AN EXTENSION OF HILL CIPHER USING GENERALISED INVERSES AND mth RESIDUE MODULO n

Abstract

Hill Cipher is a symmetric polyalphabetic block cipher that enciphers a string of letters into another string of the same length using the linear transformation y=xK. In deciphering, the determinant value must be less than 26 and relatively prime to 26 so that the matrix K of the linear transformation is invertible in modulo 26. The Affine Hill cipher extends the concept of Hill cipher by using the non-linear transformation y=xK+b. In this paper, we extend this concept to encrypt a plaintext of blocksize m to a ciphertext of blocksize n≥m using (a) affine transformation and (b) polynomial transformation to make this cipher more secure. Here the matrix K of the transformation need not be a square matrix. To enable decryption, we state the conditions to be satisfied by K which are as follows.Case (a): (i) For affine transformation, the generalised inverse K + of the matrix K corresponding to the transformation should satisfy the equation KK +=I in modulo p where p is a chosen prime p>26. For m=n, K + is the usual inverse of the matrix K.Case(b): (i) For polynomial transformation, the generalised inverse K + should satisfy the above condition, (ii) If r is the degree of the polynomial, then choose those values of s≤r such that the sth root of modulo p exists for all elements in Z p . In other words, choose those values of s that are relatively prime to Φ(p).