Cryptography Reference
In-Depth Information
q := Modular:-Random(modulus,dimension,dimension,integer);
d := Modular:-Determinant(modulus,q)
end do;
q
end:
Now, we use the preceding procedure to generate a 4
×
4 invertible matrix
over
Z
27
:
> k: = PseudoRandomInvertibleMatrixMod(4);
⎡
⎣
⎤
⎦
17 10 6 19
15 16 3 5
12 22 8 5
4211318
k
:=
Using this matrix as key, we encrypt the plaintext: “this text is the message”:
> HillEnc(k, "this text is the message");
"ctbedbeyr esrtyvna hqstm"
Finally, we decrypt the ciphertext:
> HillDec(k,%);
"this text is the message"
1.4.2 Cryptanalysis of the Hill Cipher
Let us now look at the cryptanalysis of the Hill cipher. First of all, observe that
changing one letter of plaintext can potentially change
n
letters of ciphertext, namely,
all the letters in the ciphertext block corresponding to the plaintext block with the
changed letter. In the above example, look at what happens if we change just one
letter in the plaintext:
> HillEnc(k, "thin text is the message");
"jwrwdbeyr esrtyvna hqstm"
We see that all four characters in the first ciphertext block have changed. This
is in contrast with the previous encryption schemes and makes frequency analysis
ineffective if the block size
n
is not too small. The situation can be best described
by referring to the concepts of
diffusion
and
confusion
, which were introduced by
Shannon in his 1949 paper [176]. These concepts are formulated in the context of
information theory
and refer to properties that a good encryption scheme should
have to make statistical cryptanalysis difficult or even impossible. We will not give a
precise definition of these terms—which would require an excursion into information
theory—since an informal one will be sufficient for our purposes:
