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The State of Elliptic Curve Cryptography [chapter]

Neal Koblitz, Alfred Menezes, Scott Vanstone
2000 Towards a Quarter-Century of Public Key Cryptography  
integers modulo a prime, this idea can be extended to arbitrary groups and, in particular, to elliptic curve groups.  ...  The resulting public-key systems provide relatively small block size, high speed, and high security.  ...  over a finite field.  ... 
doi:10.1007/978-1-4757-6856-5_5 fatcat:6hvidisd6zdzzaarclkcqs4fsu

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Neal Koblitz, Alfred Menezes, Scott Vanstone
2012 Designs, Codes and Cryptography  
integers modulo a prime, this idea can be extended to arbitrary groups and, in particular, to elliptic curve groups.  ...  The resulting public-key systems provide relatively small block size, high speed, and high security.  ...  over a finite field.  ... 
doi:10.1023/a:1008354106356 fatcat:zeumo2bjbffabkvqohtru7j62m

Construction of Hyperelliptic Curves with CM and Its Application to Cryptosystems [chapter]

Jinhui Chao, Kazuto Matsuo, Hiroto Kawashiro, Shigeo Tsujii
2000 Lecture Notes in Computer Science  
Algorithms for CM test of Jacobian varieties of algebraic curves and to lift from small finite fields both the models and the invariants of CM curves are presented.  ...  This paper presents new algorithms to find explicit models of hyperelliptic curves with CM.  ...  In this section, we show how to lift from small finite fields the models of curves with CM defined e.g. over the class field of K.  ... 
doi:10.1007/3-540-44448-3_20 fatcat:z2gwjbvgpzdhpe6xedky7z4o5q

Solving discrete logarithms on a 170-bit MNT curve by pairing reduction [article]

Aurore Guillevic, Emmanuel Thomé
2016 arXiv   pre-print
Remaining instances are built over finite fields of large characteristic and their security relies on the fact that the embedding field of the underlying curve is relatively large.  ...  The aim of our work is to sustain the claim that the combination of degree 3 embedding and too small finite fields obviously does not provide enough security.  ...  However for MNT curves over prime fields of 160 bits, the MOV and FR reduction attacks embed to finite fields of size 480, 640, or 960 bits, none of which should be considered as having a hard enough DLP  ... 
arXiv:1605.07746v2 fatcat:kwaeic6iqrfw3ekowzhojf6rru

Software Implementation of Finite Fields of Characteristic Three, for Use in Pairing-based Cryptosystems

K. Harrison, D. Page, N. P. Smart
2002 LMS Journal of Computation and Mathematics  
Issues related to the arithmetic of supersingular elliptic curves over fields of characteristic three are also examined.  ...  Three alternative representations of the field elements are examined, and the resulting algorithms for the field addition, multiplication and cubing are compared.  ...  we assume is a general finite field of arbitrary characteristic.  ... 
doi:10.1112/s1461157000000747 fatcat:wfnx3jcgy5bp7puk7xihyjblxi

Design of Information Security for Large System Development Projects [chapter]

James Murphy
2012 Information Security Management Handbook, Sixth Edition, Volume 6  
., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps.  ...  The dimension of F over K is called the degree of the extension of F over K. Polynomial Rings Let F he an arbitrary ring.  ...  to F q , is called a normal basis of FQm over F q • For every extension field of finite degree of a finite field there is a normal basis.  ... 
doi:10.1201/b11802-24 fatcat:xk3kajawgjfrbbwc4k6eojwzzy

Elliptic Curve Cryptosystems [chapter]

Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, Tomik Yaghoobian
1993 Applications of Finite Fields  
The application of elliptic curves to the field of cryptography has been relatively recent. It has opened up a wealth of possibilities in terms of security, encryption, and real-world applications.  ...  The objective of this thesis is to assemble the most important facts and findings into a broad, unified overview of this field.  ...  Some well-known cryptosystems work with multiplicative groups of fields, and as it turns out, elliptic curves over finite fields are a rich source of finite abelian groups.  ... 
doi:10.1007/978-1-4757-2226-0_8 fatcat:qolt5n2aibetfedkcdw6dkxvaa

Elliptic Curve Cryptosystems [chapter]

2005 Series on Coding Theory and Cryptology  
The application of elliptic curves to the field of cryptography has been relatively recent. It has opened up a wealth of possibilities in terms of security, encryption, and real-world applications.  ...  The objective of this thesis is to assemble the most important facts and findings into a broad, unified overview of this field.  ...  Some well-known cryptosystems work with multiplicative groups of fields, and as it turns out, elliptic curves over finite fields are a rich source of finite abelian groups.  ... 
doi:10.1142/9789812703309_0006 fatcat:qtjvt6vivvcftlwgfwd7wjhree

