The fastest matrices multiplication using involutory matrix in AES MixColumns-InvMixcolumns transformation

Abstract

The traditional computer will be difference attacks by future quantum computer. Now, AES seems a resistant primitive in the post quantum world, with a bigger security margin against quantum computer attacks. In this paper, the key idea here is to propose a method with a variations 2^k×2^k=n×n involutory matrix for enhancing diffusion data in AES MixColumns-InvMixColumns step that the Branch Number of the confusion capability is increased n+1 where k>=1 integer number, but the matrix multiplication is required a lot of the finite field multiplications. A 16×16 involutory matrix for matrix multiplication needs 256 multiplications and 240 additions for using encryption and decryption in AES MixColumns transformation. By utilizing both properties, the addition of the same elements over GF(2^m) results in zero properties, and dividing the involutory matrix into four sets of submatrices circulant matrix properties; the matrix multiplication can be simplified by Scheme 3 (16×16 matrix) that matrix multiplications can use 81 multiplications and 260 additions with good branch number 17. Using Scheme 3 and the proposed method of the multiplication running on Intel CPU, to compare traditional matrix multiplication, the computational cost of matrix multiplication can be reduced by ~67%. Finally, using Scheme 1, Scheme 2, and Scheme 3 into AES Cipher and InvCipher procedure that the methods can increase encryption and decryption speed for data transmission.