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Arithmetic Operations in the Polynomial Modular Number System

J.-C. Bajard, L. Imbert, T. Plantard
<i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/n5ba5d6pxzhy3flbxdipvlyzrq" style="color: black;">17th IEEE Symposium on Computer Arithmetic (ARITH&#39;05)</a> </i> &nbsp;
The socalled Polynomial Modular Number System (PMNS) allows for fast polynomial arithmetic and easy parallelization.  ...  The most important contribution of this paper is the fundamental theorem of a Modular Number System, which provides a bound for the coefficients of the polynomials used to represent the set Z p .  ...  Imbert leave of absence at the university of Calgary, with the Centre for Information Security and Cryptography (CISaC) and the Advanced Technology Information Processing Systems (ATIPS) Laboratory.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/arith.2005.11">doi:10.1109/arith.2005.11</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/arith/BajardIP05.html">dblp:conf/arith/BajardIP05</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5avjov4o2netbisgf5673a6h6a">fatcat:5avjov4o2netbisgf5673a6h6a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808164818/http://www-pequan.lip6.fr/~bajard/MesPublis/ARITH17b.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9e/5c/9e5ce8e1a26c3e89167c9f7e419d893e47972d1a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/arith.2005.11"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Modular arithmetic and finite field theory

E. Horowitz
<span title="">1971</span> <i title="ACM Press"> Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC &#39;71 </i> &nbsp;
In addition, attention will be paid to certain practical problems which arise in the construction of a modular arithmetic system.  ...  Now, a modular arithmetic system for symbol manipulation must first provide a reasonable number of single precision primes, say 50-100.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/800204.806287">doi:10.1145/800204.806287</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dhyhlimhefcshjsgacfnbmptie">fatcat:dhyhlimhefcshjsgacfnbmptie</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20140628181125/http://f3.tiera.ru/3/M_Mathematics/MT_Number%20theory/Horowitz.%20Finite%20fields%20and%20modular%20arithmetic,%20tutorial(7s).pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/58/f1/58f1aa091a082e2a3fff445759cf64851eca2001.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/800204.806287"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

The basic polynomial algebra subprograms

Changbo Chen, Svyatoslav Covanov, Farnam Mansouri, Robert H. C. Moir, Marc Moreno Maza, Ning Xie, Yuzhen Xie
<span title="2016-11-04">2016</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/km5ozgn5evaidlgq5bosq7rm5q" style="color: black;">ACM Communications in Computer Algebra</a> </i> &nbsp;
The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer  ...  One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers.  ...  Acknowledgments This work was supported by the NSFC (11301524) and the CSTC (cstc2013jjys0002).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3015306.3015312">doi:10.1145/3015306.3015312</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cgsssin4ebht7mtcik3lumafe4">fatcat:cgsssin4ebht7mtcik3lumafe4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808003702/http://www.csd.uwo.ca/~moreno/Publications/ICMS-2014-BPAS.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b5/7e/b57ea3a28f4e9e59bad1c2068b35d71abe1dcc06.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3015306.3015312"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

The Basic Polynomial Algebra Subprograms [chapter]

Changbo Chen, Svyatoslav Covanov, Farnam Mansouri, Marc Moreno Maza, Ning Xie, Yuzhen Xie
<span title="">2014</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer  ...  One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers.  ...  This work was supported by the NSFC (11301524) and the CSTC (cstc2013jjys0002).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-44199-2_100">doi:10.1007/978-3-662-44199-2_100</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ix7c4a7t75c2zaqkf5pirh2ptq">fatcat:ix7c4a7t75c2zaqkf5pirh2ptq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20141014070058/http://www.csd.uwo.ca:80/~nxie6/papers/bpas.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b6/da/b6da117763b9f54d6d9a981a6727aae20a8a0f55.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-44199-2_100"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Basic Polynomial Algebra Subprograms

