Abstract
Multiplications in G F ( 2 N ) can be securely optimized for cryptographic applications when the integer N is small and does not match machine words (i.e., N < 32 ). In this paper, we present a set of optimizations applied to DAGS, a code-based post-quantum cryptographic algorithm and one of the submissions to the National Institute of Standards and Technology’s (NIST) Post-Quantum Cryptography (PQC) standardization call.
Highlights
Arithmetic in GF (2 N ) is very attractive since addition is carry-less
We show that computations can be faster when mapping elements from tower fields GF ((2` )m ) to isomorphic fields GF (2 N ), where N = `m
We start by presenting state-of-the-art multiplication algorithms in GF (2 N ) for small values of N, i.e., when N is smaller than the machine word
Summary
Arithmetic in GF (2 N ) is very attractive since addition is carry-less. This is why it is adopted in many cryptographic algorithms, which are efficient both in hardware (no carry means no long delays) and in software implementations. N. When N is smaller than a machine word size (that is, N < 32 or 64, on typical smartphones or desktops), all known window-based computational optimizations become irrelevant. Our method is not to come up with novel algorithms for multiplication, but to organize the computations in such a way that the resources of the computer are utilized optimally. Our contribution is to explore the way to load the machine in the most efficient way while remaining regular
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