Hill Cipher

This cipher was used for the encryption of radio call signs in World War Two. Fix a block

size N and choose an invertible N ´N matrix K with entries in Z+%26

Now the input signal might consist of series of elements such as ‘y’. Then output will

also be series of vectors like ‘ky’. The independent variable for both input & output

signals will be position of particular vector in the series.

Properties:

Homogenity: The Hill cipher is homogeneous.

Ek(x)=Kx

Ek(cx)=K(cx)=cKx=cEk(x)

Linearity: As Hill cipher is homogeneous and additive, it is linear.

Memory: The definition of memory from the time domain is not applicable here.

However we can say that the system has memory in the sense that the encryption of

each character is dependent of the other characters because

Stability: String length at the output = String length of the input. For a finite string as

input the output is also a finite string of the same length.

Shift Invariance: The Hill Cipher system is Shift – Invariant. This can be shown as

follows :- Consider input signal, x ={x(i)} , series of vectors . Let the corresponding

output signal be y ={y(i)} .

Then y(i) = Kx(i) .

Now if x(l)->x(i + i0 )

Then corresponding y(l) = kx(l) = kx(i + i0 ) = y(i + i0 ).

Causality: The system is indeed causal .The output can not be nonzero before the input

deviates from zero.