Changbo Chen, Svyatoslav Covanov, Farnam Mansouri, Marc Moreno Maza, Ning Xie, Yuzhen Xie
<span title="2015-02-05">2015</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/km5ozgn5evaidlgq5bosq7rm5q" style="color: black;">ACM Communications in Computer Algebra</a> </i> &nbsp;
The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer  ...  One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers.  ...  Acknowledgments This work was supported by the NSFC (11301524) and the CSTC (cstc2013jjys0002).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2733693.2733723">doi:10.1145/2733693.2733723</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v5cunis54nbgxmh7ttf5tcsbm4">fatcat:v5cunis54nbgxmh7ttf5tcsbm4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808003702/http://www.csd.uwo.ca/~moreno/Publications/ICMS-2014-BPAS.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b5/7e/b57ea3a28f4e9e59bad1c2068b35d71abe1dcc06.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2733693.2733723"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

A Bit-Serial Multiplier Architecture for Finite Fields Over Galois Fields

Modares
<span title="2010-11-01">2010</span> <i title="Science Publications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wake4w3hqzd4ndiy2zowpciv64" style="color: black;">Journal of Computer Science</a> </i> &nbsp;
In cryptography the most common finite field used is binary field GF (2 m ). Conclusion: Our design performs all basic binary polynomial operations in Galois Field (GF) using a microcode structure.  ...  Results: Thus, it plays an important role in cryptography. As a result of their carry free arithmetic property, they are suitable to be used in hardware implementation in ECC.  ...  In Modular arithmetic, over a number 'p' arithmetic covers the number in the interval [0 and p-1]. Modular Arithmetic Multiplication Inverse and Finding x mod y operations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3844/jcssp.2010.1237.1246">doi:10.3844/jcssp.2010.1237.1246</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/olzg7ziv3rfcvhiswwcvzlqnii">fatcat:olzg7ziv3rfcvhiswwcvzlqnii</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170923004303/http://thescipub.com/PDF/jcssp.2010.1237.1246.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/44/a2/44a29472c65da4e308d8af7d4a567f74edfb1403.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3844/jcssp.2010.1237.1246"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Dual-Field Arithmetic Unit for GF(p) and GF(2m) [chapter]

Johannes Wolkerstorfer
<span title="">2003</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Redundant number representation and efficient modular reduction make it ready for future cryptographic bitlengths and allow operation at high clock frequency on moderate hardware resources.  ...  In this article we present a hardware solution for finite field arithmetic with application in asymmetric cryptography. It supports calculation in GF (p) as well as in GF (2 m ).  ...  The required operations can be summarized in the following categories: integer arithmetic, modular integer arithmetic, modular polynomial arithmetic, and comparisons.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-36400-5_36">doi:10.1007/3-540-36400-5_36</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vretwkbzzngbhlymdrsehfzvce">fatcat:vretwkbzzngbhlymdrsehfzvce</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190503015744/https://link.springer.com/content/pdf/10.1007%2F3-540-36400-5_36.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d5/e4/d5e43b4cacd4ce6cc1314b8bd81d4cb2a591df0a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-36400-5_36"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Multifunction Residue Architectures for Cryptography

Dimitrios Schinianakis, Thanos Stouraitis
<span title="">2014</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/l6r6h4xpzvh6jkuzrx565pgwgu" style="color: black;">IEEE Transactions on Circuits and Systems Part 1: Regular Papers</a> </i> &nbsp;
A design methodology for incorporating Residue Number System (RNS) and Polynomial Residue Number System (PRNS) in Montgomery modular multiplication in or respectively, as well as a VLSI architecture of  ...  A versatile architecture is derived that supports all operations of Montgomery multiplication in and , input/output conversions, Mixed Radix Conversion (MRC) for integers and polynomials, dual-field modular  ...  , Fellow, IEEE Abstract-A design methodology for incorporating Residue Number System (RNS) and Polynomial Residue Number System (PRNS) in Montgomery modular multiplication in or respectively, as well as  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcsi.2013.2283674">doi:10.1109/tcsi.2013.2283674</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qvr6hohdl5hpjet5ouhyzrwd2y">fatcat:qvr6hohdl5hpjet5ouhyzrwd2y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170317003445/http://kresttechnology.com/krest-academic-projects/krest-mtech-projects/ECE/M.Tech%20VLSI%20PAPERS%202016-2017/papers/75.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/34/27/34278df0b3d49ccee28e1a09dc40c376ade8c7c0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcsi.2013.2283674"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Verifying Arithmetic Assembly Programs in Cryptographic Primitives (Invited Talk)

