Fast Power-of-Two RNS Scaling Algorithm for Large Dynamic Ranges

Isupov, Konstantin; Knyazkov, Vladimir; Kuvaev, Alexander

doi:10.1109/icent.2017.36

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…All known scaling techniques can be classified into four main categories: scaling by the product of some RNS moduli [ 32 , 35 , 36 , 37 , 38 ], scaling by an integer from the RNS number range [ 39 , 40 ], scaling by a common fraction [ 41 ], scaling by a power of two [ 42 , 43 , 44 ]. …”

Section: The Main Types Of Scaling Algorithms In Rns Arithmeticmentioning

confidence: 99%

“…In [ 43 ], the power-of-two scaling technique is applied to realize a digital filter in quadratic RNS. The scaling algorithm presented in [ 44 ] focuses on arbitrary moduli sets with large dynamic ranges and requires only machine-precision integer and floating-point operations. At the same time, it is used for software implementation of rounding and exponent alignment procedures in a multiple-precision RNS-based arithmetic library for parallel CPU-GPU systems.…”

Section: The Main Types Of Scaling Algorithms In Rns Arithmeticmentioning

confidence: 99%

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An Efficient CRT-Base Power-of-Two Scaling in Minimally Redundant Residue Number System

Selianinau

Povstenko

2022

Entropy

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In this paper, we consider one of the key problems in modular arithmetic. It is known that scaling in the residue number system (RNS) is a rather complicated non-modular procedure, which requires expensive and complex operations at each iteration. Hence, it is time consuming and needs too much hardware for implementation. We propose a novel approach to power-of-two scaling based on the Chinese Remainder Theorem (CRT) and rank form of the number representation in RNS. By using minimal redundancy of residue code, we optimize and speed up the rank calculation and parity determination of divisible integers in each iteration. The proposed enhancements make the power-of-two scaling simpler and faster than the currently known methods. After calculating the rank of the initial number, each iteration of modular scaling by two is performed in one modular clock cycle. The computational complexity of the proposed method of scaling by a constant Sl=2l associated with both required modular addition operations and lookup tables is estimeted as k and 2k+1, respectively, where k equals the number of primary non-redundant RNS moduli. The time complexity is log2k+l modular clock cycles.

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Section: The Main Types Of Scaling Algorithms In Rns Arithmeticmentioning

confidence: 99%

Section: The Main Types Of Scaling Algorithms In Rns Arithmeticmentioning

confidence: 99%

An Efficient CRT-Base Power-of-Two Scaling in Minimally Redundant Residue Number System

Selianinau

Povstenko

2022

Entropy

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“…5) It is necessary to perform the expensive conversion to binary representation after each arithmetic operation to be able to extract the state of the arithmetic results. 6) Division by a constant is difficult to implement with RNS representation [34]- [37]. 7) It is inefficient to perform the division operation with RNS representation due to the combination of iterated subtractions and comparisons operations [38]- [43].…”

Section: Related Workmentioning

confidence: 99%

Fast Large Integer Modular Addition in GF(p) Using Novel Attribute-Based Representation

Alhazmi

Gebali

2019

IEEE Access

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Addition is an essential operation in all cryptographic algorithms. Higher levels of security require larger key sizes and this becomes a limiting factor in GF(p) using large integers because of the carry propagation problem. We propose a novel and efficient attribute-based large integer representation scheme suitable for large integers commonly used in cryptography such as the five NIST primes and the Pierpont primes used in supersingular isogeny Diffie-Hellman (SIDH) for post-quantum cryptography. Algorithms are proposed for this new representation to implement arithmetic operations such as two's complement, addition/subtraction, comparison, sign detection, and modular reduction. Algorithms are also developed for converting binary numbers to attribute representation and vice versa. The extensive numerical simulations were done to verify the performance of the new number representation. Results show that addition is done faster in our proposed representation when compared with binary and residue number system (RNS)-based additions. Attribute addition outperformed RNS addition for all values of m where 128 ≤ m ≤ 32 768 bits for all machine word sizes w where 4 ≤ w ≤ 128 bits. Attribute-based addition outperforms Kogge-Stone binary adders for a wide range of m when w is small. For increasing values of w, the speed advantages are evident only for large values of m. This makes the proposed number representation suitable for implementing cryptographic applications in embedded processors for IoT and consumer electronic devices where w is small. INDEX TERMS Prime fields GF(p), large integer arithmetic, modular arithmetic, Kogge-Stone adder, number representation, post-quantum cryptography, SIDH, cryptographic processor, embedded systems, NIST primes, generalized pierpont prime, parallel algorithms.BADER ALHAZMI received the B.Sc. degree in electrical and computer engineering from King Abdulaziz University, Jeddah, Saudi Arabia, and the master's degree in information systems security from Concordia University, Montreal, QC, Canada. He is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, University of Victoria, Canada. His research interests include cryptosystems, hardware security, computer arithmetic, and parallel algorithms.FAYEZ GEBALI received the B.Sc. degree (Hons.) in electrical engineering from Cairo University, the B.Sc. degree (Hons.) in mathematics from Ain Shams University, and the Ph.D. degree in electrical engineering from the University of British Columbia. He is currently a Professor of electrical and computer engineering with the University of Victoria. His research interests include parallel algorithms, dataflow computing, finite-field arithmetic, and radar signal processing.

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Multiple-Precision Scaled Vector Addition on Graphics Processing Unit

Isupov

Kuvaev

2019

Lecture Notes in Computer Science

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Fast Power-of-Two RNS Scaling Algorithm for Large Dynamic Ranges

Cited by 6 publications

References 21 publications

An Efficient CRT-Base Power-of-Two Scaling in Minimally Redundant Residue Number System

An Efficient CRT-Base Power-of-Two Scaling in Minimally Redundant Residue Number System

Fast Large Integer Modular Addition in GF(p) Using Novel Attribute-Based Representation

Multiple-Precision Scaled Vector Addition on Graphics Processing Unit

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