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A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF(q)

1999
*
Theoretical Computer Science
*

In this paper

doi:10.1016/s0304-3975(99)00061-4
fatcat:2dp47wynkzcsbbendhljcpfdlq
*a**subexponential**algorithm*is presented*for*ÿnding*discrete**logarithms*in the group*of*rational points on the Jacobians*of**large**genus**hyperelliptic**curves**over*prime ÿelds. ...*logarithms*in the group*of*rational points on the Jacobian*of**a**genus*g*hyperelliptic**curve**over**GF*(*q*)*of*the form y 2 = f(x) with deg(f) = 2g+1, within expected time L*q*2g+1 [1=2; c], or equivalently ... Somewhat surprisingly, the high*genus**hyperelliptic**curve*case may be easiest to solve. We present an*algorithm*that is*subexponential**for**curves**of*high*genus*. ...##
###
Page 676 of Mathematical Reviews Vol. , Issue 96b
[page]

1996
*
Mathematical Reviews
*

*subexponential*

*algorithm*

*for*

*discrete*

*logarithms*

*over*the rational subgroup

*of*the Jacobians

*of*

*large*

*genus*

*hyperelliptic*

*curves*

*over*finite fields. ... Let K be

*a*finite field, and H

*a*

*hyperelliptic*

*curve*

*over*K

*of*

*genus*g. ...

##
###
Mutual Authentication Protocol Using Hyperelliptic Curve Cryptosystem in Constrained Devices

2013
*
International Journal of Network Security
*

This paper proposes

dblp:journals/ijnsec/ChatterjeeDG13
fatcat:7hqdpxtjjbe77kgjuuxapqnhf4
*a*mutual authentication protocol based on*Hyperelliptic**Curve*Digital Signature*Algorithm**for**a*secure access in constrained devices which allows both the entities to verify each other's ...*Hyperelliptic**Curve*Cryptosystem (HECC) is well suited*for*secure communication in wireless sensor network as limited resources (storage, time or power) on sensor nodes restrict the use*of*conventional ... [4] found*a**subexponential*time*algorithm*to solve the DL in the Jacobian*of*HEC*of**a*big*genus**over**a*finite field. ...##
###
Page 6868 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

*algorithm*

*for*

*discrete*

*logarithms*

*over*

*hyperelliptic*

*curves*

*of*

*large*

*genus*

*over*

*GF*(

*q*). ... These groups are more intractable than ones associated to

*hyperelliptic*

*curves*which in some cases allow

*a*reducibility

*of*the

*discrete*

*logarithm*problem. See,

*for*instance, the paper by T. ...

##
###
Index

1999
*
Theoretical Computer Science
*

Huang,

doi:10.1016/s0304-3975(99)00245-5
fatcat:wgpkcnp6yrhktjv3dxs5epu5wu
*A**subexponential**algorithm**for**discrete**logarithms**over**hyperelliptic**curves**of**large**genus**over**GF*(*q*) ( 1 }2) 7} 18 Chen, S., see R. Tao (1}2) 143}172 Chua, S.K., K. H. Leung and S. ... Vielhaber, An*algorithm**for*shifted continued fraction expansions in parallel linear time (1}2) 93}104 Niemi, V. and*A*. ...##
###
Page 6747 of Mathematical Reviews Vol. , Issue 2000i
[page]

2000
*
Mathematical Reviews
*

Adleman, Jonathan DeMarrais and Ming-Deh Huang,

*A**subexponential**algorithm**for**discrete**logarithms**over**hyperelliptic**curves**of**large**genus**over**GF*(*q*) (7-18); Seng Kiat Chua, Ka Hin Leung and San Ling, ... Attack on RSA-type cryp- tosystems based on singular cubic*curves**over*Z/nZ (19-27); Thomas W. ...##
###
Bit Security of the Hyperelliptic Curves Diffie-Hellman Problem
[chapter]

2017
*
Lecture Notes in Computer Science
*

Due to the recent cryptanalytic results that the best known

doi:10.1007/978-3-319-68637-0_13
fatcat:hgeon5ok3bckzeum7svlmfedzi
*algorithms*to attack*hyperelliptic**curve*cryptosystems*of**genus*g < 3 are the generic methods and the recent implementation results that*hyperelliptic*... Finally, we extend our techniques and results to*hyperelliptic**curves**of*any*genus*. ... Part*of*this work was done during the author was visiting the UbiSeC lab at University at Buffalo, State University*of*New York. ...##
###
Open problems in number theoretic complexity, II
[chapter]

1994
*
Lecture Notes in Computer Science
*

, Ming-Deh Huang, and Andrew Granville

doi:10.1007/3-540-58691-1_70
fatcat:3netr75u7nhsncaz3gsliuncm4
*for*their contributions. ... The work*of*the rst author was supported by NSF grant CCR-9214671. ... Adleman, Huang, and DeMarrais AHDar] discovered*a*heuristic*subexponential*probabilistic*algorithm**for*the*discrete**logarithm*problem in these subgroups when the*genus**of*the*curve*is*large*with respect ...##
###
Isomorphism classes of hyperelliptic curves of genus 3 over finite fields