Andy Polyakov, Ming-Hsien Tsai, Bow-Yaw Wang, Bo-Yin Yang, Michael Wagner
<span title="2018-08-13">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iv4yo5vao5ctfjfushi4akt5xi" style="color: black;">International Conference on Concurrency Theory</a> </i> &nbsp;
Arithmetic over large finite fields is indispensable in modern cryptography. For efficienty, these operations are often implemented in manually optimized assembly programs.  ...  We develop techniques to verify such programs automatically in this paper. Using our techniques, we have successfully verified a number of assembly programs in OpenSSL.  ...  In modular polynomial abstraction, program behaviors are modeled by solutions to systems of modular polynomial equations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.concur.2018.4">doi:10.4230/lipics.concur.2018.4</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/concur/PolyakovTWY18.html">dblp:conf/concur/PolyakovTWY18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/okbarjdsuvcpbnfaxclg4xobqy">fatcat:okbarjdsuvcpbnfaxclg4xobqy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220121221422/https://drops.dagstuhl.de/opus/volltexte/2018/9542/pdf/LIPIcs-CONCUR-2018-4.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b5/3c/b53c0dab8f7cec3839f36ac8cb28f8f1f463ea74.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.concur.2018.4"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Toward high-performance polynomial system solvers based on triangular decompositions

Xin Li
<span title="2010-06-24">2010</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hab6y3yvcbd2hpuybkxo2pp34a" style="color: black;">ACM SIGSAM Bulletin</a> </i> &nbsp;
GCD without fast arithmetic. AXIOM modular GCD use fast arithmetic. ◮ Huge factor comparing with the Maple's latest implementation. ◮ In AXIOM, replacing only the modular multivariate operation.  ...  ◮ Multiplication time: M(d) number of coefficient operations conducted for a univariate polynomial multiplication in degree less than d.  ...  . ◮ If deg(G , y ) = 1 then V (P, Q) can be decomposed at the cost of computing R that is O ∼ (d 2 2 d 1 ) operations in k. ◮ Otherwise the decompsition is obtained within O ∼ (d 3 2 d 1 ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1823931.1823956">doi:10.1145/1823931.1823956</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xqix6emmrfgndf7rqvsspfywpu">fatcat:xqix6emmrfgndf7rqvsspfywpu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110905055858/http://www.csd.uwo.ca:80/~moreno/Publications/thesisTalk.Xin-Li.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/60/3e/603ee6526a2b71e1c0486f8fe7b572abfdb92ab6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1823931.1823956"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Polynomial Residue Number Ssystem Gf(2M) Multiplier Using Trinomials

M. Benaissa, Junfeng Chu
<span title="2009-08-24">2009</span> <i title="Zenodo"> Zenodo </i> &nbsp;
Publication in the conference proceedings of EUSIPCO, Glasgow, Scotland, 2009  ...  In the Polynomial Residue Number System (PRNS), each channel is generated by a polynomial instead of a prime number as in the typical RNS.  ...  In this paper, the work in [23] is further improved by adopting trinomial irreducible polynomials for the PRNS channels to simplify the modular reduction and conversion operations thereby resulting in  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5281/zenodo.41423">doi:10.5281/zenodo.41423</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lk6rrg6ahzcvtmnokebxkehvla">fatcat:lk6rrg6ahzcvtmnokebxkehvla</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170814193129/https://zenodo.org/record/41423/files/1569188950.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/be/12/be129b5b90ffa8f1ad367038c1135d6b205a3bcc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5281/zenodo.41423"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> zenodo.org </button> </a>

In-place arithmetic for polynomials over Zn [chapter]

