scholarly journals Practical Evaluation of Protected Residue Number System Scalar Multiplication

2018 ◽  
pp. 259-282
Author(s):  
Louiza Papachristodoulou ◽  
Apostolos P. Fournaris ◽  
Kostas Papagiannopoulos ◽  
Lejla Batina
Keyword(s):  
Public Key Cryptography ◽  
Number System ◽  
Residue Number System ◽  
Scalar Multiplication ◽  
Public Key ◽  
Side Channel ◽  
Evaluation Approach ◽  
Rings Of Integers ◽  
Residue Number ◽  
Template Attacks

The Residue Number System (RNS) arithmetic is gaining grounds in public key cryptography, because it offers fast, efficient and secure implementations over large prime fields or rings of integers. In this paper, we propose a generic, thorough and analytic evaluation approach for protected scalar multiplication implementations with RNS and traditional Side Channel Attack (SCA) countermeasures in an effort to assess the SCA resistance of RNS. This paper constitutes the first robust evaluation of RNS software for Elliptic Curve Cryptography against electromagnetic (EM) side-channel attacks. Four different countermeasures, namely scalar and point randomization, random base permutations and random moduli operation sequence, are implemented and evaluated using the Test Vector Leakage Assessment (TVLA) and template attacks. More specifically, variations of RNS-based Montgomery Powering Ladder scalar multiplication algorithms are evaluated on an ARM Cortex A8 processor using an EM probe for acquisition of the traces. We show experimentally and theoretically that new bounds should be put forward when TVLA evaluations on public key algorithms are performed. On the security of RNS, our data and location dependent template attacks show that even protected implementations are vulnerable to these attacks. A combination of RNS-based countermeasures is the best way to protect against side-channel leakage.

scholarly journals Revisiting Sum of Residues Modular Multiplication

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Yinan Kong ◽  
Braden Phillips
Keyword(s):  
Public Key Cryptography ◽  
Number System ◽  
Residue Number System ◽  
The Other ◽  
Public Key ◽  
Modular Multiplication ◽  
Residue Number ◽  
The Cost

In the 1980s, when the introduction of public key cryptography spurred interest in modular multiplication, many implementations performed modular multiplication using a sum of residues. As the field matured, 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).

scholarly journals Design of a Modular Multiplier for Public-Key Cryptography Applications Using Residue Number System and Signed-Digit Representation

2020 ◽  
Author(s):  
Mohammad Hizzani
Keyword(s):  
Public Key Cryptography ◽  
Number System ◽  
Residue Number System ◽  
Public Key ◽  
Modular Multiplication ◽  
Secret Key ◽  
Public Key Cryptosystems ◽  
Number Systems ◽  
Wide Range ◽  
Residue Number

Public-Key Cryptosystems are prone to wide range of cryptanalyses due to its property of having key pairs one of them is public. Therefore, the recommended length of these keys is extremely large (e.g. in RSA and D-H the key is at least 2048 bits long) and this leads the computation of such cryptosystems to be slower than the secret-key cryptosystems (i.e. AES and AES-family). Since, the key operation in such systems is the modular multiplication; in this research a novel design for the modular multiplication based on the Montgomery Multiplication, the Residue Number Systems for moduli of any form, and the Signed-Digit Representation is proposed. The proposed design outperforms the current designs in the literature in terms of delay with at least 28% faster for the key of 2048 bits long. Up to our knowledge, this design is the first design that utilizes Signed-Digit Representation with the Residue Number System for moduli of any form.

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

scholarly journals Computationally Efficient Approach to Implementation of the Chinese Remainder Theorem Algorithm in Minimally Redundant Residue Number System

2021 ◽  
Author(s):  
Mikhail Selianinau
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Computer Applications ◽  
Efficient Approach ◽  
Residue Modulo ◽  
Modern Computer ◽  
New Variant ◽  
Residue Number ◽  
Residue Code

AbstractIn this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. Throughout the article, an efficient approach for implementing the CRT algorithm is described. The structure of the rank of an RNS number, a principal positional characteristic of the residue code, is investigated. It is shown that the rank of a number can be represented by a sum of an inexact rank and a two-valued correction to it. We propose a new variant of minimally redundant RNS, which provides low computational complexity for the rank calculation, and its effectiveness analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the excess residue modulo 2, the complexity of the rank calculation goes down from $O\left (k^{2}\right )$ O k 2 to $O\left (k\right )$ O k with respect to required modular addition operations and lookup tables, where k equals the number of non-redundant RNS moduli.

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