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

Leonard M. Adleman, Jonathan DeMarrais, Ming-Deh Huang
1999 Theoretical Computer Science  
In this paper 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.  ... 
doi:10.1016/s0304-3975(99)00061-4 fatcat:2dp47wynkzcsbbendhljcpfdlq

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

Kakali Chatterjee, Asok De, Daya Gupta
2013 International Journal of Network Security  
This paper proposes 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.  ... 
dblp:journals/ijnsec/ChatterjeeDG13 fatcat:7hqdpxtjjbe77kgjuuxapqnhf4

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, 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.  ... 
doi:10.1016/s0304-3975(99)00245-5 fatcat:wgpkcnp6yrhktjv3dxs5epu5wu

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]

Fangguo Zhang
2017 Lecture Notes in Computer Science  
Due to the recent cryptanalytic results that the best known 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.  ... 
doi:10.1007/978-3-319-68637-0_13 fatcat:hgeon5ok3bckzeum7svlmfedzi

Open problems in number theoretic complexity, II [chapter]

Leonard M. Adleman, Kevin S. McCurley
1994 Lecture Notes in Computer Science  
, Ming-Deh Huang, and Andrew Granville 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  ... 
doi:10.1007/3-540-58691-1_70 fatcat:3netr75u7nhsncaz3gsliuncm4

Isomorphism classes of hyperelliptic curves of genus 3 over finite fields

Yingpu Deng
2006 Finite Fields and Their Applications  
In this paper we present 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.  ... 
doi:10.1016/j.ffa.2005.05.006 fatcat:gtwlwdqsljdfbcs6tg2kwj66me

On Polynomial Systems Arising from a Weil Descent [chapter]

Christophe Petit, Jean-Jacques Quisquater
2012 Lecture Notes in Computer Science  
As 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.  ... 
doi:10.1007/978-3-642-34961-4_28 fatcat:we5w7xhlfjdpjmq77qwfwp7rdi

Breaking `128-bit Secure' Supersingular Binary Curves (or how to solve discrete logarithms in F_2^4 · 1223 and F_2^12 · 367) [article]

Robert Granger and Thorsten Kleinjung and Jens Zumbrägel
2014 arXiv   pre-print
In particular, they suggested that the new 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  ... 
arXiv:1402.3668v3 fatcat:xzbrdmc3zzhrdkoiqwcawnrbtu

Breaking '128-bit Secure' Supersingular Binary Curves [chapter]

Robert Granger, Thorsten Kleinjung, Jens Zumbrägel
2014 Lecture Notes in Computer Science  
In particular, they suggested that the new 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  ... 
doi:10.1007/978-3-662-44381-1_8 fatcat:ujh33j54hbhvloocykpbfjv7eu

Cantor versus Harley: optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems

T. Wollinger, J. Pelzl, C. Paar
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.  ... 
doi:10.1109/tc.2005.109 fatcat:nwxfbah7dbfy7hutxx3sutz4wq

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)

Harald Niederreiter, Andrew Odlyzko, Michael E. Pohst
1999
part 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.  ... 
doi:10.4230/dagsemrep.225 fatcat:hwqxwfc2uzavtoicifqk6fhgqq
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