Algebraic Extension Ring Framework for Non-Commutative Asymmetric Cryptography

Abstract: Post-Quantum Cryptography PQC attempts to find cryptographic protocols resistant to attacks using Shors polynomial time algorithm for numerical field problems or Grovers algorithm to find the unique input to a black-box function that produces a particular output value. The use of non-standard algebraic structures like non-commutative or non-associative structures, combined with one-way trapdoor functions derived from combinatorial group theory, are mainly unexplored choices for these new kinds of protocols and overlooked in current PQC solutions. In this paper, we develop an algebraic extension ring framework who could be applied to different asymmetric protocols, i.e. key exchange, key transport, enciphering, digital signature, zero-knowledge authentication, oblivious transfer, secret sharing etc.. A valuable feature is that there is no need for big number libraries as all arithmetic is performed in F256 extension field operations (precisely the AES field). We assume that the new framework is cryptographical secure against strong classical attacks like the sometimes-useful length-based attack, Romankovs linearization attacks and Tsabans algebraic span attack. This statement is based on the non-linear structure of the selected platform which proved to be useful protecting the AES protocol. Otherwise, it could resist post-quantum attacks Grover, Shor and be particularly useful for computational platforms with limited capabilities like USB cryptographic keys or smartcards. Semantic security IND-CCA2 could also be inferred for this new platform.