Finite Field Arithmetic in Large Characteristic for Classical and Post-quantum Cryptography

Both classical and post-quantum cryptography massively use large characteristic finite fields or rings. Consequently, basic arithmetic on these fields or rings (integer or polynomial multiplication, modular reduction) may significantly impact cryptographic devices' efficiency and power consumption. In this paper, we will present the most used and the less common methods, clarify their advantages and drawbacks and explain which ones are the more relevant depending on the implementation context and the chosen cryptographic primitive. We also explain why recent proposals such as RNS, PMNS or Montgomery-friendly primes may be a good alternative to classical methods depending on the context and suggest directions for further research to improve them.