Modular multiplication using the core function in the residue number system

2015 ◽  
Vol 27 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Yinan Kong ◽  
Shahzad Asif ◽  
Mohammad A. U. Khan
Keyword(s):  
Number System ◽  
Residue Number System ◽  
Modular Multiplication ◽  
The Core ◽  
Core Function ◽  
Residue Number

scholarly journals Improved Modular Division Implementation with the Akushsky Core Function

Computation ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 9
Author(s):  
Mikhail Babenko ◽  
Andrei Tchernykh ◽  
Viktor Kuchukov
Keyword(s):  
Approximate Method ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Division Algorithm ◽  
Incorrect Result ◽  
Core Function ◽  
Residue Number ◽  
Division Operation

The residue number system (RNS) is widely used in different areas due to the efficiency of modular addition and multiplication operations. However, non-modular operations, such as sign and division operations, are computationally complex. A fractional representation based on the Chinese remainder theorem is widely used. In some cases, this method gives an incorrect result associated with round-off calculation errors. In this paper, we optimize the division operation in RNS using the Akushsky core function without critical cores. We show that the proposed method reduces the size of the operands by half and does not require additional restrictions on the divisor as in the division algorithm in RNS based on the approximate method.

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 Efficient Montgomery Modular Multiplication by using Residue Number System

2012 ◽  
Vol 2 (1) ◽  
pp. 56-62
Author(s):  
Elham Khani
Keyword(s):  
Elliptic Curve Cryptography ◽  
Arithmetic Operation ◽  
Digital Signal ◽  
Number System ◽  
Residue Number System ◽  
Binary Number ◽  
Modular Multiplication ◽  
Free System ◽  
Montgomery Modular Multiplication ◽  
Residue Number

Residue number system is a carry free system that performs arithmetic operation on residues instead of the weighted binary number. By applying Residue Number System (RNS) to Montgomery modular multiplication the delay of modular multiplication will be decreased. Modular multiplication over large number is frequently used in some application such as Elliptic Curve Cryptography, digital signal processing, and etc.By choosing appropriate RNS moduli sets the time consuming operation of multiplication can be replaced by smaller operations.  In addition because of the property of RNS, arithmetic operations are done over smaller numbers called residues. In this paper by choosing appropriate moduli sets the efficiency of conversion from RNS to RNS that is the most time consuming part of the Montgomery modular multiplication will be increased.

Fast modular multiplication execution in residue number system

2016 ◽  
Author(s):  
Nikolai I. Chervyakov ◽  
Mikhail G. Babenko ◽  
Viktor A. Kuchukov ◽  
Maxim A. Deryabin ◽  
Nataliya N. Kuchukova ◽  
...  
Keyword(s):  
Number System ◽  
Residue Number System ◽  
Modular Multiplication ◽  
Residue Number

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.

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