# New frameworks for Montgomery's modular multiplication method

@article{McLaughlin2004NewFF, title={New frameworks for Montgomery's modular multiplication method}, author={Philip B. McLaughlin}, journal={Math. Comput.}, year={2004}, volume={73}, pages={899-906} }

We present frameworks for fast modular multiplication based on a modification of Montgomery's original method. For (fixed) large integers, our algorithms may be significantly faster than conventional methods. Our techniques may also be extended to modular polynomial arithmetic.

## 16 Citations

Spectral arithmetic in Montgomery modular multiplication

- Computer Science, MathematicsJournal of Cryptographic Engineering
- 2017

This survey paper introduces the development of spectral-based MMM, as well as its two important properties: high parallelism and low complexity, and compares these algorithms in terms of digit-level complexity.

A Generalized RNS Mclaughlin Modular Multiplication with Non-Coprime Moduli Sets

- Mathematics, Computer ScienceIEEE Transactions on Computers
- 2019

A set of moduli that are non-coprime for RNS in the algorithm to take both the advantage of the fewer multiplications required for a modular multiplication in McLaughlin modular multiplication and theadvantage of the moduli sets of similar sizes in classic Montgomery modularmultiplication in RNS.

Trinomial bases and Chinese remaindering for modular polynomial multiplication

- Computer Science, Mathematics
- 2008

This paper shows how to keep the same residue basis for modular polynomial multiplication, inspired by l’Hospital rule, and obtains improved estimates on the number of additions for some useful degree ranges.

FFT-Based McLaughlin's Montgomery Exponentiation without Conditional Selections

- Computer ScienceIEEE Transactions on Computers
- 2018

This work restricts the parameters of MLM by a set of new bounds and presents a modified MLM algorithm involving no conditional selection, which improves both area-time efficiency and security against timing attacks.

Subquadratic Polynomial Multiplication over GF(2m) Using Trinomial Bases and Chinese Remaindering

- Computer Science, MathematicsSelected Areas in Cryptography
- 2008

This paper shows how to keep the same residue basis for modular polynomial multiplication, inspired by l'Hospital rule, and obtains improved estimates on the number of additions for some useful degree ranges.

Fast convolutions meet Montgomery

- MathematicsMath. Comput.
- 2008

This paper gives a method for understanding and bypassing the short multiplication problem, thus reducing the costs of ring arithmetic to roughly 2M(R) when also using fast convolutions.

Area-Time Efficient Architecture of FFT-Based Montgomery Multiplication

- Computer ScienceIEEE Transactions on Computers
- 2017

This paper integrates the fast Fourier transform (FFT) method into the McLaughlin’s framework, and presents an improved FFT-based Montgomery modular multiplication (MMM) algorithm achieving high area-time efficiency.

Modern Computer Arithmetic

- Computer Science
- 2010

Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics…

Parameter Space for the Architecture of FFT-Based Montgomery Modular Multiplication

- Computer Science, MathematicsIEEE Transactions on Computers
- 2016

Improvements to FFT-based Montgomery Modular Multiplication (FFTM3) using carry-save arithmetic and pre-computation techniques are presented and pseudo-Fermat number transform is used to enrich the supported operand sizes for the FFTM3.

An efficient FPGA-based accelerator design for convolution

- Computer Science2017 IEEE 8th International Conference on Awareness Science and Technology (iCAST)
- 2017

A new efficient architecture of the transform have been proposed which support a various operand size and a variant constant geometry architecture is used which the forward and backward sub-stage used the same computation pattern.

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