An Approximate Method for Comparing Modular Numbers and its Application to the Division of Numbers in Residue Number Systems*

Chervyakov, Nikolay I.; Babenko, Mikhail; Lyakhov, Pavel; Lavrinenko, Irina

doi:10.1007/s10559-014-9689-2

Cited by 35 publications

(20 citation statements)

References 7 publications

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“…The authors of [13] offer to use RNS to improve the performance of the convolutional neural network developed for pattern recognition tasks. Reference [14] describes the method of construction for finite impulse response filers using RNS.However, the limitations of RNS include some operations such as reverse conversion into positional form, magnitude comparison and division of numbers in RNS [15,16]. These limitations exist because RNS is a non-positional number system, and magnitude comparison of numbers in RNS form is impossible, so the division operation consists of a magnitude comparison operation that is also a problematic operation.…”

mentioning

confidence: 99%

“…However, the limitations of RNS include some operations such as reverse conversion into positional form, magnitude comparison and division of numbers in RNS [15,16]. These limitations exist because RNS is a non-positional number system, and magnitude comparison of numbers in RNS form is impossible, so the division operation consists of a magnitude comparison operation that is also a problematic operation.…”

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confidence: 99%

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Construction of Residue Number System Using Hardware Efficient Diagonal Function

et al. 2019

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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. such as field-programmable gate array (FPGA) and application-specific integrated circuit (ASIC). All these attractive features increase interest to RNS in the areas where large amounts of computation are needed. The applications of RNS are digital signal processing [2][3][4], cryptography [5][6][7], digital image processing [8], cloud computing [9], Internet of Things [10] and others. In [11], the authors propose a technique to estimate real-valued numbers by means of the Chinese remainder theorem (CRT), employing for this goal a Kroenecker based M-Estimation, to improve robustness. A new method based on the Chinese remainder theorem (CRT) is proposed for absolute position computation in [12]. This has advantages in terms of hardware and flexibility because it does not use memory. The authors of [13] offer to use RNS to improve the performance of the convolutional neural network developed for pattern recognition tasks. Reference [14] describes the method of construction for finite impulse response filers using RNS.However, the limitations of RNS include some operations such a...

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confidence: 99%

Construction of Residue Number System Using Hardware Efficient Diagonal Function

et al. 2019

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“…When RNS numbers are replaced by their approximate characteristics, an important issue is the accuracy of the representation [F(a)] 2 −N that guarantees correct results of the division operation. In accordance with the experimental studies [30,32,37], the values of N that are used for rounding and also for restoring the positional representation of numbers can be insufficient in several cases. This aspect may considerably restrict the performance of the device.…”

Section: Experimental Performance Analysis: New Modular Division Algomentioning

confidence: 82%

“…As was demonstrated in [32], N = log 2 (Pρ) bits after the decimal point have to be used for rounding F(a) without considerable errors that would affect calculation accuracy. In other words, there exists a bijection between the set of numbers in the RNS representation and the set of numbers rounded to the Nth bit, i.e., [F(a)] 2 −N .…”

Section: The Rounding Of F(a) Causes Inevitable Errors Introduce Thementioning

confidence: 99%

A High-Speed Division Algorithm for Modular Numbers Based on the Chinese Remainder Theorem with Fractions and Its Hardware Implementation

et al. 2019

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In this paper, a new simplified iterative division algorithm for modular numbers that is optimized on the basis of the Chinese remainder theorem (CRT) with fractions is developed. It requires less computational resources than the CRT with integers and mixed radix number systems (MRNS). The main idea of the algorithm is (a) to transform the residual representation of the dividend and divisor into a weighted fixed-point code and (b) to find the higher power of 2 in the divisor written in a residue number system (RNS). This information is acquired using the CRT with fractions: higher power is defined by the number of zeros standing before the first significant digit. All intermediate calculations of the algorithm involve the operations of right shift and subtraction, which explains its good performance. Due to the abovementioned techniques, the algorithm has higher speed and consumes less computational resources, thereby being more appropriate for the multidigit division of modular numbers than the algorithms described earlier. The new algorithm suggested in this paper has O (log2 Q) iterations, where Q is the quotient. For multidigit numbers, its modular division complexity is Q(N), where N denotes the number of bits in a certain fraction required to restore the number by remainders. Since the number N is written in a weighed system, the subtraction-based comparison runs very fast. Hence, this algorithm might be the best currently available.

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“…Next, we consider a modification of the Chinese remainder theorem using fractional quantities, which we will denote CRTd [10][11][12]. If both parts of formula (3) are divided by , then we obtain the relation…”

Section: Xf • X =mentioning

confidence: 99%

Computationally secure threshold secret sharing scheme with minimal redundancy

Babenko¹,

Tchernykh²,

Golimblevskaia³

et al. 2020

The International Workshop on Information, Computation, and Control Systems for Distributed Environments

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When designing and using distributed storage systems with cloud technology, the security issues become crucial. One of the promising mechanisms is the computationally secure threshold secret sharing scheme. We propose a computationally secure secret sharing scheme based on the minimally redundant modular code. It reduces the computational complexity of data encoding and decoding and reduce data redundancy. We show that it is computationally secure and provides data redundancy equivalent to the redundancy of the Rabin system. We demonstrate that the minimally redundant modular code does not satisfy the criterion of compactness of a sequence, but it can be used as an asymptotically ideal secret sharing scheme.

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An Approximate Method for Comparing Modular Numbers and its Application to the Division of Numbers in Residue Number Systems*

Cited by 35 publications

References 7 publications

Construction of Residue Number System Using Hardware Efficient Diagonal Function

Construction of Residue Number System Using Hardware Efficient Diagonal Function

A High-Speed Division Algorithm for Modular Numbers Based on the Chinese Remainder Theorem with Fractions and Its Hardware Implementation

Computationally secure threshold secret sharing scheme with minimal redundancy

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