scholarly journals A Division Algorithm in a Redundant Residue Number System Using Fractions

2020 ◽  
Vol 10 (2) ◽  
pp. 695
Author(s):  
Nikolay Chervyakov ◽  
Pavel Lyakhov ◽  
Mikhail Babenko ◽  
Irina Lavrinenko ◽  
Maxim Deryabin ◽  
...  
Keyword(s):  
Chinese Remainder Theorem ◽  
Arithmetic Operation ◽  
Number System ◽  
Residue Number System ◽  
Experimental Simulation ◽  
Division Algorithm ◽  
Reverse Conversion ◽  
Base Extension ◽  
Mixed Radix Conversion ◽  
Residue Number

The residue number system (RNS) is widely used for data processing. However, division in the RNS is a rather complicated arithmetic operation, since it requires expensive and complex operators at each iteration, which requires a lot of hardware and time. In this paper, we propose a new modular division algorithm based on the Chinese remainder theorem (CRT) with fractional numbers, which allows using only one shift operation by one digit and subtraction in each iteration of the RNS division. The proposed approach makes it possible to replace such expensive operations as reverse conversion based on CRT, mixed radix conversion, and base extension by subtraction. Besides, we optimized the operation of determining the most significant bit of divider with a single shift operation of the modular divider. The proposed enhancements make the algorithm simpler and faster in comparison with currently known algorithms. The experimental simulation using Kintex-7 showed that the proposed method is up to 7.6 times faster than the CRT-based approach and is up to 10.1 times faster than the mixed radix conversion approach.

scholarly journals New Algorithm for Reverse Conversion in Residue Number System

2021 ◽  
Vol 10 (1) ◽  
pp. 1-4
Author(s):  
Daniel Asiedu ◽  
Abdul-Mumin Salifu
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Multiplicative Inverse ◽  
Computational Overhead ◽  
Reverse Conversion ◽  
Mixed Radix Conversion ◽  
Residue Number

Reverse conversion is an important exercise in achieving the properties of Residue Number System (RNS). Current algorithms available for reverse conversion exhibits greater computational overhead in terms of speed and area. In this paper, we have developed a new algorithm for reverse conversion for two-moduli set and three-moduli set that are very simple and with fewer multiplicative inverse operations than there are in the traditional algorithms like the Chinese Remainder Theorem (CRT) and Mixed Radix Conversion (MRC).

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 Reverse convertor design for the 4-moduli set {2ⁿ -1,2ⁿ, 2ⁿ +1,2²ⁿ+¹ -1} based on the mixed-radix conversion

2011 ◽  
Vol 24 (1) ◽  
pp. 89-103
Author(s):  
Negovan Stamenkovic ◽  
Bojan Jovanovic
Keyword(s):  
High Speed ◽  
Number System ◽  
Residue Number System ◽  
Hardware Architecture ◽  
Reverse Conversion ◽  
Reverse Converter ◽  
Mixed Radix Conversion ◽  
Residue Number

The residue number system (RNS) is an integer system capable of supporting high speed concurrent arithmetic. One of the most important consideration when designing RNS system is reverse conversion. The reverse converter for recently proposed for the four-moduli set {2? -1,2?, 2? +1,2??+? -1} is based on new Chinese remainder theorems II (New CRT-II) [6]. This paper presents an alternative architecture derived by Mixed-Radix conversion for this four-moduli set. Due to the using simple multiplicative inverses of the proposed moduli set, it can considerably reduce the complexity of the RNS to binary converter based on the Mixed-Radix conversion. The hardware architecture for the proposed converter is based on the adders and subtractors, without the needed ROM or multipliers.

scholarly journals Neural network method for base extension in residue number system

2020 ◽  
Author(s):  
M. Babenko ◽  
E. Shiriaev ◽  
A. Tchernykh ◽  
E. Golimblevskaia
Keyword(s):  
Neural Network ◽  
Elliptic Curve ◽  
Chinese Remainder Theorem ◽  
Homomorphic Encryption ◽  
Computational Cost ◽  
Number System ◽  
Residue Number System ◽  
Pseudorandom Sequence ◽  
Base Extension ◽  
Residue Number

Confidential data security is associated with the cryptographic primitives, asymmetric encryption, elliptic curve cryptography, homomorphic encryption, cryptographic pseudorandom sequence generators based on an elliptic curve, etc. For their efficient implementation is often used Residue Number System that allows executing additions and multiplications on parallel computing channels without bit carrying between channels. A critical operation in Residue Number System implementations of asymmetric cryptosystems is base extension. It refers to the computing a residue in the extended moduli without the application of the traditional Chinese Remainder Theorem algorithm. In this work, we propose a new way to perform base extensions using a Neural Network of a final ring. We show that it reduces 11.7% of the computational cost, compared with state-of-the-art approaches.

Area Efficient Memoryless Reverse Converter for New Four Moduli Set {2n−1,2n−1,2n+1,22n+1−1}

2018 ◽  
Vol 27 (05) ◽  
pp. 1850075 ◽  
Author(s):  
Ritesh Kumar Jaiswal ◽  
Raj Kumar ◽  
Ram Awadh Mishra
Keyword(s):  
Dynamic Range ◽  
Number System ◽  
Residue Number System ◽  
Hardware Complexity ◽  
Reverse Conversion ◽  
Reverse Converter ◽  
Mixed Radix Conversion ◽  
Residue Number ◽  
Least Area ◽  
State Of Art

