CSX—CSX16=4 × 4SX—SX16=4 × 4SX—SX8=2 × 4WC—WC16=4 × 4The key word might contain 2, 4 or 8 letters from the evidence but we may eliminate 2 as unlikely and preparation of frequency tables of each of the four alphabets would soon show that 4 is the correct number.A later and more extensive example (Case7-a) will show pairs not separated by multiples of the number of alphabets used, but the evidence in nearly every case will be practically conclusive. Especially is this so if chance assists us by giving groups of three or more letters like the groupCSXin the above example. The number of alphabets having been determined each alphabet is handled by the methods of Case6already discussed.
We will now consider the class of substitution ciphers where a number of alphabets are used, the number and choice of alphabets depending on a key word or equivalent and being used periodically throughout the message.In this class belong the methods of Vigenere, Porta, Beaufort, St. Cyr, and many others. These methods date back several hundred years, but variations of them are constantly appearing as new ciphers. The Larrabee cipher, used for communication between government departments, is the Vigenere cipher of the 17th Century. The cipher disk method is practically the Vigenere cipher with reversed alphabets.In using these ciphers, there is provided a number of different cipher alphabets, usually twenty-six, and each cipher alphabet is identified by a different letter or number. A key word or phrase (or key number) is agreed upon by the correspondents. The message to be enciphered is written in lines containing a number of letters which is a multiple of the number of letters of the key. The key is written as the first line. Then each column under a letter of the key is enciphered by the cipher alphabet pertaining to that letter of the key. For example, let us take the message, “All radio messages must hereafter be put in cipher,” with the keyGrant, using the Vigenere cipher alphabets given below. Each of these alphabets isidentifiedby the first or left hand letter which representsAof the text. We thus willuse in turn the alphabets beginning withG, withR, withA, withN, and withT.GRANTGRANTALLRADIOMESSAGESMUSTHEREAFTERBEPUTINCIPHERUsing the alphabet indicated byG, we getGJYYNLKTKContinuing for the other alphabets, we getGCLETJZOZXYJATXYDUFMNVRRTLKEEUKGUGBTTICAKIThis method of arranging the message into lines and columns and then enciphering whole columns with each cipher alphabet is much shorter than the method of handling each letter of the message separately. The chance of error is also greatly reduced.All these cipher methods can be operated by means of squares containing the various alphabets, cipher disks or arrangements of fixed and sliding alphabets. For example, this was the original cipher of Vigenere:ABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZACDEFGHIJKLMNOPQRSTUVWXYZABDEFGHIJKLMNOPQRSTUVWXYZABCEFGHIJKLMNOPQRSTUVWXYZABCDFGHIJKLMNOPQRSTUVWXYZABCDEGHIJKLMNOPQRSTUVWXYZABCDEFHIJKLMNOPQRSTUVWXYZABCDEFGIJKLMNOPQRSTUVWXYZABCDEFGHJKLMNOPQRSTUVWXYZABCDEFGHIKLMNOPQRSTUVWXYZABCDEFGHIJLMNOPQRSTUVWXYZABCDEFGHIJKMNOPQRSTUVWXYZABCDEFGHIJKLNOPQRSTUVWXYZABCDEFGHIJKLMOPQRSTUVWXYZABCDEFGHIJKLMNPQRSTUVWXYZABCDEFGHIJKLMNOQRSTUVWXYZABCDEFGHIJKLMNOPRSTUVWXYZABCDEFGHIJKLMNOPQSTUVWXYZABCDEFGHIJKLMNOPQRTUVWXYZABCDEFGHIJKLMNOPQRSUVWXYZABCDEFGHIJKLMNOPQRSTVWXYZABCDEFGHIJKLMNOPQRSTUWXYZABCDEFGHIJKLMNOPQRSTUVXYZABCDEFGHIJKLMNOPQRSTUVWYZABCDEFGHIJKLMNOPQRSTUVWXZABCDEFGHIJKLMNOPQRSTUVWXYThe first horizontal alphabet is the alphabet of the plain text. Each substitution alphabet is designated by the letter at the left of a horizontal line. For example, if the key word isBAD, the second, first and fourth alphabets are used in turn and the wordWILLis encipheredXIOM.The Larrabee cipher is merely