Improvement of FPPR method to solve ECDLP

Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Huang, Yun-Ju [verfasserIn] Petit, Christophe Shinohara, Naoyuki Takagi, Tsuyoshi

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2015

Schlagwörter:	Elliptic curve Discrete logarithm problem Index calculus Multivariable polynomial system Gröbner basis

Anmerkung:	© Huang et al. 2015

Übergeordnetes Werk:	Enthalten in: Pacific journal of mathematics for industry - Heidelberg : Springer, 2015, 7(2015), 1 vom: 25. März
Übergeordnetes Werk:	volume:7 ; year:2015 ; number:1 ; day:25 ; month:03

Links:	Volltext

DOI / URN:	10.1186/s40736-015-0012-6

Katalog-ID:	SPR03720355X

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520			\|a Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough.
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allfields	10.1186/s40736-015-0012-6 doi (DE-627)SPR03720355X (SPR)s40736-015-0012-6-e DE-627 ger DE-627 rakwb eng Huang, Yun-Ju verfasserin aut Improvement of FPPR method to solve ECDLP 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Huang et al. 2015 Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. Elliptic curve (dpeaa)DE-He213 Discrete logarithm problem (dpeaa)DE-He213 Index calculus (dpeaa)DE-He213 Multivariable polynomial system (dpeaa)DE-He213 Gröbner basis (dpeaa)DE-He213 Petit, Christophe aut Shinohara, Naoyuki aut Takagi, Tsuyoshi aut Enthalten in Pacific journal of mathematics for industry Heidelberg : Springer, 2015 7(2015), 1 vom: 25. März (DE-627)815914628 (DE-600)2806666-2 2198-4115 nnns volume:7 year:2015 number:1 day:25 month:03 https://dx.doi.org/10.1186/s40736-015-0012-6 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2111 GBV_ILN_2129 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2015 1 25 03
spelling	10.1186/s40736-015-0012-6 doi (DE-627)SPR03720355X (SPR)s40736-015-0012-6-e DE-627 ger DE-627 rakwb eng Huang, Yun-Ju verfasserin aut Improvement of FPPR method to solve ECDLP 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Huang et al. 2015 Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. Elliptic curve (dpeaa)DE-He213 Discrete logarithm problem (dpeaa)DE-He213 Index calculus (dpeaa)DE-He213 Multivariable polynomial system (dpeaa)DE-He213 Gröbner basis (dpeaa)DE-He213 Petit, Christophe aut Shinohara, Naoyuki aut Takagi, Tsuyoshi aut Enthalten in Pacific journal of mathematics for industry Heidelberg : Springer, 2015 7(2015), 1 vom: 25. März (DE-627)815914628 (DE-600)2806666-2 2198-4115 nnns volume:7 year:2015 number:1 day:25 month:03 https://dx.doi.org/10.1186/s40736-015-0012-6 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2111 GBV_ILN_2129 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2015 1 25 03
allfields_unstemmed	10.1186/s40736-015-0012-6 doi (DE-627)SPR03720355X (SPR)s40736-015-0012-6-e DE-627 ger DE-627 rakwb eng Huang, Yun-Ju verfasserin aut Improvement of FPPR method to solve ECDLP 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Huang et al. 2015 Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. Elliptic curve (dpeaa)DE-He213 Discrete logarithm problem (dpeaa)DE-He213 Index calculus (dpeaa)DE-He213 Multivariable polynomial system (dpeaa)DE-He213 Gröbner basis (dpeaa)DE-He213 Petit, Christophe aut Shinohara, Naoyuki aut Takagi, Tsuyoshi aut Enthalten in Pacific journal of mathematics for industry Heidelberg : Springer, 2015 7(2015), 1 vom: 25. März (DE-627)815914628 (DE-600)2806666-2 2198-4115 nnns volume:7 year:2015 number:1 day:25 month:03 https://dx.doi.org/10.1186/s40736-015-0012-6 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2111 GBV_ILN_2129 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2015 1 25 03
allfieldsGer	10.1186/s40736-015-0012-6 doi (DE-627)SPR03720355X (SPR)s40736-015-0012-6-e DE-627 ger DE-627 rakwb eng Huang, Yun-Ju verfasserin aut Improvement of FPPR method to solve ECDLP 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Huang et al. 2015 Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. Elliptic curve (dpeaa)DE-He213 Discrete logarithm problem (dpeaa)DE-He213 Index calculus (dpeaa)DE-He213 Multivariable polynomial system (dpeaa)DE-He213 Gröbner basis (dpeaa)DE-He213 Petit, Christophe aut Shinohara, Naoyuki aut Takagi, Tsuyoshi aut Enthalten in Pacific journal of mathematics for industry Heidelberg : Springer, 2015 7(2015), 1 vom: 25. März (DE-627)815914628 (DE-600)2806666-2 2198-4115 nnns volume:7 year:2015 number:1 day:25 month:03 https://dx.doi.org/10.1186/s40736-015-0012-6 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2111 GBV_ILN_2129 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2015 1 25 03