Michael Monagan
<span title="">1993</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We present space and time efficient algorithms for univariate polynomial arithmetic operations over Z mod n where the modulus n does not necessarily fit into is not a machine word.  ...  These algorithms provide the key tools for the efficient implementation of polynomial resultant gcd and factorization computation over Z, without having to write large amounts of code in a systems implementation  ...  The second method [14] is a multiple modular method. Gcds are computed over Zp, [x]/(a) for word size prime moduli pi and combined by application of the Chinese remainder theorem.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-57272-4_21">doi:10.1007/3-540-57272-4_21</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cnum46gw4bbkhjy7tvuae6uenq">fatcat:cnum46gw4bbkhjy7tvuae6uenq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20120103123627/http://www.cecm.sfu.ca/personal/monaganm/papers/DISCO92.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f6/1e/f61ef988992800faafdf01e56e3525c162774557.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-57272-4_21"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Automatic Library Generation for Modular Polynomial Multiplication [article]

Lingchuan Meng
<span title="2016-09-05">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Polynomial multiplication is a key algorithm underlying computer algebra systems (CAS) and its efficient implementation is crucial for the performance of CAS.  ...  In this paper we design and implement algorithms for polynomial multiplication using approaches based the fast Fourier transform (FFT) and the truncated Fourier transform (TFT).  ...  [39] also contains hand-optimized implementations of modular arithmetic in its standard libraries for large numbers, polynomials, etc. based on FFT and other fast algorithms.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1609.01010v1">arXiv:1609.01010v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/73syghptxja5tjb5ifmfomv7uu">fatcat:73syghptxja5tjb5ifmfomv7uu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191023220036/https://arxiv.org/pdf/1609.01010v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b0/11/b011d2a5e2321cbc2542b64b60d563886e56e891.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1609.01010v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Efficient Implementation of NTRU Cryptography using Residue Number System

Azin Zalekian, Mohammad Esmaeildoust, Amer Kaabi
<span title="2015-08-18">2015</span> <i title="Foundation of Computer Science"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/b637noqf3vhmhjevdfk3h5pdsu" style="color: black;">International Journal of Computer Applications</a> </i> &nbsp;
In order to fast implementation of NTRU, hardware implementation of NTRU by employing residue number system is presented.  ...  The NTRU cryptography is a lattice-based public key cryptography. Encryption and decryption process in NTRU are based on polynomial multiplication.  ...  Residue Number System (RNS) is a non-weighted number system, which arithmetic operation like addition, multiplication and subtraction can be done over small moduli [7] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5120/ijca2015905527">doi:10.5120/ijca2015905527</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zjqds22jgrfbhlabl7gydxzk7i">fatcat:zjqds22jgrfbhlabl7gydxzk7i</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180603143155/https://www.ijcaonline.org/research/volume124/number7/zalekian-2015-ijca-905527.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ad/69/ad69dfcd4cb6bea09b18720852c0297c6ec13635.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5120/ijca2015905527"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Randomization of Arithmetic Over Polynomial Modular Number System

Laurent-Stephane Didier, Fangan-Yssouf Dosso, Nadia El Mrabet, Jeremy Marrez, Pascal Veron
<span title="">2019</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/n5ba5d6pxzhy3flbxdipvlyzrq" style="color: black;">2019 IEEE 26th Symposium on Computer Arithmetic (ARITH)</a> </i> &nbsp;
The Polynomial Modular Number System (PMNS) is an integer number system designed to speed up arithmetic operations modulo a prime p.  ...  We show how to randomize the modular multiplication in order to be safe against existing SCA and we demonstrate the resistance of our construction.  ...  In order to speed up modular arithmetic, specific representations of integers such as the Residue Number System (RNS [17] , [3] ) or the Polynomial Modular Number System (PMNS [5] , [6] , [25] ,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/arith.2019.00048">doi:10.1109/arith.2019.00048</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/arith/DidierDMMV19.html">dblp:conf/arith/DidierDMMV19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/coindfhgjjfp3e6fzpw4asd7mm">fatcat:coindfhgjjfp3e6fzpw4asd7mm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200311135845/https://hal.archives-ouvertes.fr/hal-02099713/document" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c7/17/c717fc10c50add0821ab00a896d89348287eb58b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/arith.2019.00048"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>
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