a slightly different arrangement of the Vigenere cipher and is printed on a card in this form:AABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyzBABCDEFGHIJKLMNOPQRSTUVWXYZbcdefghijklmnopqrstuvwxyzaCABCDEFGHIJKLMNOPQRSTUVWXYZcdefghijklmnopqrstuvwxyzabetc.YABCDEFGHIJKLMNOPQRSTUVWXYZyzabcdefghijklmnopqrstuvwxZABCDEFGHIJKLMNOPQRSTUVWXYZzabcdefghijklmnopqrstuvwxyThe large letters at the left are the letters of the key word. It will be noted that these letters correspond to the first letters of the cipher alphabets (in small letters) as in the Vigenere cipher.A much simpler arrangement of the Vigenere cipher is the use of a fixed and sliding alphabet. Either the fixed or sliding alphabet must be double in order to get coincidence for every letter whenAis set to the letter of the key word.Fixed Alphabet of TextABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZMovable Alphabet of CipherAs shown here,Aof the fixed or text alphabet coincides withTof the movable cipher alphabet. This is the setting whereTis the letter of the key word in use. The lower movable alphabet is moved for each letter of the message and theAof the fixed alphabet is made to coincide in turn with each letter of the key before the corresponding letter of the text is enciphered. It is obviously only a step from this arrangement to that of a cipher disk, where thefixed alphabet, (a single one will now serve) is printed in a circle and the movable alphabet, also in a circle, is on a separate rotatable disk. Coincidence of any letter on the disk withAof the fixed alphabet is obtained by rotating the disk.The well known U. S. Army Cipher Disk has just such an arrangement of the fixed alphabet but the alphabet of the disk is reversed. This has several advantages in simplicity of operation but none in increasing the indecipherability of the cipher prepared with it. The arrangement of fixed and sliding alphabets which is equivalent to the U. S. Army cipher disk is this:Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBAMovable AlphabetIt will be noticed that with this arrangement of running the alphabets in opposite directions, it becomes immaterial which alphabet is used for the text and which for the cipher for ifA=GthenG=A. This is not true of the Vigenere cipher.It is perfectly feasible to substitute a card for the U. S. Army cipher disk. It would have this form:ABCDEFGHIJKLMNOPQRSTUVWXYZ1AZYXWVUTSRQPONMLKJIHGFEDCB2BAZYXWVUTSRQPONMLKJIHGFEDC3CBAZYXWVUTSRQPONMLKJIHGFEDetc.25YXWVUTSRQPONMLKJIHGFEDCBAZ26ZYXWVUTSRQPONMLKJIHGFEDCBAThe first horizontal line is the alphabet of the text. The other twenty-six lines are the cipher alphabets each corresponding to the letter of the key word which is at the left of the line.One of the ciphers of Porta was prepared with a card of this kind:ABABCDEFGHIJKLMNOPQRSTUVWXYZCDABCDEFGHIJKLMZNOPQRSTUVWXYEFABCDEFGHIJKLMYZABCDEFGHIJKetc.WXABCDEFGHIJKLMPQRSTUVWXYZNOYZABCDEFGHIJKLMOPQRSTUVWXYZNIn this cipher the large letters at the left correspond to the letters of the key and, in each alphabet, the lower letter is substituted for the upper and vice versa. For example, with keyBADto encipherWILLwe would getJVXY. Note that with eitherBorAas the key letter, the first alphabet would be used.A combination of the Vigenere and Porta ciphers is this:AABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZBCABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZADEABCDEFGHIJKLMNOPQRSTUVWXYZCDEFGHIJKLMNOPQRSTUVWXYZABetc.VWABCDEFGHIJKLMNOPQRSTUVWXYZLMNOPQRSTUVWXYZABCDEFGHIJKXYABCDEFGHIJKLMNOPQRSTUVWXYZMNOPQRSTUVWXYZABCDEFGHIJKLZABCDEFGHIJKLMNOPQRSTUVWXYZNOPQRSTUVWXYZABCDEFGHIJKLMHere again the large letters at the left correspond to the letters of the key and, in each pair of alphabets, the upper one is that of the plain text and the lower is that of the cipher.This cipher can also be operated by a fixed and sliding alphabet.Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZSliding Alphabet of Cipher\/IndexBDFHJLNPRTVXAZLetters of Key.CEGIKMOQSUWYThe other ciphers mentioned are merely variations of these that have been discussed. It is immaterial, in the following analysis, which variety has been used. The analysis is really based on what can be done with a cipher made up with a mixed cipher alphabet which may be moved with reference to the fixed alphabet of the text, (See Case 7-b). Clearly this is a much more difficult proposition than dealing with a cipher in which the cipher alphabets run in their regular sequence, either backward or forward. In fact, in the analysis of Case 7, we may consider any cipher prepared by the method of Vigenere or any of its variations as a special and simple case.It was long ago discovered that, in any cipher of this class, (1) two like groups of letters in the cipher are most probably the result of two like groups of letters of the text enciphered by the same alphabets and (2) the number of letters in one group plus the number of letters to the beginning of the second group is a multiple of the number of alphabets used. It is evident, of course, that we may have similar groups in the cipher which are not the result of encipheringsimilar groups of the text by the same alphabets but if we take all recurring groups in a message and investigate the number of intervening letters, we will find that the majority of such cases will conform to these two principles.Changing the key word and message to illustrate more clearly the above points, the following is quoted from the Signal Book, 1914, with reference to the use of the cipher disk in preparing a message with a key word.1“—This simple disk can be used with a cipher word or, preferably, cipher words, known only to the correspondents.... Using the key word ‘disk’ to encipher the message ‘Artillery commander will order all guns withdrawn,’ we will proceed as follows: Write out the message to be enciphered and above it write the key word ... letter over letter, thus:DISKDISKDISKDISKDISKDISKDISKDISKDISKDISKDISARTILLERYCOMMANDERWILLORDERALLGUNSWITHDRAWNDRZCSXOTFGEYRIFHZRWCSXETAEBKSXMQQQWCXBPTDMP“Now bring the ‘a’ of the upper disk under the first letter of the key word on the lower disk, in this case ‘D’. The first letter of the message to be enciphered is ‘A’: ‘d’ is found to be the letter connected with ‘A’, and it is put down as the first cipher letter. The letter ‘a’ is then brought under ‘I’ which is the second letter of the key word. ‘R’ is to be enciphered and ‘r’ is found to be the second cipher letter.... Proceed in this manner until the last letter of the key word is used and beginning again with the letter ‘D’, so continue until all letters ofthe message have been enciphered. Divided into groups of five letters, it will be as follows:“DRZCS XOTFG EYRIF HZRWC SXETA EBKSX MQQQW CKBPT DMF.”So much for the Signal Book; now let us examine the above message for pairs or similar groups and count the intervening letters to demonstrate principles (1) and (2);CSX—CSX16=4 × 4SX—SX16=4 × 4SX—SX8=2 × 4WC—WC16=4 × 4The key word might contain 2, 4 or 8 letters from the evidence but we may eliminate 2 as unlikely and preparation of frequency tables of each of the four alphabets would soon show that 4 is the correct number.A later and more extensive example (Case7-a) will show pairs not separated by multiples of the number of alphabets used, but the evidence in nearly every case will be practically conclusive. Especially is this so if chance assists us by giving groups of three or more letters like the groupCSXin the above example. The number of alphabets having been determined each alphabet is handled by the methods of Case6already discussed.