allfieldsSound	10.1186/s40736-015-0012-6 doi (DE-627)SPR03720355X (SPR)s40736-015-0012-6-e DE-627 ger DE-627 rakwb eng Huang, Yun-Ju verfasserin aut Improvement of FPPR method to solve ECDLP 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Huang et al. 2015 Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. Elliptic curve (dpeaa)DE-He213 Discrete logarithm problem (dpeaa)DE-He213 Index calculus (dpeaa)DE-He213 Multivariable polynomial system (dpeaa)DE-He213 Gröbner basis (dpeaa)DE-He213 Petit, Christophe aut Shinohara, Naoyuki aut Takagi, Tsuyoshi aut Enthalten in Pacific journal of mathematics for industry Heidelberg : Springer, 2015 7(2015), 1 vom: 25. März (DE-627)815914628 (DE-600)2806666-2 2198-4115 nnns volume:7 year:2015 number:1 day:25 month:03 https://dx.doi.org/10.1186/s40736-015-0012-6 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2111 GBV_ILN_2129 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2015 1 25 03
language	English
source	Enthalten in Pacific journal of mathematics for industry 7(2015), 1 vom: 25. März volume:7 year:2015 number:1 day:25 month:03
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topic_facet	Elliptic curve Discrete logarithm problem Index calculus Multivariable polynomial system Gröbner basis
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container_title	Pacific journal of mathematics for industry
authorswithroles_txt_mv	Huang, Yun-Ju @@aut@@ Petit, Christophe @@aut@@ Shinohara, Naoyuki @@aut@@ Takagi, Tsuyoshi @@aut@@
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author	Huang, Yun-Ju
spellingShingle	Huang, Yun-Ju misc Elliptic curve misc Discrete logarithm problem misc Index calculus misc Multivariable polynomial system misc Gröbner basis Improvement of FPPR method to solve ECDLP
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topic_title	Improvement of FPPR method to solve ECDLP Elliptic curve (dpeaa)DE-He213 Discrete logarithm problem (dpeaa)DE-He213 Index calculus (dpeaa)DE-He213 Multivariable polynomial system (dpeaa)DE-He213 Gröbner basis (dpeaa)DE-He213
topic	misc Elliptic curve misc Discrete logarithm problem misc Index calculus misc Multivariable polynomial system misc Gröbner basis
topic_unstemmed	misc Elliptic curve misc Discrete logarithm problem misc Index calculus misc Multivariable polynomial system misc Gröbner basis
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title	Improvement of FPPR method to solve ECDLP
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title_full	Improvement of FPPR method to solve ECDLP
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title_sort	improvement of fppr method to solve ecdlp
title_auth	Improvement of FPPR method to solve ECDLP
abstract	Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. © Huang et al. 2015
abstractGer	Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. © Huang et al. 2015
abstract_unstemmed	Abstract Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gröbner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faugère, Perret, Petit and Renault proposed a new method (FPPR method) using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of FPPR method that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in FPPR method to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than FPPR method when the extension degree is large enough. © Huang et al. 2015
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title_short	Improvement of FPPR method to solve ECDLP
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author2	Petit, Christophe Shinohara, Naoyuki Takagi, Tsuyoshi
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Improvement of FPPR method to solve ECDLP

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