Fast modular reduction and squaring in GF(2m)
We present an efficient bit-parallel algorithm for squaring in GF(2m) using polynomial basis. This algorithm achieves competitive efficiency while being aimed at any choice of low-weight irreducible polynomial. For a large class of irreducible polynomials it is more efficient than the previously bes...
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| Vydáno v: | Information processing letters Ročník 132; s. 33 - 38 |
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Elsevier B.V
01.04.2018
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | We present an efficient bit-parallel algorithm for squaring in GF(2m) using polynomial basis. This algorithm achieves competitive efficiency while being aimed at any choice of low-weight irreducible polynomial. For a large class of irreducible polynomials it is more efficient than the previously best general squarer. In contrast, other efficient squarers often require a change of basis or are suitable for only a small number of irreducible polynomials. Additionally, we present a simple algorithm for modular reduction with equivalent cost to the state of the art for general irreducible polynomials. This fast reduction is used in our squaring method.
•We propose a new algorithm for squaring elements in binary finite field extensions.•The algorithm is very efficient for squaring elements defined using any low weight polynomial.•A general efficient algorithm for polynomial modular reduction of any weight is described.•The square algorithm is extended to finite field extensions of any odd characteristic. |
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| AbstractList | We present an efficient bit-parallel algorithm for squaring in GF(2m) using polynomial basis. This algorithm achieves competitive efficiency while being aimed at any choice of low-weight irreducible polynomial. For a large class of irreducible polynomials it is more efficient than the previously best general squarer. In contrast, other efficient squarers often require a change of basis or are suitable for only a small number of irreducible polynomials. Additionally, we present a simple algorithm for modular reduction with equivalent cost to the state of the art for general irreducible polynomials. This fast reduction is used in our squaring method.
•We propose a new algorithm for squaring elements in binary finite field extensions.•The algorithm is very efficient for squaring elements defined using any low weight polynomial.•A general efficient algorithm for polynomial modular reduction of any weight is described.•The square algorithm is extended to finite field extensions of any odd characteristic. |
| Author | Custódio, R. Niehues, L. Boppre Panario, D. |
| Author_xml | – sequence: 1 givenname: L. Boppre orcidid: 0000-0003-1285-492X surname: Niehues fullname: Niehues, L. Boppre email: lucasboppre@inf.ufsc.br organization: Department of Informatics and Statistics, Federal University of Santa Catarina, Brazil – sequence: 2 givenname: R. surname: Custódio fullname: Custódio, R. email: ricardo.custodio@ufsc.br organization: Department of Informatics and Statistics, Federal University of Santa Catarina, Brazil – sequence: 3 givenname: D. surname: Panario fullname: Panario, D. email: daniel@math.carleton.ca organization: School of Mathematics and Statistics, Carleton University, Canada |
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| Cites_doi | 10.1109/TC.2002.1017695 10.1109/TC.2009.70 10.1016/j.ffa.2014.10.008 10.1049/el.2014.0006 10.1016/j.vlsi.2011.07.005 |
