Residue Number Systems - Theory and Implementation

Omondi, Amos R.; Premkumar, Benjamin

doi:10.1142/9781860948671

Cited by 218 publications

(288 citation statements)

References 12 publications

Supporting

Mentioning

281

Contrasting

Unclassified

Order By: Relevance

“…[10,11]). Readers interested in learning more about the theoretical and practical aspects of RNS, as well as some of its applications, can consult the book by Omondi and Premkumar [4].…”

Section: Residue Number Systemsmentioning

confidence: 99%

“…Thus, it can be accomplished by standard table-lookup, followed by modular multioperand addition [4]. The fact that the moduli may be nonintegers does not complicate either of the two steps.…”

Section: Arithmetic In Cd-rnsmentioning

confidence: 99%

“…Unfortunately, standard methods based on the CRT and its derivatives [4,13] are inapplicable to CD-RNS. It suffices to show the inapplicability of CRT in the special case of a CD-RNS using moduli that are scaled-down versions of a conventional RNS with integer moduli.…”

Section: Arithmetic In Cd-rnsmentioning

confidence: 99%

“…1d [3]), would, in principle, allow the rat to derive full location information. Again, the analog in digital arithmetic is the identification of an integer R from several residues, via the Chinese remainder theorem (CRT) [4].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Digital Arithmetic in Nature: Continuous-Digit RNS

Parhami

2014

The Computer Journal

View full text Add to dashboard Cite

It has been reported in the literature on computational neuroscience that a rat's uncanny ability to dash back to a home position in the absence of any visual clues (or in total darkness, for that matter) may stem from its distinctive method of position representation. More specifically, it is hypothesized that the rat uses a multimodular method akin to residue number system (RNS), but with continuous residues or digits, to encode position information. After a brief review of the evidence in support of this hypothesis, and how it relates to RNS, we discuss the properties of continuous-digit RNS, and derive results on the dynamic range, representational accuracy and factors affecting the choice of the moduli, which are themselves real numbers. We conclude with suggestions for further research on important open problems concerning the process of selection, or evolutionary refinement, of the set of moduli in such a representation.

show abstract

“…[10,11]). Readers interested in learning more about the theoretical and practical aspects of RNS, as well as some of its applications, can consult the book by Omondi and Premkumar [4].…”

Section: Residue Number Systemsmentioning

confidence: 99%

Section: Arithmetic In Cd-rnsmentioning

confidence: 99%

Section: Arithmetic In Cd-rnsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Digital Arithmetic in Nature: Continuous-Digit RNS

Parhami

2014

The Computer Journal

View full text Add to dashboard Cite

show abstract

“…Residue number systems (RNS) have been studied for nearly as long as electronic digital computers [Svob59], and are still attracting considerable attention from the research community [Omon07]. They have found niche applications for certain arithmetic-intensive digital signal processing (DSP) problems [Sode86].…”

Section: Introductionmentioning

confidence: 99%

On equivalences and fair comparisons among residue number systems with special moduli

Parhami

2010

2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers

View full text Add to dashboard Cite

show abstract

An acceleration of quasigroup operations by residue arithmetic

Krömer

Platoš

Nowaková

et al. 2017

Concurrency and Computation

View full text Add to dashboard Cite

Quasigroup operations are essential for a wide range of cryptographic procedures that includes cryptographic hash functions, electronic signatures, pseudorandom number generators, and stream and block ciphers. Quasigroup cryptography achieves high levels of security at low memory and computational costs by an iterative application of quasigroup operations to streams and blocks of data. The use of large quasigroups can further improve the strength of cryptographic operations. However, the order of used quasigroups is the main factor affecting the memory requirements of quasigroup cryptographic schemes. Alternative quasigroup representations that do not store their multiplication tables in computer memory yield increased computational costs.In any case, an efficient implementation of quasigroup operations is critical for practical applications of quasigroup cryptography. Residue number systems allow a fast, concurrent realization of addition and multiplication. In this work, residue arithmetic is used to accelerate quasigroup operations, and an efficient computational approach to their implementation, designed with respect to the extended instruction sets of modern processors, is proposed. KEYWORDSacceleration, quasigroup operations, residue arithmetic, residue number systems, security INTRODUCTIONQuasigroups are algebraic structures with a large number of applications in cryptography and computer security. 1-5 A quasigroup consists of a set of elements (carrier) and a binary operation (multiplication) that maps every 2 of its elements onto a third element. Mathematically, they are grupoids closely related to the combinatorial concept of latin squares 2,3,6,7 and the geometric notion of k−nets (3−nets). 2,8 Multiplication (Cayley) tables of finite quasigroups can be expressed as latin squares, and k−nets can be coordinatized by systems of orthogonal quasigroups.In general, quasigroups are noncommutative, nonassociative, nonidempotent, and noninvolutory and do not have neutral elements. 9 These properties make them suitable for applications in cryptography, and they have been used as fundamental elements of a wide range of cryptographic algorithms with success. Their applications in this area include novel high-performance stream 4-6,10-13 and block 4,5,14-22 ciphers, cryptographic hash 4,5,23-27 and trapdoor 4,14,23 functions, fast and secure electronic signatures (eg, message authentication codes), 4,14,23,28 pseudorandom number generators, 5,29-31 error-detecting 2 and error-correcting 2,9,32 codes, and simultaneous encryption and error-correction coding (cryptcoding). 4Quasigroups can be also used to model cryptosystems based on other cryptographic primitives. A quasigroup representation has been devised for block ciphers based on Feistel networks (ie, Feistel and generalized Feistel ciphers). 33The attractiveness of quasigroups for practical cryptographic applications consists in the combination of provably strong security and inexpensive realization of quasigroup-based cryptographic operations. They support la...

show abstract

Residue Number Systems - Theory and Implementation

Cited by 218 publications

References 12 publications

Digital Arithmetic in Nature: Continuous-Digit RNS

Digital Arithmetic in Nature: Continuous-Digit RNS

On equivalences and fair comparisons among residue number systems with special moduli

An acceleration of quasigroup operations by residue arithmetic

Contact Info

Product

Resources

About