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arXiv2026

Residue Number System Comparison revisited, a software perspective

This paper presents a novel method to compare two numbers in Residue Number System (RNS) using an additional modulus, which is often already available because it is required in modular computations and digital signal processing scaling.Our method provides the comparison of two integers in the full range of the RNS base. It does not require moduli of a special form, unlike other state-of-the-art methods that are restricted to specific RNS bases or require bounds on input numbers. Our approach only requires one single conversion to a mixed radix representation with a complexity of O(n2), which can be reduced to O(log(n)) in time with parallelization. This provides a significant advantage over classical methods and more recent competitive methods which work under restrictions. This opens perspectives for advancements in challenging RNS operations such as division, scaling, and cryptographic applications.

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arXiv2024

Truncated multiplication and batch software SIMD AVX512 implementation for faster Montgomery multiplications and modular exponentiation

This paper presents software implementations of batch computations, dealing with multi-precision integer operations. In this work, we use the Single Instruction Multiple Data (SIMD) AVX512 instruction set of the x86-64 processors, in particular the vectorized fused multiplier-adder VPMADD52. We focus on batch multiplications, squarings, modular multiplications, modular squarings and constant time modular exponentiations of 8 values using a word-slicing storage. We explore the use of Schoolbook and Karatsuba approaches with operands up to 4108 and 4154 bits respectively. We also introduce a truncated multiplication that speeds up the computation of the Montgomery modular reduction in the context of software implementation. Our Truncated Montgomery modular multiplication improvement offers speed gains of almost 20 % over the conventional non-truncated versions. Compared to the state-of-the-art GMP and OpenSSL libraries, our speedup modular operations are more than 4 times faster. Compared to OpenSSL BN_mod_exp_mont_consttimex2 using AVX512 and madd52* (madd52hi or madd52lo) in 256-bit registers, in fixed-window exponentiations of sizes 1024 and 2048 , our 512-bit implementation provides speedups of respectively 1.75 and 1.38, while the 256-bit version speedups are 1.51 and 1.05 for 1024 and 2048 -bit sizes (batch of 4 values in this case).

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