Optimized Algorithms and Architectures for Montgomery Multiplication for Post-quantum Cryptography

Finite field multiplication plays the main role determining the efficiency of public key cryptography systems based on RSA and elliptic curve cryptography (ECC). Most recently, quantum-safe cryptographic systems are proposed based on supersingular isogenies on elliptic curves which require large integer multiplications over extended prime fields. In this work, we present two Montgomery multiplication architectures for special primes used in a post-quantum cryptography system known as supersingular isogeny key encapsulation (SIKE). We optimize two existing Montgomery multiplication algorithms and develop areaefficient and time-efficient Montgomery multiplication architectures for hardware implementations of post-quantum cryptography. Our proposed time-efficient architecture is 32% to 42% faster than the leading one (depending on the prime size) available in the literature which has been used in original SIKE submission to the NIST standardization process. The area-efficient architecture is 42% to 50% smaller than the counterparts and is about 3% to 11% faster depending on the NIST security level.