Residue number system arithmetic: modern applications in digital signal processing
Residue number system arithmetic: modern applications in digital signal processing
Proceedings on Advances in cryptology---CRYPTO '86
A fast modular arithmetic algorithm using a residue table
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
Comparison of three modular reduction functions
CRYPTO '93 Proceedings of the 13th annual international cryptology conference on Advances in cryptology
A fast modular multiplication algorithm for calculating the product AB modulo N
Information Processing Letters
Architectural tradeoff in implementing RSA processors
ACM SIGARCH Computer Architecture News
Modular Exponentiation Using Recursive Sums of Residues
CRYPTO '89 Proceedings of the 9th Annual International Cryptology Conference on Advances in Cryptology
Faster Modular Multiplication by Operand Scaling
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
Montgomery's Multiplication Technique: How to Make It Smaller and Faster
CHES '99 Proceedings of the First International Workshop on Cryptographic Hardware and Embedded Systems
Implementation of RSA Algorithm Based on RNS Montgomery Multiplication
CHES '01 Proceedings of the Third International Workshop on Cryptographic Hardware and Embedded Systems
Simplifying Quotient Determination in High-Radix Modular Multiplication
ARITH '95 Proceedings of the 12th Symposium on Computer Arithmetic
Modular Multiplication and Base Extensions in Residue Number Systems
ARITH '01 Proceedings of the 15th IEEE Symposium on Computer Arithmetic
A Full RNS Implementation of RSA
IEEE Transactions on Computers
A Computer Algorithm for Calculating the Product AB Modulo M
IEEE Transactions on Computers
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In the 1980s, when the introduction of public key cryptography spurred interest in modular multiplication, many implementations performed modular multiplication using a sumof residues. As the fieldmatured, sum of residues modular multiplication lost favor to the extent that all recent surveys have either overlooked it or incorporated it within a larger class of reduction algorithms. In this paper, we present a new taxonomy of modular multiplication algorithms. We include sum of residues as one of four classes and argue why it should be considered different to the other, now more common, algorithms. We then apply techniques developed for other algorithms to reinvigorate sum of residues modular multiplication. We compare FPGA implementations of modular multiplication up to 24 bits wide. The Sum of Residues multipliers demonstrate reduced latency at nearly 50% compared to Montgomery architectures at the cost of nearly doubled circuit area. The new multipliers are useful for systems based on the Residue Number System (RNS).