Post-Quantum Cryptography(PQC): Generalized ElGamal Cipher over GF(251^8)

Shor's polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). Other aspects are the backdoors discovered in deterministic random generators or recent advances in solving some instances of DLP. The use of alternative algebraic structures like non-commutative or non-associative partial groupoids, magmas, monoids, semigroups, quasigroups or groups, are valid choices for these new kinds of protocols. In this paper, we focus in an asymmetric cipher based on a generalized ElGamal non-arbitrated protocol using a non-commutative general linear group. The developed protocol forces a hard subgroup membership search problem into a non-commutative structure. The protocol involves at first a generalized Diffie-Hellman key interchange and further on the private and public parameters are recursively updated each time a new cipher session is launched. Security is based on a hard variation of the Generalized Symmetric Decomposition Problem (GSDP). Working with GF(251^8) a 64-bits security is achieved, and if GF(251^16) is chosen, the security rises to 127-bits. An appealing feature is that there is no need for big number libraries as all arithmetic if performed in ℤ and therefore the new protocol is particularly useful for computational platforms with very limited capabilities like smartphones or smartcards.