The efficiency of residue number system depends on the reverse converter due to several modulo operations like addition, subtraction and multiplication. In this paper, a design of new four moduli set [Formula: see text], reverse converter is presented. The moduli set have moduli with length ranging from ([Formula: see text]) to ([Formula: see text])-bits. The reverse conversion for moduli set [Formula: see text] has been optimized in existing state of art. Thus, proposed converter is based on two new moduli set [Formula: see text] and utilizes the mixed radix conversion. This converter is memoryless, and occupies least area. The proposed converter is based on carry save adder (CSA) and modulo adder enabling more speed and less hardware complexity for dynamic range of [Formula: see text]-bit, offering good area-delay product.

scholarly journals Construction of Residue Number System Using Hardware Efficient Diagonal Function

Electronics ◽  
2019 ◽  
Vol 8 (6) ◽  
pp. 694 ◽  
Author(s):  
Maria Valueva ◽  
Georgii Valuev ◽  
Nataliya Semyonova ◽  
Pavel Lyakhov ◽  
Nikolay Chervyakov ◽  
...  
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Magnitude Comparison ◽  
Reverse Conversion ◽  
Circuit Delay ◽  
Residue Number ◽  
Hardware Costs ◽  
Positional Number System ◽  
Diagonal Function

The residue number system (RNS) is a non-positional number system that allows one to perform addition and multiplication operations fast and in parallel. However, because the RNS is a non-positional number system, magnitude comparison of numbers in RNS form is impossible, so a division operation and an operation of reverse conversion into a positional form containing magnitude comparison operations are impossible too. Therefore, RNS has disadvantages in that some operations in RNS, such as reverse conversion into positional form, magnitude comparison, and division of numbers are problematic. One of the approaches to solve this problem is using the diagonal function (DF). In this paper, we propose a method of RNS construction with a convenient form of DF, which leads to the calculations modulo 2 n , 2 n − 1 or 2 n + 1 and allows us to design efficient hardware implementations. We constructed a hardware simulation of magnitude comparison and reverse conversion into a positional form using RNS with different moduli sets constructed by our proposed method, and used different approaches to perform magnitude comparison and reverse conversion: DF, Chinese remainder theorem (CRT) and CRT with fractional values (CRTf). Hardware modeling was performed on Xilinx Artix 7 xc7a200tfbg484-2 in Vivado 2016.3 and the strategy of synthesis was highly area optimized. The hardware simulation of magnitude comparison shows that, for three moduli, the proposed method allows us to reduce hardware resources by 5.98–49.72% in comparison with known methods. For the four moduli, the proposed method reduces delay by 4.92–21.95% and hardware costs by twice as much by comparison to known methods. A comparison of simulation results from the proposed moduli sets and balanced moduli sets shows that the use of these proposed moduli sets allows up to twice the reduction in circuit delay, although, in several cases, it requires more hardware resources than balanced moduli sets.

LARGE DYNAMIC RANGE RNS SYSTEMS AND THEIR RESIDUE TO BINARY CONVERTERS

2007 ◽  
Vol 16 (02) ◽  
pp. 267-286 ◽  
Author(s):  
ALEXANDER SKAVANTZOS ◽  
MOHAMMAD ABDALLAH ◽  
THANOS STOURAITIS
Keyword(s):  
High Speed ◽  
Dynamic Range ◽  
Chinese Remainder Theorem ◽  
Digital Signal ◽  
Number System ◽  
Residue Number System ◽  
New Class ◽  
Mixed Radix Conversion ◽  
Residue Number ◽  
Signal Processors

The Residue Number System (RNS) is an integer system appropriate for implementing fast digital signal processors. It can be used for supporting high-speed arithmetic by operating in parallel channels without need for exchanging information among the channels. In this paper, two novel RNS are proposed. First, a new RNS system based on the modulus set {2n+1, 2n - 1, 2n + 1, 2n + 2(n+1)/2 + 1, 2n - 2(n+1)/2 + 1}, n odd, is developed, along with an efficient implementation of its residue-to-weighted converter. The new RNS is a balanced five-modulus system, appropriate for large dynamic ranges. The proposed residue-to-binary converter is fast and hardware efficient and is based on a one's complement multi-operand adder that adds operands of size only 80% of the size dictated by the system's dynamic range. Second, a new class of multi-modulus RNS systems is proposed. These systems are based on sets consisting of two groups of moduli with the modulus product within one group being of the form 2a(2b - 1), while the modulus product within the other group is of the form 2c - 1. Their RNS-to-weighted converters are based on efficient combinations of the Chinese Remainder Theorem and Mixed Radix Conversion decoding techniques. Systems based on four, five, and seven moduli are constructed and analyzed. The new systems allow efficient implementations for their RNS-to-weighted decoders, imply fast and balanced RNS arithmetic, and may achieve large dynamic ranges. The presented residue-to-weighted converters for these systems rely on simple mod (2x - 1) hardware, which can be easily implemented as one's complement hardware.

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.

scholarly journals Zero-Aware Low-Precision RNS Scaling Scheme

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 5
Author(s):  
Amir Sabbagh Molahosseini
Keyword(s):  
High Precision ◽  
Deep Neural Networks ◽  
State Of The Art ◽  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Scaling Factor ◽  
Improve Performance ◽  
High Area ◽  
Residue Number

Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2k modulo as the scaling factor, which results in a high-precision output with a high area and delay. Therefore, low-precision scaling based on multi-moduli scaling factors should be used to improve performance. However, low-precision scaling for numbers less than the scale factor results in zero output, which makes the subsequent operation result faulty. This paper first presents the formulation and hardware architecture of low-precision RNS scaling for four-moduli sets using new Chinese remainder theorem 2 (New CRT-II) based on a two-moduli scaling factor. Next, the low-precision scaler circuits are reused to achieve a high-precision scaler with the minimum overhead. Therefore, the proposed scaler can detect the zero output after low-precision scaling and then transform low-precision scaled residues to high precision to prevent zero output when the input number is not zero.

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