We will now consider the class of substitution ciphers where a number of alphabets are used, the number and choice of alphabets depending on a key word or equivalent and being used periodically throughout the message.In this class belong the methods of Vigenere, Porta, Beaufort, St. Cyr, and many others. These methods date back several hundred years, but variations of them are constantly appearing as new ciphers. The Larrabee cipher, used for communication between government departments, is the Vigenere cipher of the 17th Century. The cipher disk method is practically the Vigenere cipher with reversed alphabets.In using these ciphers, there is provided a number of different cipher alphabets, usually twenty-six, and each cipher alphabet is identified by a different letter or number. A key word or phrase (or key number) is agreed upon by the correspondents. The message to be enciphered is written in lines containing a number of letters which is a multiple of the number of letters of the key. The key is written as the first line. Then each column under a letter of the key is enciphered by the cipher alphabet pertaining to that letter of the key. For example, let us take the message, “All radio messages must hereafter be put in cipher,” with the keyGrant, using the Vigenere cipher alphabets given below. Each of these alphabets isidentifiedby the first or left hand letter which representsAof the text. We thus willuse in turn the alphabets beginning withG, withR, withA, withN, and withT.GRANTGRANTALLRADIOMESSAGESMUSTHEREAFTERBEPUTINCIPHERUsing the alphabet indicated byG, we getGJYYNLKTKContinuing for the other alphabets, we getGCLETJZOZXYJATXYDUFMNVRRTLKEEUKGUGBTTICAKIThis method of arranging the message into lines and columns and then enciphering whole columns with each cipher alphabet is much shorter than the method of handling each letter of the message separately. The chance of error is also greatly reduced.All these cipher methods can be operated by means of squares containing the various alphabets, cipher disks or arrangements of fixed and sliding alphabets. For example, this was the original cipher of Vigenere:ABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZACDEFGHIJKLMNOPQRSTUVWXYZABDEFGHIJKLMNOPQRSTUVWXYZABCEFGHIJKLMNOPQRSTUVWXYZABCDFGHIJKLMNOPQRSTUVWXYZABCDEGHIJKLMNOPQRSTUVWXYZABCDEFHIJKLMNOPQRSTUVWXYZABCDEFGIJKLMNOPQRSTUVWXYZABCDEFGHJKLMNOPQRSTUVWXYZABCDEFGHIKLMNOPQRSTUVWXYZABCDEFGHIJLMNOPQRSTUVWXYZABCDEFGHIJKMNOPQRSTUVWXYZABCDEFGHIJKLNOPQRSTUVWXYZABCDEFGHIJKLMOPQRSTUVWXYZABCDEFGHIJKLMNPQRSTUVWXYZABCDEFGHIJKLMNOQRSTUVWXYZABCDEFGHIJKLMNOPRSTUVWXYZABCDEFGHIJKLMNOPQSTUVWXYZABCDEFGHIJKLMNOPQRTUVWXYZABCDEFGHIJKLMNOPQRSUVWXYZABCDEFGHIJKLMNOPQRSTVWXYZABCDEFGHIJKLMNOPQRSTUWXYZABCDEFGHIJKLMNOPQRSTUVXYZABCDEFGHIJKLMNOPQRSTUVWYZABCDEFGHIJKLMNOPQRSTUVWXZABCDEFGHIJKLMNOPQRSTUVWXYThe first horizontal alphabet is the alphabet of the plain text. Each substitution alphabet is designated by the letter at the left of a horizontal line. For example, if the key word isBAD, the second, first and fourth alphabets are used in turn and the wordWILLis encipheredXIOM.The Larrabee cipher is merely a slightly different arrangement of the Vigenere cipher and is printed on a card in this form:AABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyzBABCDEFGHIJKLMNOPQRSTUVWXYZbcdefghijklmnopqrstuvwxyzaCABCDEFGHIJKLMNOPQRSTUVWXYZcdefghijklmnopqrstuvwxyzabetc.YABCDEFGHIJKLMNOPQRSTUVWXYZyzabcdefghijklmnopqrstuvwxZABCDEFGHIJKLMNOPQRSTUVWXYZzabcdefghijklmnopqrstuvwxyThe large letters at the left are the letters of the key word. It will be noted that these letters correspond to the first letters of the cipher alphabets (in small letters) as in the Vigenere cipher.A much simpler arrangement of the Vigenere cipher is the use of a fixed and sliding alphabet. Either the fixed or sliding alphabet must be double in order to get coincidence for every letter whenAis set to the letter of the key word.Fixed Alphabet of TextABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZMovable Alphabet of CipherAs shown here,Aof the fixed or text alphabet coincides withTof the movable cipher alphabet. This is the setting whereTis the letter of the key word in use. The lower movable alphabet is moved for each letter of the message and theAof the fixed alphabet is made to coincide in turn with each letter of the key before the corresponding letter of the text is enciphered. It is obviously only a step from this arrangement to that of a cipher disk, where thefixed alphabet, (a single one will now serve) is printed in a circle and the movable alphabet, also in a circle, is on a separate rotatable