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Fast Integer Multiplication using Modular Arithmetic
[article]

2008
*
arXiv
*
pre-print

Thus, we show that the two seemingly different approaches to

arXiv:0801.1416v3
fatcat:7vvehkcxmnferjdimsqdz2s2gi
*integer**multiplication*,*modular*and complex*arithmetic*, are similar. ... We give an O(N· N· 2^O(^*N)) algorithm for multiplying two N-bit*integers*that improves the O(N· N· N) algorithm by Schönhage-Strassen. Both these algorithms*use**modular**arithmetic*. ... Schönhage and Strassen introduced two seemingly different approaches to*integer**multiplication*-*using*complex and*modular**arithmetic*. ...##
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Fast Integer Multiplication Using Modular Arithmetic

2013
*
SIAM journal on computing (Print)
*

We give an N · log N · 2 O(log * N ) time algorithm to multiply two N -bit

doi:10.1137/100811167
fatcat:7jy5vrfat5gmjbuuhfeu3cm5xu
*integers*that*uses**modular**arithmetic*for intermediate computations instead of*arithmetic*over complex numbers as in Fürer's algorithm ... The previous best algorithm*using**modular**arithmetic*(by Schönhage and Strassen) has complexity O(N · log N · log log N ). ... The Motivation Schönhage and Strassen introduced two seemingly different approaches to*integer**multiplication*-*using*complex and*modular**arithmetic*. ...##
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Fast integer multiplication using modular arithmetic

2008
*
Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08
*

We give an N · log N · 2 O(log * N ) time algorithm to multiply two N -bit

doi:10.1145/1374376.1374447
dblp:conf/stoc/DeKSS08
fatcat:y2dslsj3jfewjav4ulseq4hm3u
*integers*that*uses**modular**arithmetic*for intermediate computations instead of*arithmetic*over complex numbers as in Fürer's algorithm ... The previous best algorithm*using**modular**arithmetic*(by Schönhage and Strassen) has complexity O(N · log N · log log N ). ... The Motivation Schönhage and Strassen introduced two seemingly different approaches to*integer**multiplication*-*using*complex and*modular**arithmetic*. ...##
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Automatic Generation of Vectorized Montgomery Algorithm
[article]

2016
*
arXiv
*
pre-print

*Modular*

*arithmetic*is widely

*used*in crytography and symbolic computation. ... This paper presents a vectorized Montgomery algorithm for

*modular*

*multiplication*, the key to

*fast*

*modular*

*arithmetic*, that fully utilizes the SIMD instructions. ... with

*integer*blend

*Fast*Algorithms In this section, we review the

*fast*algorithms for

*modular*

*arithmetic*operations, with a particular focus on the pivotal

*modular*

*multiplication*, since the

*modular*...

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More Generalized Mersenne Numbers
[chapter]

2004
*
Lecture Notes in Computer Science
*

We also show that it is possible to perform long

doi:10.1007/978-3-540-24654-1_24
fatcat:nbfyp7dmrrfatagquqtvg6k4ae
*integer**modular**arithmetic*without*using**multiple*precision operations when t is chosen properly. ... It is shown that such p's lead to*fast**modular*reduction methods which*use*only a few*integer*additions and subtractions. We further generalize this idea by allowing any*integer*for t. ... Such representations lead to*fast**modular*reduction which*uses*only a few*integer*additions and subtractions. ...##
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Optimal extension fields for fast arithmetic in public-key algorithms
[chapter]

1998
*
Lecture Notes in Computer Science
*

This contribution introduces a class of Galois eld

doi:10.1007/bfb0055748
fatcat:7fclfpqsyrej3jqy4h6excogdi
*used*to achieve*fast*nite eld*arithmetic*which we call an Optimal Extension Field (OEF). ... Our construction employs well-known techniques for*fast*nite eld*arithmetic*which fully exploit the*fast**integer**arithmetic*found on these processors. ... It is well known that*fast**modular*reduction is possible with moduli of the form 2 n c, where c is a \small"*integer*.*Integers*of this form allow*modular*reduction without division. ...##
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A Review on Fast FPGA Development of RSD based ECC Processor

2018
*
International Journal of Trend in Scientific Research and Development
*

Furthermore, an pwerfull smodular adder without comparison and a highthroughput

doi:10.31142/ijtsrd7166
fatcat:fak5lseqizbi7okb24firkhqay
*modular*divider, which results in a short datapath for maximized frequency, are implemented. ... The processor employs extensive pipelining techniques for Karatsuba-Ofman method to achieve high throughput*multiplication*. ... Left-right point*multiplication*method: The RSD delgation, first popularized by Avizienisis a carry free*arithmetic*where*integers*are expressed by the difference of two other*integers*. ...##
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Parallel computation of residue number system

