Montgomery reduction with even modulus

1994 ◽  
Vol 141 (5) ◽  
pp. 314 ◽  
Author(s):  
C.K. Koc
Keyword(s):  
Montgomery Reduction

scholarly journals Montgomery reduction within the context of residue number system arithmetic

2017 ◽  
Vol 8 (3) ◽  
pp. 189-200 ◽  
Author(s):  
Jean-Claude Bajard ◽  
Julien Eynard ◽  
Nabil Merkiche
Keyword(s):  
Number System ◽  
Residue Number System ◽  
Residue Number ◽  
Montgomery Reduction

Hybrid Montgomery Reduction

2016 ◽  
Vol 15 (3) ◽  
pp. 1-13 ◽  
Author(s):  
Hwajeong Seo ◽  
Zhe Liu ◽  
Yasuyuki Nogami ◽  
Jongseok Choi ◽  
Howon Kim
Keyword(s):  
Montgomery Reduction

scholarly journals Montgomery Reduction for Gaussian Integers

Cryptography ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 6
Author(s):  
Malek Safieh ◽  
Jürgen Freudenberger
Keyword(s):  
Elliptic Curve ◽  
Initial Point ◽  
Modular Arithmetic ◽  
Complex Numbers ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Point Multiplication ◽  
Prime Fields ◽  
Integer Rings ◽  
Montgomery Reduction

Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

scholarly journals Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods

2010 ◽  
Vol 59 (12) ◽  
pp. 1715-1721 ◽  
Author(s):  
Miroslav Knezevic ◽  
Frederik Vercauteren ◽  
Ingrid Verbauwhede
Keyword(s):  
Modular Multiplication ◽  
Reduction Methods ◽  
Montgomery Reduction

scholarly journals RNS Montgomery reduction algorithms using quadratic residuosity

2018 ◽  
Vol 9 (4) ◽  
pp. 313-331 ◽  
Author(s):  
Shinichi Kawamura ◽  
Yuichi Komano ◽  
Hideo Shimizu ◽  
Tomoko Yonemura
Keyword(s):  
Quadratic Residuosity ◽  
Montgomery Reduction

scholarly journals SIDH on ARM: Faster Modular Multiplications for Faster Post-Quantum Supersingular Isogeny Key Exchange

2018 ◽  
pp. 1-20 ◽  
Author(s):  
Hwajeong Seo ◽  
Zhe Liu ◽  
Patrick Longa ◽  
Zhi Hu
Keyword(s):  
High Speed ◽  
High Performance ◽  
Key Exchange ◽  
Modular Arithmetic ◽  
Diffie Hellman ◽  
Real World Applications ◽  
Diffie Hellman Key Exchange ◽  
Memory Accesses ◽  
Montgomery Reduction ◽  
Cache Attacks

We present high-speed implementations of the post-quantum supersingular isogeny Diffie-Hellman key exchange (SIDH) and the supersingular isogeny key encapsulation (SIKE) protocols for 32-bit ARMv7-A processors with NEON support. The high performance of our implementations is mainly due to carefully optimized multiprecision and modular arithmetic that finely integrates both ARM and NEON instructions in order to reduce the number of pipeline stalls and memory accesses, and a new Montgomery reduction technique that combines the use of the UMAAL instruction with a variant of the hybrid-scanning approach. In addition, we present efficient implementations of SIDH and SIKE for 64-bit ARMv8-A processors, based on a high-speed Montgomery multiplication that leverages the power of 64-bit instructions. Our experimental results consolidate the practicality of supersingular isogeny-based protocols for many real-world applications. For example, a full key-exchange execution of SIDHp503 is performed in about 176 million cycles on an ARM Cortex-A15 from the ARMv7-A family (i.e., 88 milliseconds @2.0GHz). On an ARM Cortex-A72 from the ARMv8-A family, the same operation can be carried out in about 90 million cycles (i.e., 45 milliseconds @1.992GHz). All our software is protected against timing and cache attacks. The techniques for modular multiplication presented in this work have broad applications to other cryptographic schemes.

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