disk. Coincidence of any letter on the disk withAof the fixed alphabet is obtained by rotating the disk.The well known U. S. Army Cipher Disk has just such an arrangement of the fixed alphabet but the alphabet of the disk is reversed. This has several advantages in simplicity of operation but none in increasing the indecipherability of the cipher prepared with it. The arrangement of fixed and sliding alphabets which is equivalent to the U. S. Army cipher disk is this:Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBAMovable AlphabetIt will be noticed that with this arrangement of running the alphabets in opposite directions, it becomes immaterial which alphabet is used for the text and which for the cipher for ifA=GthenG=A. This is not true of the Vigenere cipher.It is perfectly feasible to substitute a card for the U. S. Army cipher disk. It would have this form:ABCDEFGHIJKLMNOPQRSTUVWXYZ1AZYXWVUTSRQPONMLKJIHGFEDCB2BAZYXWVUTSRQPONMLKJIHGFEDC3CBAZYXWVUTSRQPONMLKJIHGFEDetc.25YXWVUTSRQPONMLKJIHGFEDCBAZ26ZYXWVUTSRQPONMLKJIHGFEDCBAThe first horizontal line is the alphabet of the text. The other twenty-six lines are the cipher alphabets each corresponding to the letter of the key word which is at the left of the line.One of the ciphers of Porta was prepared with a card of this kind:ABABCDEFGHIJKLMNOPQRSTUVWXYZCDABCDEFGHIJKLMZNOPQRSTUVWXYEFABCDEFGHIJKLMYZABCDEFGHIJKetc.WXABCDEFGHIJKLMPQRSTUVWXYZNOYZABCDEFGHIJKLMOPQRSTUVWXYZNIn this cipher the large letters at the left correspond to the letters of the key and, in each alphabet, the lower letter is substituted for the upper and vice versa. For example, with keyBADto encipherWILLwe would getJVXY. Note that with eitherBorAas the key letter, the first alphabet would be used.A combination of the Vigenere and Porta ciphers is this:AABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZBCABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZADEABCDEFGHIJKLMNOPQRSTUVWXYZCDEFGHIJKLMNOPQRSTUVWXYZABetc.VWABCDEFGHIJKLMNOPQRSTUVWXYZLMNOPQRSTUVWXYZABCDEFGHIJKXYABCDEFGHIJKLMNOPQRSTUVWXYZMNOPQRSTUVWXYZABCDEFGHIJKLZABCDEFGHIJKLMNOPQRSTUVWXYZNOPQRSTUVWXYZABCDEFGHIJKLMHere again the large letters at the left correspond to the letters of the key and, in each pair of alphabets, the upper one is that of the plain text and the lower is that of the cipher.This cipher can also be operated by a fixed and sliding alphabet.Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZSliding Alphabet of Cipher\/IndexBDFHJLNPRTVXAZLetters of Key.CEGIKMOQSUWYThe other ciphers mentioned are merely variations of these that have been discussed. It is immaterial, in the following analysis, which variety has been used. The analysis is really based on what can be done with a cipher made up with a mixed cipher alphabet which may be moved with reference to the fixed alphabet of the text, (See Case 7-b). Clearly this is a much more difficult proposition than dealing with a cipher in which the cipher alphabets run in their regular sequence, either backward or forward. In fact, in the analysis of Case 7, we may consider any cipher prepared by the method of Vigenere or any of its variations as a special and simple case.It was long ago discovered that, in any cipher of this class, (1) two like groups of letters in the cipher are most probably the result of two like groups of letters of the text enciphered by the same alphabets and (2) the number of letters in one group plus the number of letters to the beginning of the second group is a multiple of the number of alphabets used. It is evident, of course, that we may have similar groups in the cipher which are not the result of encipheringsimilar groups of the text by the same alphabets but if we take all recurring groups in a message and investigate the number of intervening letters, we will find that the majority of such cases will conform to these two principles.Changing the key word and message to illustrate more clearly the above points, the following is quoted from the Signal Book, 1914, with reference to the use of the cipher disk in preparing a message with a key word.1“—This simple disk can be used with a cipher word or, preferably, cipher words, known only to the correspondents.... Using the key word ‘disk’ to encipher the message ‘Artillery commander will order all guns withdrawn,’ we will proceed as follows: Write out the message to be enciphered and above it write the key word ... letter over letter, thus:DISKDISKDISKDISKDISKDISKDISKDISKDISKDISKDISARTILLERYCOMMANDERWILLORDERALLGUNSWITHDRAWNDRZCSXOTFGEYRIFHZRWCSXETAEBKSXMQQQWCXBPTDMP“Now bring the ‘a’ of the upper disk under the first letter