Solving Discrete Logarithms on a 170-Bit MNT Curve by Pairing Reduction [chapter]

Aurore Guillevic, François Morain, Emmanuel Thomé
2017 Lecture Notes in Computer Science  
, with degree-4 embedding [26] .  ...  This includes the so-called MNT curves defined by Miyaji-Nakabayashi-Takano, e.g. [40, Example 1], an elliptic curve defined over a 170-bit prime p, and of 508-bit embedding field F p 3 .  ...  However for MNT curves over prime fields of 160 bits, the MOV and FR reduction attacks embed to finite fields of size 480, 640, or 960 bits, none of which should be considered as having a hard enough DLP  ... 
doi:10.1007/978-3-319-69453-5_30 fatcat:qhxg3rrwe5c37lratgpzk5jihy

Efficient implementation of elliptic curve cryptography in wireless sensors

Diego Aranha, Ricardo Dahab, Julio López, Leonardo Oliveira
2010 Advances in Mathematics of Communications  
Finite field arithmetic was implemented in C and Assembly and elliptic curve arithmetic was implemented in Koblitz and generic binary curves.  ...  We illustrate the performance of our implementation with timings for key agreement and digital signature protocols.  ...  prime fields and 1.27 seconds over binary fields.  ... 
doi:10.3934/amc.2010.4.169 fatcat:6wdwdw6uujgtlh4t6ksj2oitmm

An Extension of Kedlaya's Point-Counting Algorithm to Superelliptic Curves [chapter]

Pierrick Gaudry, Nicolas Gürel
2001 Lecture Notes in Computer Science  
We present an algorithm for counting points on superelliptic curves y r = f (x) over a finite field Fq of small characteristic different from r.  ...  We give some numerical examples obtained with our first implementation, thus proving that cryptographic sizes are now reachable.  ...  We took a randomly chosen curve over the finite field F q with q = 3 37 .  ... 
doi:10.1007/3-540-45682-1_28 fatcat:5uncot4mpzfytdd77acs3ivx7i

Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography [chapter]

Nadia El Mrabet, Christophe Negre
2009 Lecture Notes in Computer Science  
Pairings over ellitpic curve use fields F p k with p ≥ 2 160 and 6 < k ≤ 32. In this paper we propose to represent elements in Fp with AMNS sytem of [1].  ...  The DFT/FFT approach for multiplication in extension field F p k is thus optimized.  ...  hal-00360280, version 1 -10 Aug 2009 -The second method which could be used in order to build curves with arbitrary embedding degree k is the Cocks-Pinch method [7] .  ... 
doi:10.1007/978-3-642-02620-1_29 fatcat:wxddxvvexjghzk7els4lbsyfs4

Short Signatures from the Weil Pairing

Dan Boneh, Ben Lynn, Hovav Shacham
2004 Journal of Cryptology  
We use E/F q to denote an elliptic curve with coefficients in F q . For r ≥ 1, we use E(F q r ) to denote the group of points on E in F q r .  ...  Our short signature scheme is designed for systems where signatures are typed in by a human or are sent over a low-bandwidth channel.  ...  over a finite field.  ... 
doi:10.1007/s00145-004-0314-9 fatcat:4yytx4rgqfevzddcvwwr56l544

Genetic improvement of Pacific white shrimp [Penaeus (Litopenaeus) vannamei]: perspectives for genomic selection

Héctor Castillo-Juárez, Gabriel R. Campos-Montes, Alejandra Caballero-Zamora, Hugo H. Montaldo
2015 Frontiers in Genetics  
We use E/F q to denote an elliptic curve with coefficients in F q . For r ≥ 1, we use E(F q r ) to denote the group of points on E in F q r .  ...  Our short signature scheme is designed for systems where signatures are typed in by a human or are sent over a low-bandwidth channel.  ...  over a finite field.  ... 
doi:10.3389/fgene.2015.00093 pmid:25852740 pmcid:PMC4371756 fatcat:37tajiuaibd3tjaqujynnodf44

Short Signatures from the Weil Pairing [chapter]

Dan Boneh, Ben Lynn, Hovav Shacham
2001 Lecture Notes in Computer Science  
We use E/F q to denote an elliptic curve with coefficients in F q . For r ≥ 1, we use E(F q r ) to denote the group of points on E in F q r .  ...  Our short signature scheme is designed for systems where signatures are typed in by a human or are sent over a low-bandwidth channel.  ...  over a finite field.  ... 
doi:10.1007/3-540-45682-1_30 fatcat:2klih4ovcjftdcfmnzcl4wyfii
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