Post-Quantum Cryptography: S381 Cyclic Subgroup of High Order
This provides a post-quantum cryptographic solution suitable for low-performance devices like smart cards and cellphones, though it appears incremental as it adapts existing protocols to a new mathematical structure.
The authors tackled the need for post-quantum cryptography by developing a Diffie-Hellman key exchange and ElGamal ciphering procedure based on a high-order permutation subgroup in S381, using combinatorial operations to achieve conceptual simplicity, fast throughput, and high security without arithmetic operations.
Currently there is an active Post-Quantum Cryptography (PQC) solutions search, which attempts to find cryptographic protocols resistant to attacks by means of for instance Shor polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). The use of non-commutative or non-associative structures are, among others, valid choices for these kinds of protocols. In our case, we focus on a permutation subgroup of high order and belonging to the symmetric group S381. Using adequate one-way functions (OWF), we derived a Diffie-Hellman key exchange and an ElGamal ciphering procedure that only relies on combinatorial operations. Both OWF pose hard search problems which are assumed as not belonging to BQP time-complexity class. Obvious advantages of present protocols are their conceptual simplicity, fast throughput implementations, high cryptanalytic security and no need for arithmetic operations and therefore extended precision libraries. Such features make them suitable for low performance and low power consumption platforms like smart cards, USB-keys and cellphones.