of the key word on the lower disk, in this case ‘D’. The first letter of the message to be enciphered is ‘A’: ‘d’ is found to be the letter connected with ‘A’, and it is put down as the first cipher letter. The letter ‘a’ is then brought under ‘I’ which is the second letter of the key word. ‘R’ is to be enciphered and ‘r’ is found to be the second cipher letter.... Proceed in this manner until the last letter of the key word is used and beginning again with the letter ‘D’, so continue until all letters ofthe message have been enciphered. Divided into groups of five letters, it will be as follows:“DRZCS XOTFG EYRIF HZRWC SXETA EBKSX MQQQW CKBPT DMF.”So much for the Signal Book; now let us examine the above message for pairs or similar groups and count the intervening letters to demonstrate principles (1) and (2);CSX—CSX16=4 × 4SX—SX16=4 × 4SX—SX8=2 × 4WC—WC16=4 × 4The key word might contain 2, 4 or 8 letters from the evidence but we may eliminate 2 as unlikely and preparation of frequency tables of each of the four alphabets would soon show that 4 is the correct number.A later and more extensive example (Case7-a) will show pairs not separated by multiples of the number of alphabets used, but the evidence in nearly every case will be practically conclusive. Especially is this so if chance assists us by giving groups of three or more letters like the groupCSXin the above example. The number of alphabets having been determined each alphabet is handled by the methods of Case6already discussed.
We will now consider the class of substitution ciphers where a number of alphabets are used, the number and choice of alphabets depending on a key word or equivalent and being used periodically throughout the message.
In this class belong the methods of Vigenere, Porta, Beaufort, St. Cyr, and many others. These methods date back several hundred years, but variations of them are constantly appearing as new ciphers. The Larrabee cipher, used for communication between government departments, is the Vigenere cipher of the 17th Century. The cipher disk method is practically the Vigenere cipher with reversed alphabets.
In using these ciphers, there is provided a number of different cipher alphabets, usually twenty-six, and each cipher alphabet is identified by a different letter or number. A key word or phrase (or key number) is agreed upon by the correspondents. The message to be enciphered is written in lines containing a number of letters which is a multiple of the number of letters of the key. The key is written as the first line. Then each column under a letter of the key is enciphered by the cipher alphabet pertaining to that letter of the key. For example, let us take the message, “All radio messages must hereafter be put in cipher,” with the keyGrant, using the Vigenere cipher alphabets given below. Each of these alphabets isidentifiedby the first or left hand letter which representsAof the text. We thus willuse in turn the alphabets beginning withG, withR, withA, withN, and withT.
GRANTGRANTALLRADIOMESSAGESMUSTHEREAFTERBEPUTINCIPHER
Using the alphabet indicated byG, we get
GJYYNLKTK
Continuing for the other alphabets, we get
GCLETJZOZXYJATXYDUFMNVRRTLKEEUKGUGBTTICAKI
This method of arranging the message into lines and columns and then enciphering whole columns with each cipher alphabet is much shorter than the method of handling each letter of the message separately. The chance of error is also greatly reduced.
All these cipher methods can be operated by means of squares containing the various alphabets, cipher disks or arrangements of fixed and sliding alphabets. For example, this was the original cipher of Vigenere:
ABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZACDEFGHIJKLMNOPQRSTUVWXYZABDEFGHIJKLMNOPQRSTUVWXYZABCEFGHIJKLMNOPQRSTUVWXYZABCDFGHIJKLMNOPQRSTUVWXYZABCDEGHIJKLMNOPQRSTUVWXYZABCDEFHIJKLMNOPQRSTUVWXYZABCDEFGIJKLMNOPQRSTUVWXYZABCDEFGHJKLMNOPQRSTUVWXYZABCDEFGHIKLMNOPQRSTUVWXYZABCDEFGHIJLMNOPQRSTUVWXYZABCDEFGHIJKMNOPQRSTUVWXYZABCDEFGHIJKLNOPQRSTUVWXYZABCDEFGHIJKLMOPQRSTUVWXYZABCDEFGHIJKLMNPQRSTUVWXYZABCDEFGHIJKLMNOQRSTUVWXYZABCDEFGHIJKLMNOPRSTUVWXYZABCDEFGHIJKLMNOPQSTUVWXYZABCDEFGHIJKLMNOPQRTUVWXYZABCDEFGHIJKLMNOPQRSUVWXYZABCDEFGHIJKLMNOPQRSTVWXYZABCDEFGHIJKLMNOPQRSTUWXYZABCDEFGHIJKLMNOPQRSTUVXYZABCDEFGHIJKLMNOPQRSTUVWYZABCDEFGHIJKLMNOPQRSTUVWXZABCDEFGHIJKLMNOPQRSTUVWXY
The first horizontal alphabet is the alphabet of the plain text. Each substitution alphabet is designated by the letter at the left of a horizontal line. For example, if the key word isBAD, the second, first and fourth alphabets are used in turn and the wordWILLis encipheredXIOM.
