Vernam Cipher

Here we analyze the working of the encryption algorithm of the Vernam

Cipher (also known as One Time Pad) in the realm of Signals and Systems. It

is the only known cipher in the world that can keep messages secret no

matter how long an adversary attacks, and no matter what machinery the

adversary has.

System Formulation:

For a string of m numbers, a string of m random numbers is generated using a

key r which is “large prime number”. Here the term “large” is in a sense that it

should have as many bits as the message to be transmitted has.

Encrypted output E (i)= (x (i) + k (i))%26

x(i) = Number at the ith position in input string

k(i) = Corresponding random number generated

Hence ‘m’ random numbers + ‘m’ meaningful numbers give rise to set of

m numbers which form the encrypted message. Decrypted output D( i)= (x (i)

- k( i))%26

The position of the number in the string is the independent variable for both

input and output signals. We define following operations on the signals:

Addition: Defined and meaningful if and only if length of the two strings being

added is the same.

Strings a={a(i)} b={b(i)}

Then a+b={[a(i)+b(i)]%26}

Scaling: ca={ca(i)}

Now the properties of Vernam Cipher can be discussed as follows:


2 ) Memory: The system is memoryless as the encryption of each character is

independent of the previous or next characters.

3 ) Stability: The system is stable .Here stability is examined in the sense that for a finite

input character, the encrypted character in the output is also finite.

4 ) Causality: The system is non-causal because if a signal is shifted in terms of its

position of characters the corresponding output signal is not the shifted version of

original output as the random number generated are characteristic of positions.

The difficulty with the Vernam Cipher is its requirement of very large prime number. But

this difficulty also renders almost unbreakability to Vernam Cipher. Hence it is used in

securing hotline communication between Moscow & Washington.