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A Subexponential Algorithm for Discrete Logarithms Over all Finite Fields

Leonard M. Adleman, Jonathan Demarrais
1993 Mathematics of Computation  
However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields.  ...  There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields.  ...  Acknowledgments We would like to thank Dennis Estes and Bob Guralnick for their help. In an earlier version of this algorithm the authors had chosen m = [nx/2¡ .  ... 
doi:10.2307/2152932 fatcat:uxauutc3ljbbhm3zlhuvizz2ve

A subexponential algorithm for discrete logarithms over all finite fields

Leonard M. Adleman, Jonathan DeMarrais
1993 Mathematics of Computation  
However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields.  ...  There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields.  ...  Acknowledgments We would like to thank Dennis Estes and Bob Guralnick for their help. In an earlier version of this algorithm the authors had chosen m = [nx/2¡ .  ... 
doi:10.1090/s0025-5718-1993-1225541-3 fatcat:ppcwprlr2jh63ecch5dpmpgrd4

A Subexponential Algorithm for Discrete Logarithms over All Finite Fields [chapter]

Leonard M. Adleman, Jonathan DeMarrais
Advances in Cryptology — CRYPTO' 93  
However, there appears to be no published subexponential algorithm for computing discrete logarithms over all finite fields.  ...  There are numerous subexponential algorithms for computing discrete logarithms over certain classes of finite fields.  ...  Acknowledgments We would like to thank Dennis Estes and Bob Guralnick for their help. In an earlier version of this algorithm the authors had chosen m = [nx/2¡ .  ... 
doi:10.1007/3-540-48329-2_13 dblp:conf/crypto/AdlemanD93 fatcat:57c26ihcqndhheuhg6rmgmmj4m

Discrete logarithms in curves over finite fields [article]

Andreas Enge
2007 arXiv   pre-print
A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.  ...  The first subexponential algorithm for computing discrete logarithms in hyperelliptic curves of large genus defined over a finite field K = F q is due to Adleman, DeMarrais and Huang [3] .  ...  Subexponential algorithms of complexity L(1/3) Following the progress for factorisation algorithms, a complexity of L(1/3) has also been established for discrete logarithm computations in finite fields  ... 
arXiv:0712.3916v1 fatcat:yt776ztyrbacdi3rauevpcwjka

A general framework for subexponential discrete logarithm algorithms

Andreas Enge, Pierrick Gaudry
2002 Acta Arithmetica  
The first examples of such groups were the multiplicative groups of finite fields; it turned out that in this case the discrete logarithm problem can be solved in expected subexponential time by so-called  ...  Other important examples are class groups of number fields [24, 6] and Jacobians, i.e. divisor class groups, of elliptic [25, 19] and hyperelliptic [20] curves over finite fields.  ...  The results of this article form part of the authors' doctoral theses; we thank our respective supervisors, Dieter Jungnickel and François Morain, for their continuous support and useful suggestions.  ... 
doi:10.4064/aa102-1-6 fatcat:x3fkfj6sl5cvng5j5bxrmeu4na

Page 2503 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
There are numerous subexponential algorithms for the problem over fields of special form. The paper presents a subexponential method for computing discrete logarithms over all finite fields.  ...  over all finite fields.  ... 

On the Bounded Sum-of-digits Discrete Logarithm Problem in Kummer and Artin-Schreier Extensions [article]

Qi Cheng
2003 arXiv   pre-print
In this paper, we study the discrete logarithm problem in the finite fields _q^n where n|q-1. The field is called a Kummer field or a Kummer extension of _q.  ...  Since every finite field has an extension of reasonable degree which is a Kummer field, our result reveals an unexpected property of the discrete logarithm problem, namely, the bounded sum-of-digits discrete  ...  Acknowledgments We thank Professor Pedro Berrizbeitia for very helpful discussions.  ... 
arXiv:math/0311120v1 fatcat:6guf7oj7bjaxli6vercrofc6fy

Page 2507 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews  
subexponential algorithm for discrete logarithms over all finite fields.  ...  However, there appears to be no published subexponential algo- rithm for computing discrete logarithms over all finite fields.  ... 

On the Bounded Sum-of-Digits Discrete Logarithm Problem in Finite Fields

Qi Cheng
2005 SIAM journal on computing (Print)  
In this paper, we study the bounded sum-of-digits discrete logarithm problem in finite fields. Our results concern primarily with fields Fqn where n|q − 1.  ...  We also prove that in the field F q q−1 , the bounded sum-of-digits discrete logarithm with respect to g can be computed in random time O(f (w) log 4 (q q−1 )), where f is a subexponential function and  ...  Acknowledgments We thank Professor Pedro Berrizbeitia for very helpful discussions.  ... 
doi:10.1137/s0097539704446037 fatcat:wxmpgefqkbbg7nnxgxyfmdjeke

On the Bounded Sum-of-Digits Discrete Logarithm Problem in Finite Fields [chapter]

Qi Cheng
2004 Lecture Notes in Computer Science  
In this paper, we study the bounded sum-of-digits discrete logarithm problem in finite fields. Our results concern primarily with fields Fqn where n|q − 1.  ...  We also prove that in the field F q q−1 , the bounded sum-of-digits discrete logarithm with respect to g can be computed in random time O(f (w) log 4 (q q−1 )), where f is a subexponential function and  ...  Acknowledgments We thank Professor Pedro Berrizbeitia for very helpful discussions.  ... 
doi:10.1007/978-3-540-28628-8_12 fatcat:yjt5sj3i7nhppfqgmfbzp4x47e

An L (1/3 + ε) Algorithm for the Discrete Logarithm Problem for Low Degree Curves [chapter]

Andreas Enge, Pierrick Gaudry
2007 Lecture Notes in Computer Science  
The discrete logarithm problem in Jacobians of curves of high genus g over finite fields _q is known to be computable with subexponential complexity L_q^g(1/2, O(1)).  ...  For this family, the group structure can be computed in subexponential time of L_q^g(1/3, O(1)), and a discrete logarithm computation takes subexponential time of L_q^g(1/3+ϵ, o(1)) for any positive ϵ.  ...  We thank Claus Diem for his careful reading of our article and many useful remarks.  ... 
doi:10.1007/978-3-540-72540-4_22 fatcat:3chevkqcn5e5phscimwrveq574

Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time

Andreas Enge
2001 Mathematics of Computation  
We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of high-genus hyperelliptic curves over finite fields.  ...  Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given by The algorithm works over any finite field, and its running  ...  I thank Alfred Menezes and Scott Vanstone for the invitation and their hospitality, and Michael Jacobson, Andreas Stein and Edlyn Teske for fruitful discussions.  ... 
doi:10.1090/s0025-5718-01-01363-1 fatcat:lhvat6ftdre4njrnd3wtrgheam

Page 831 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews  
The algorithms for computing discrete logarithms in F,» run in subexponential-time only in the case that m grows and p is fixed.  ...  Summary: “We present a subexponential algorithm of the dis- crete logarithm in GF(p*), p= 1 mod 3, which runs in time O(exp(24,/log ploglog p)).  ... 

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).  ...  ; La Jolla, CA) New algorithms for generating Conway polynomials over finite fields.  ... 

Smooth ideals in hyperelliptic function fields

Andreas Enge, Andreas Stein
2001 Mathematics of Computation  
Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields.  ...  Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived.  ...  Most of this contribution is a result of a research visit of the first author's at the Centre for Applied Cryptographic Research.  ... 
doi:10.1090/s0025-5718-01-01352-7 fatcat:y7dvvxlpand3tpl36v6p3jhylq
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