The Larrabee cipher is merely a slightly different arrangement of the Vigenere cipher and is printed on a card in this form:
AABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyzBABCDEFGHIJKLMNOPQRSTUVWXYZbcdefghijklmnopqrstuvwxyzaCABCDEFGHIJKLMNOPQRSTUVWXYZcdefghijklmnopqrstuvwxyzabetc.YABCDEFGHIJKLMNOPQRSTUVWXYZyzabcdefghijklmnopqrstuvwxZABCDEFGHIJKLMNOPQRSTUVWXYZzabcdefghijklmnopqrstuvwxy
The large letters at the left are the letters of the key word. It will be noted that these letters correspond to the first letters of the cipher alphabets (in small letters) as in the Vigenere cipher.
A much simpler arrangement of the Vigenere cipher is the use of a fixed and sliding alphabet. Either the fixed or sliding alphabet must be double in order to get coincidence for every letter whenAis set to the letter of the key word.
Fixed Alphabet of TextABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZMovable Alphabet of Cipher
As shown here,Aof the fixed or text alphabet coincides withTof the movable cipher alphabet. This is the setting whereTis the letter of the key word in use. The lower movable alphabet is moved for each letter of the message and theAof the fixed alphabet is made to coincide in turn with each letter of the key before the corresponding letter of the text is enciphered. It is obviously only a step from this arrangement to that of a cipher disk, where thefixed alphabet, (a single one will now serve) is printed in a circle and the movable alphabet, also in a circle, is on a separate rotatable disk. Coincidence of any letter on the disk withAof the fixed alphabet is obtained by rotating the disk.
The well known U. S. Army Cipher Disk has just such an arrangement of the fixed alphabet but the alphabet of the disk is reversed. This has several advantages in simplicity of operation but none in increasing the indecipherability of the cipher prepared with it. The arrangement of fixed and sliding alphabets which is equivalent to the U. S. Army cipher disk is this:
Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBAMovable Alphabet
It will be noticed that with this arrangement of running the alphabets in opposite directions, it becomes immaterial which alphabet is used for the text and which for the cipher for ifA=GthenG=A. This is not true of the Vigenere cipher.
It is perfectly feasible to substitute a card for the U. S. Army cipher disk. It would have this form:
ABCDEFGHIJKLMNOPQRSTUVWXYZ1AZYXWVUTSRQPONMLKJIHGFEDCB2BAZYXWVUTSRQPONMLKJIHGFEDC3CBAZYXWVUTSRQPONMLKJIHGFEDetc.25YXWVUTSRQPONMLKJIHGFEDCBAZ26ZYXWVUTSRQPONMLKJIHGFEDCBA
The first horizontal line is the alphabet of the text. The other twenty-six lines are the cipher alphabets each corresponding to the letter of the key word which is at the left of the line.
One of the ciphers of Porta was prepared with a card of this kind:
ABABCDEFGHIJKLMNOPQRSTUVWXYZCDABCDEFGHIJKLMZNOPQRSTUVWXYEFABCDEFGHIJKLMYZABCDEFGHIJKetc.WXABCDEFGHIJKLMPQRSTUVWXYZNOYZABCDEFGHIJKLMOPQRSTUVWXYZN
In this cipher the large letters at the left correspond to the letters of the key and, in each alphabet, the lower letter is substituted for the upper and vice versa. For example, with keyBADto encipherWILLwe would getJVXY. Note that with eitherBorAas the key letter, the first alphabet would be used.
