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Affine Cipher
Computer Science (ICS 2103)
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Math 135 (Summer 2006) Affine Ciphers, Decimation Ciphers, and Modular Arithmetic
Affine Ciphers
An encipherment scheme (or algorithm) of the form
E(x) = (ax+b) MOD 26
is called anaffine cipher. Herexis the numerical equivalent of the given plaintext letter, andaand bare (appropriately chosen) integers.
Recall that the numerical equivalents of the letters are as follows:
A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 N O P Q R S T U V W X Y Z 13 14 15 16 17 18 19 20 21 22 23 24 25
Example: EncipherITS COOLwith
E(x) = (5x+ 8) MOD 26.
Solution: Filling in the following table gives
plain I T S C O O L x 8 19 18 2 14 14 11 5 x+ 8 48 103 98 18 78 78 63 (5x+ 8) MOD 26 22 25 20 18 0 0 11 cipher W Z U S A A L
Ify=E(x) = (ax+b) MOD 26, then we can “solve forxin terms ofy” and so determineE− 1 (y). That is, ify≡(ax+b) (mod 26), theny−b≡ax(mod 26), or equivalentlyax≡(y−b) (mod 26). Using our earlier results, we see that if we multiply both sides bya− 1 (mod 26), thenx≡a− 1 (y−b) (mod 26) and so our decipherment function is
E− 1 (y) =a− 1 (y−b) MOD 26.
Example: DecipherHPCCXAQif the encipherment function isE(x) = (5x+ 8) MOD 26.
Solution: We begin by finding the decipherment function. Since 5x ≡ 1 (mod 26) is solved with x≡21 (mod 26) we see 5− 1 (mod 26) = 21. Therefore,
E− 1 (y) = 21(y−8) MOD 26
and so filling in our table gives
cipher H P C C X A Q y 7 15 2 2 23 0 16 y− 8 -1 7 -6 -6 15 -8 8 21(y−8) -21 147 -126 -126 315 -168 168 21(y−8) MOD 26 5 17 4 4 3 14 12 plain F R E E D O M
Example: Suppose that an affine cipherE(x) = (ax+b) MOD 26 enciphersHasXandQasY. Find the cipher (that is, determineaandb).
Solution: We see that
H7→XmeansE(7) = 23 andQ7→YmeansE(16) = 24.
That is,
a·7 +b≡23 (mod 26) anda·16 +b≡24 (mod 26).
Subtracting gives 16a− 7 a≡1 (mod 26) so that 9a≡1 (mod 26). Therefore,a= 9− 1 (mod 26) = 3. Finally, we substitutea= 3 into either of the earlier equations and solve forb,
i., 3·7 +b≡23 (mod 26) impliesb= 2.
In summary, E(x) = (3x+ 2) MOD 26.
Remark: (The “Mod-mod Connection”) The least non-negative solution of the congruencex≡b(modm) isx=bMODm.
Decimation Ciphers
In the special case whereb= 0, the affine cipherE(x) =axMODmis called adecimation cipher. This is discussed in detail on pages 70–73. The key idea in this subsection is that certain choices ofa andmdo not lead to valid substitutions.
Example: Suppose thatE(x) = 4xMOD 26. Determine the ciphertext alphabet.
Solution: We begin with our table of numerical equivalents, and then determine 4xMOD 26.
plain A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 4 xx 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 4 xMOD 26 0 4 8 12 16 20 24 2 6 10 14 18 22 0 4 8 12 16 20 24 2 6 10 14 18 22 cipher A E I M Q U Y C G K O S W A E I M Q U Y C G K O S W
The problem, of course, is that 4 and 26 arenotrelatively prime, and so this cyclic phenomenon occurs in the cipher alphabet. Since the numbers 0, 2 , 4 , 6 , 8 , 10 , 12 , 13 , 14 , 16 , 18 , 20 , 22 ,24 are not relatively prime with respect to the 26, the only possible choices for the decimation cipherE(x) =axMOD 26 area= 1, 3 , 5 , 7 , 9 , 11 , 15 , 17 , 19 , 21 , 23 ,25. Therefore, we conclude that the decimation cipher is weaker than the simple shift cipher. If the cryptanalyst knows that a shift cipher has been used, then there are 25 possible shifts that need to be checked. However, if it is knownthat a decimation cipher has been used, then there are only 12 possible ciphers that need to be checked.
Summary of Valid Affine Ciphers
The functionE(x) = (ax+b) MOD 26 defines a valid affine cipher ifais relatively prime to 26, andb is an integer between 0 and 25, inclusive. Ifb= 0, then we refer to this cipher as a decimation cipher. (Note that since there are 12 valid choices ofaand 26 valid choices ofb, there are 12×26 = 312 possible valid affine ciphers.)
Also note that ifa= 1, thenE(x) = (x+b) MOD 26 is simply a Caesar (+b) shift cipher.
Affine Cipher
Course: Computer Science (ICS 2103)
University: Meru University of Science and Technology
