A Practical Example for McEliece Cryptography

Using the (7,4) Hamming code with one bit error correction

Robert McEliece, in 1978, outlined the McEliece cryptosystem [here]:

Overall it is asymmetric encryption method (with a public key and a private key), and, at the time, looked to be a serious contender. Unfortunately for the method, RSA became the King of the Hill, and the McEliece method was pushed to the end of the queue for designers. It has basically drifted in the 38 years since. But, as an era of quantum computers is dawning, it is now being reconsidered, as it is seen to be immune to attacks using Shor’s algorithm.

This is based on the example [here]. We illustrate the process with a (7,4) Hamming code which corrects one error. The generator matrix is then:

    1  0  0  0  1  1  0
G = 0  1  0  0  1  0  1
    0  0  1  0  0  1  1
    0  0  0  1  1  1  1

Bob then selectes a scrambler matrix (S):

    1  1  0  1
S = 1  0  0  1
    0  1  1  1
    1  1  0  0

And also a permutation matrix (P):

     0  1  0  0  0  0  0 
     0…