A Public Key Cryptosystem Using a Group of Permutation Polynomials

Abstract

In this paper we propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group ℒ(q, m) of permutation polynomials over the finite field Fqm. We construct a trapdoor function for the cryptosystem using polynomials in ℒ(2, m), where m =2k for some k ≥ 0. The complexity of encryption in our public key cryptosystem is O(m3) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most 5m2+3m – 4 left cyclic shifts, 5m2 +3m + 4 xor operations and 7 permutations on bits for decryption.

Some Proposed Problems on Permutation Polynomials over Finite Fields

“…Singh, Saikia and Sarma [9][10][11][12][13][14][15] designed efficient multivariate public key cryptosystem using permutation polynomials over finite fields. The same authors used a group of linearized permutation polynomials to design an efficient multivariate public key cryptosystem [16].…”

“…Orthomorphism polynomials can be used in check digit systems to detect single errors and adjacent transpositions whereas complete permutation polynomials to detect single and twin errors. For more details on complete mappings and orthomorphisms over finite fields, we refer to the reader [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In addition, complete permutation polynomials are very useful in the study of orthogonal latin squares and orthomorphism polynomials are useful in close connection to hyperovals in finite projective plane.…”