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| References | Fan, Hasan (br0010) 2015; 32 Hariri, Reyhani-Masoleh (br0080) 2009; 58 Wu (br0050) 2002; 51 Doche (br0060) 2005 Adrian (br0030) 2015 Hankerson, Menezes, Vanstone (br0040) 2006 Park (br0070) 2012; 45 Xiong, Fan (br0090) 2014; 50 Bernstein, Lange, Niederhagen (br0020) 2016 Bernstein (10.1016/j.ipl.2017.12.002_br0020) 2016 Hankerson (10.1016/j.ipl.2017.12.002_br0040) 2006 Wu (10.1016/j.ipl.2017.12.002_br0050) 2002; 51 Fan (10.1016/j.ipl.2017.12.002_br0010) 2015; 32 Adrian (10.1016/j.ipl.2017.12.002_br0030) 2015 Doche (10.1016/j.ipl.2017.12.002_br0060) 2005 Park (10.1016/j.ipl.2017.12.002_br0070) 2012; 45 Hariri (10.1016/j.ipl.2017.12.002_br0080) 2009; 58 Xiong (10.1016/j.ipl.2017.12.002_br0090) 2014; 50 |
| References_xml | – volume: 51 start-page: 750 year: 2002 end-page: 758 ident: br0050 article-title: Bit-parallel finite field multiplier and squarer using polynomial basis publication-title: IEEE Trans. Comput. – volume: 32 start-page: 5 year: 2015 end-page: 43 ident: br0010 article-title: A survey of some recent bit-parallel publication-title: Finite Fields Appl. – start-page: 122 year: 2005 end-page: 133 ident: br0060 article-title: Redundant trinomials for finite fields of characteristic 2 publication-title: Information Security and Privacy – volume: 45 start-page: 205 year: 2012 end-page: 210 ident: br0070 article-title: Explicit formulae of polynomial basis squarer for pentanomials using weakly dual basis publication-title: Integr. VLSI J. – volume: 50 start-page: 655 year: 2014 end-page: 657 ident: br0090 article-title: bit-parallel squarer using generalised polynomial basis for new class of irreducible pentanomials publication-title: Electron. Lett. – start-page: 5 year: 2015 end-page: 17 ident: br0030 article-title: Imperfect forward secrecy: how Diffie–Hellman fails in practice publication-title: Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security – year: 2006 ident: br0040 article-title: Guide to Elliptic Curve Cryptography – volume: 58 start-page: 1332 year: 2009 end-page: 1345 ident: br0080 article-title: Bit-serial and bit-parallel Montgomery multiplication and squaring over publication-title: IEEE Trans. Comput. – start-page: 256 year: 2016 end-page: 281 ident: br0020 article-title: Dual EC: a standardized back door publication-title: The New Codebreakers – year: 2006 ident: 10.1016/j.ipl.2017.12.002_br0040 – volume: 51 start-page: 750 issue: 7 year: 2002 ident: 10.1016/j.ipl.2017.12.002_br0050 article-title: Bit-parallel finite field multiplier and squarer using polynomial basis publication-title: IEEE Trans. Comput. doi: 10.1109/TC.2002.1017695 – volume: 58 start-page: 1332 issue: 10 year: 2009 ident: 10.1016/j.ipl.2017.12.002_br0080 article-title: Bit-serial and bit-parallel Montgomery multiplication and squaring over GF(2m) publication-title: IEEE Trans. Comput. doi: 10.1109/TC.2009.70 – volume: 32 start-page: 5 year: 2015 ident: 10.1016/j.ipl.2017.12.002_br0010 article-title: A survey of some recent bit-parallel GF(2n) multipliers publication-title: Finite Fields Appl. doi: 10.1016/j.ffa.2014.10.008 – start-page: 122 year: 2005 ident: 10.1016/j.ipl.2017.12.002_br0060 article-title: Redundant trinomials for finite fields of characteristic 2 – start-page: 256 year: 2016 ident: 10.1016/j.ipl.2017.12.002_br0020 article-title: Dual EC: a standardized back door – volume: 50 start-page: 655 issue: 9 year: 2014 ident: 10.1016/j.ipl.2017.12.002_br0090 article-title: GF(2n) bit-parallel squarer using generalised polynomial basis for new class of irreducible pentanomials publication-title: Electron. Lett. doi: 10.1049/el.2014.0006 – start-page: 5 year: 2015 ident: 10.1016/j.ipl.2017.12.002_br0030 article-title: Imperfect forward secrecy: how Diffie–Hellman fails in practice – volume: 45 start-page: 205 issue: 2 year: 2012 ident: 10.1016/j.ipl.2017.12.002_br0070 article-title: Explicit formulae of polynomial basis squarer for pentanomials using weakly dual basis publication-title: Integr. VLSI J. doi: 10.1016/j.vlsi.2011.07.005 |
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| SubjectTerms | Cryptography Finite field Number theory Polynomial Squaring |
| Title | Fast modular reduction and squaring in GF(2m) |
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