2006
*
2006 International Conference on Computing & Informatics
*

Some recently proposed

doi:10.1109/icoci.2006.5276434
fatcat:5vpjn3m6njdyvo4hquftq55qzu
*modular**arithmetic*operations are stated and employed to accelerate the computation. ... Chinese remainder theorem (CRT), an old and famous theorem, is widely*used*in many modern computer applications. ... Montgomery Reduction and the*Fast**Modular**Arithmetic*The division operation is intuitionally necessary for*modular**arithmetic*. However, a division is much more expansive than a*multiplication*. ...##
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CNOT-count optimized quantum circuit of the Shor's algorithm
[article]

2021
*
arXiv
*
pre-print

We present improved quantum circuit for

arXiv:2112.11358v1
fatcat:yzzjnpv6e5dpthmkjn6hkg7uni
*modular*exponentiation of a constant, which is the most expensive operation in Shor's algorithm for*integer*factorization. ... First, we give the implementation of basic*arithmetic*with known lowest number of CNOT gate and the construction of improved*modular*exponentiation of a constant by accumulating intermediate date and windowing ... Roetteler et al.[22] showed the quantum cir- cuits of two approaches to compute*modular**multiplication*of two factors both in the form of quantum state:*Fast**modular**multiplication*[24] and Montgomery*modular*...##
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Special issue in honor of Peter Lawrence Montgomery

2018
*
Journal of Cryptographic Engineering
*

Cheung present a tutorial on spectral

doi:10.1007/s13389-017-0168-3
fatcat:4a5fgu6bezepjna4nd6lodvkgy
*arithmetic**using*Montgomery*modular**multiplication*. ... Such algorithms require*integer*and/or*modular**arithmetic*on large*integer*operands, and this means that the computational efficiency of public-key cryptography heavily depends on the computational cost ...##
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Modular arithmetic and finite field theory

1971
*
Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC '71
*

A second

doi:10.1145/800204.806287
fatcat:dhyhlimhefcshjsgacfnbmptie
*use*of*modular**arithmetic*has been in the area of polynomial factorization over the field of rationals. However, the advantage gained here is not the ability for*fast**multiplication*. ... for constructing*fast*algorithms. Related to the theory of*modular**arithmetic*is the theory of finite fields. ...##
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Efficient FPGA-based ECDSA Verification Engine for Permissioned Blockchains
[article]

2021
*
arXiv
*
pre-print

We propose several optimizations for

arXiv:2112.02229v1
fatcat:r3bw4kn4tnhzvhh3cjqugnj65m
*modular**arithmetic*(e.g., custom multipliers and*fast**modular*reduction) and point*arithmetic*(e.g., reduced number of point double and addition operations, and optimal ... Based on these optimized*modular*and point*arithmetic*modules, we propose an ECDSA verification engine that can be*used*by any application for*fast*verification of ECDSA signatures. ... 8: end if FPGA, we implement all*modular**arithmetic*modules*using*multi- 9: Return(𝑐) word*integer**arithmetic*[12]. ...##
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Low-Power Elliptic Curve Cryptography Using Scaled Modular Arithmetic
[chapter]

2004
*
Lecture Notes in Computer Science
*

The scaling technique may be

doi:10.1007/978-3-540-28632-5_7
fatcat:m434epm46bgxlmdp7it4mogxdy
*used*to improve*multiplication*and inversion in finite fields. We present an efficient inversion algorithm that utilizes the structure of scaled modulus. ... We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables efficient*arithmetic*. ... To alleviate the reduction problem in*integer**modular**multiplications*, Crandall proposed [3]*using*special primes, primes of the form p = 2 k − u, where u is a small*integer*constant. ...##
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Computing A*B (mod N) efficiently in ANSI C

1992
*
SIGPLAN notices
*

The

doi:10.1145/130722.130735
fatcat:5rmmpzh3g5fzjlbt36i6rug5ou
*modular*product computation A*B (mod N) is a bottleneck for some public-key encryption algorithms, as well as many exact computations implemented*using*the Chinese Remainder Theorem. ... INTRODUCTION Many exact*integer*computations, rather than being performed*using**multiple*-precision*arithmetic*, are performed instead over a number of single-precision*modular*rings or fields, with the ... However, since 0≤R<N<W/2<W, we can more efficiently calculate R*using*single-precision ANSI C unsigned (*modular*)*arithmetic*. ...##
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Optimal Extension Fields for XTR
[chapter]

2003
*
Lecture Notes in Computer Science
*

in GF (p 2m ) to achieve

doi:10.1007/3-540-36492-7_24
fatcat:qn2xh5z3h5fs7bjgk4pgd3u3uu
*fast*finite field*arithmetic*in GF (p 2m ). ... In order to select such fields, we introduce a new notion of Generalized Optimal Extension Fields(GOEFs) and suggest a condition of prime p, a defining polynomial of GF (p 2m ) and a*fast*method of*multiplication*...*Fast*Subfield*Multiplication*with*Modular*Reduction In general,*fast*subfield*multiplication*is essential for*fast**multiplication*in GF (p m ). ...
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