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4.5. BREAKING DECIMATION AND LINEAR CIPHERS 65
(4) Decipher JWNIC G. It was enciphered with the Linear Cipher 13m + 4
modulo 33.
Ok, now we do need Euclid’s algorithm:
q
r
c
33
0
2
13
1
1
7
−2
1
6
3
6
1
−5
0
Since −5%33 = 28 the inverse of 13 is 28.
subtracting by 4 and multiplying by 28. 5
Now we may proceed by
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
As one last example, recall from Chapter 3 that the Kid-RSA cipher has
as public information an enciphering key e and modulus n. In order for this
cipher to be worthwhile, it must be infeasible for an adversary to compute the
secret deciphering key d from the public keys e and n. Because of the Euclidean
Algorithm, not only is computing d not infeasible, it is simple.
To illustrate, if Albert has published e = 893 and n = 8106 as his public
keys, we simply compute gcd(8106, 893):
q
r
c
8106
0
9
893
1
12
69
−9
1
65
109
16
4
−118
4
1
1997
0
So in a very few steps, the “secret” deciphering key is d = 1997. Even though
there is no real security in Koblitz’s system, it did allow us to introduce the
important concepts related to public keys.
4.5 Breaking Decimation and Linear Ciphers
Earlier we saw that decimation and linear ciphers cause letters which are adjacent
in the plaintext alphabet to be separated in the ciphertext alphabet. How
much does this improve security Consider the ciphertext from Section 3.5 that
was enciphered with a linear cipher.
UQESF YFTGW SGPVS PPVQX QEDGR PMQFP YJSFG EORVQ DQBQF PVQWO
MRTQW PUOTT SPPOM QWOFE TIXQS FIMLQ DYJUY FXQDO FCW
It has the following letter frequency table.
5 cipher.
0 1 1 4 4 9 4 0 2 2 0 1 4 0 6 9 13 3 6 5 3 4 5 3 4 0
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