A combination of the Vigenere and Porta ciphers is this:
AABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZBCABCDEFGHIJKLMNOPQRSTUVWXYZBCDEFGHIJKLMNOPQRSTUVWXYZADEABCDEFGHIJKLMNOPQRSTUVWXYZCDEFGHIJKLMNOPQRSTUVWXYZABetc.VWABCDEFGHIJKLMNOPQRSTUVWXYZLMNOPQRSTUVWXYZABCDEFGHIJKXYABCDEFGHIJKLMNOPQRSTUVWXYZMNOPQRSTUVWXYZABCDEFGHIJKLZABCDEFGHIJKLMNOPQRSTUVWXYZNOPQRSTUVWXYZABCDEFGHIJKLM
Here again the large letters at the left correspond to the letters of the key and, in each pair of alphabets, the upper one is that of the plain text and the lower is that of the cipher.
This cipher can also be operated by a fixed and sliding alphabet.
Fixed AlphabetABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZSliding Alphabet of Cipher\/IndexBDFHJLNPRTVXAZLetters of Key.CEGIKMOQSUWY
BDFHJLNPRTVXAZLetters of Key.CEGIKMOQSUWY
The other ciphers mentioned are merely variations of these that have been discussed. It is immaterial, in the following analysis, which variety has been used. The analysis is really based on what can be done with a cipher made up with a mixed cipher alphabet which may be moved with reference to the fixed alphabet of the text, (See Case 7-b). Clearly this is a much more difficult proposition than dealing with a cipher in which the cipher alphabets run in their regular sequence, either backward or forward. In fact, in the analysis of Case 7, we may consider any cipher prepared by the method of Vigenere or any of its variations as a special and simple case.
It was long ago discovered that, in any cipher of this class, (1) two like groups of letters in the cipher are most probably the result of two like groups of letters of the text enciphered by the same alphabets and (2) the number of letters in one group plus the number of letters to the beginning of the second group is a multiple of the number of alphabets used. It is evident, of course, that we may have similar groups in the cipher which are not the result of encipheringsimilar groups of the text by the same alphabets but if we take all recurring groups in a message and investigate the number of intervening letters, we will find that the majority of such cases will conform to these two principles.
Changing the key word and message to illustrate more clearly the above points, the following is quoted from the Signal Book, 1914, with reference to the use of the cipher disk in preparing a message with a key word.1
“—This simple disk can be used with a cipher word or, preferably, cipher words, known only to the correspondents.... Using the key word ‘disk’ to encipher the message ‘Artillery commander will order all guns withdrawn,’ we will proceed as follows: Write out the message to be enciphered and above it write the key word ... letter over letter, thus:
DISKDISKDISKDISKDISKDISKDISKDISKDISKDISKDISARTILLERYCOMMANDERWILLORDERALLGUNSWITHDRAWNDRZCSXOTFGEYRIFHZRWCSXETAEBKSXMQQQWCXBPTDMP
“Now bring the ‘a’ of the upper disk under the first letter of the key word on the lower disk, in this case ‘D’. The first letter of the message to be enciphered is ‘A’: ‘d’ is found to be the letter connected with ‘A’, and it is put down as the first cipher letter. The letter ‘a’ is then brought under ‘I’ which is the second letter of the key word. ‘R’ is to be enciphered and ‘r’ is found to be the second cipher letter.... Proceed in this manner until the last letter of the key word is used and beginning again with the letter ‘D’, so continue until all letters ofthe message have been enciphered. Divided into groups of five letters, it will be as follows:
“DRZCS XOTFG EYRIF HZRWC SXETA EBKSX MQQQW CKBPT DMF.”
So much for the Signal Book; now let us examine the above message for pairs or similar groups and count the intervening letters to demonstrate principles (1) and (2);
CSX—CSX16=4 × 4SX—SX16=4 × 4SX—SX8=2 × 4WC—WC16=4 × 4
The key word might contain 2, 4 or 8 letters from the evidence but we may eliminate 2 as unlikely and preparation of frequency tables of each of the four alphabets would soon show that 4 is the correct number.
A later and more extensive example (Case7-a) will show pairs not separated by multiples of the number of alphabets used, but the evidence in nearly every case will be practically conclusive. Especially is this so if chance assists us by giving groups of three or more letters like the groupCSXin the above example. The number of alphabets having been determined each alphabet is handled by the methods of Case6already discussed.