2006
*
Finite Fields and Their Applications
*

In this paper we present

doi:10.1016/j.ffa.2005.05.006
fatcat:gtwlwdqsljdfbcs6tg2kwj66me
*a*direct method to compute the number*of*isomorphism classes*of**hyperelliptic**curves**of**genus*3*over*finite fields F*q*. ... The number*of*isomorphism classes is*a*polynomial in*q**of*degree 5. In all the cases we show an explicit formula*for*this polynomial. ... Mulan Liu*for*her encouragement. He wish to thank Prof. Huah Chu, Prof. Shou-Jen Hu and Prof. Ming-Chang Kang*for*their helpful discussions about the manuscript*of*this paper. ...##
###
On Polynomial Systems Arising from a Weil Descent
[chapter]

2012
*
Lecture Notes in Computer Science
*

As

doi:10.1007/978-3-642-34961-4_28
fatcat:we5w7xhlfjdpjmq77qwfwp7rdi
*a*main consequence, our heuristic analysis implies that Diem's variant*of*index calculus*for*ECDLP requires*a**subexponential*number*of*bit operations O(2 c n 2/3 log n )*over*the binary field F2n , ... We then revisit applications to the elliptic*curve**discrete**logarithm*problem (ECDLP)*for*binary*curves*, to the factorization problem in SL(2, F2n ) and to other*discrete**logarithm*problems. ... We also thank Jean-Charles Faugère, Ludovic Perret and Guénaël Renault*for*their useful comments on*a*preliminary version*of*this paper. ...##
###
Breaking `128-bit Secure' Supersingular Binary Curves (or how to solve discrete logarithms in F_2^4 · 1223 and F_2^12 · 367)
[article]

2014
*
arXiv
*
pre-print

In particular, they suggested that the new

arXiv:1402.3668v3
fatcat:xzbrdmc3zzhrdkoiqwcawnrbtu
*algorithms*have no impact on the security*of**a**genus*one*curve**over*F_2^1223, and reduce the security*of**a**genus*two*curve**over*F_2^367 to 94.6 bits. ... In late 2012 and early 2013 the*discrete**logarithm*problem (DLP) in finite fields*of*small characteristic underwent*a*dramatic series*of*breakthroughs, culminating in*a*heuristic quasi-polynomial time ... After the initial publication*of*[19] , Joux released*a*preprint [33] detailing an*algorithm**for*solving the*discrete**logarithm*problem*for*fields*of*the form F*q*2n , with n ≤*q*+ d*for*some very small ...##
###
Breaking '128-bit Secure' Supersingular Binary Curves
[chapter]

2014
*
Lecture Notes in Computer Science
*

In particular, they suggested that the new

doi:10.1007/978-3-662-44381-1_8
fatcat:ujh33j54hbhvloocykpbfjv7eu
*algorithms*have no impact on the security*of**a**genus*one*curve**over*F 2 1223 , and reduce the security*of**a**genus*two*curve**over*F 2 367 to 94.6 bits. ... In late 2012 and early 2013 the*discrete**logarithm*problem (DLP) in finite fields*of*small characteristic underwent*a*dramatic series*of*breakthroughs, culminating in*a*heuristic quasipolynomial time*algorithm*... After the initial publication*of*[19] , Joux released*a*preprint [33] detailing an*algorithm**for*solving the*discrete**logarithm*problem*for*fields*of*the form F*q*2n , with n ≤*q*+ d*for*some very small ...##
###
Cantor versus Harley: optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems

2005
*
IEEE transactions on computers
*

*For*certain

*genus*-3

*curves*we can gain almost 80% performance

*for*

*a*group doubling. ... In this contribution, we increase the efficiency

*of*the

*genus*-2 and

*genus*-3

*hyperelliptic*

*curve*cryptosystems (HECC). ... He optimized the sub-exponential

*algorithm*to compute the

*discrete*

*logarithm*in the Jacobian

*of*low

*genus*

*hyperelliptic*

*curves*. ...

##
###
Page 293 of Mathematical Reviews Vol. 32, Issue Index
[page]

*
Mathematical Reviews
*

(with DeMarrais, Jonathan; Huang, Ming-Deh

*A*.)*A*subexponen- tial*algorithm**for**discrete**logarithms**over**hyperelliptic**curves**of**large**genus**over**GF*(*q*). ...*A*. On the reduction*of*the calculation*of**discrete**logarithms*on an elliptic*curve*to the calculation*of**discrete**logarithms*in*a*finite field. (Russian. ...##
###
Algorithms and Number Theory (Dagstuhl Seminar 98431)

1999

part

doi:10.4230/dagsemrep.225
fatcat:hwqxwfc2uzavtoicifqk6fhgqq
*of*the new standard*for*public key cryptosystems; • primality testing and proving,*large*primes being*of*importance*for*quite*a*few cryptosystems; ...*A*major goal was always to bring together number theorists who develop the theory*for*efficient*algorithms*and people writing the corresponding software*for*applications. ... Groups are useful*for*cryptographic reasons, if the inverse operation, the "*discrete**logarithm*" n = log P*Q*(P,*Q*∈ G), is not*a*polynomial time or*subexponential*time*algorithm*. ...
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