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### Function Field Sieve Method for Discrete Logarithms over Finite Fields

Leonard M. Adleman, Ming-Deh A. Huang
1999 Information and Computation
We present a function field sieve method for discrete logarithms over finite fields. This method is an analog of the number field sieve method originally developed for factoring integers.  ...  It is asymptotically faster than the previously known algorithms when applied to finite fields F p n , where p 6 n.  ...  As noted in  that, while the number field sieve method works well for finite fields F p n , where n< <(log p) 1Â2 , the function field sieve method works well for finite fields F p n , where log  ...

### Function Field Sieve in Characteristic Three [chapter]

R. Granger, A. J. Holt, D. Page, N. P. Smart, F. Vercauteren
2004 Lecture Notes in Computer Science
In this paper we investigate the efficiency of the function field sieve to compute discrete logarithms in the finite fields F3n .  ...  This allows us to represent the function field over different base fields. Practical experiments appear to show that a function field over F3 gives the best results.  ...  The same cannot be said for discrete logarithm algorithms over finite fields. Such algorithms depend quite closely on the characteristic.  ...

### The Function Field Sieve Is Quite Special [chapter]

Antoine Joux, Reynald Lercier
2002 Lecture Notes in Computer Science
In this paper, we describe improvements to the function field sieve (FFS) for the discrete logarithm problem in Fpn , when p is small.  ...  We demonstrate the efficiency of the algorithm by successfully computing discrete logarithms in a large finite field of characteristic two, namely F 2 521 .  ...  Conclusion In this paper, we described improvements to the function field sieve for the discrete logarithm problem.  ...

### An experiment of number field sieve for discrete logarithm problem over $\text{GF}(p^n)$

Kenichiro Hayasaka, Kazumaro Aoki, Tetsutaro Kobayashi, Tsuyoshi Takagi
2014 JSIAM Letters
At CRYPTO 2006, Joux et al. proposed the number field sieve over GF(p n ), but the number field sieve needs multi-dimensional sieving.  ...  sieving.  ...  The security of pairing-based cryptography using the BN curves is based on the hardness of the discrete logarithm problem (DLP) over finite fields GF(p 12 ).  ...

### Page 10 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews
The authors present a new version of the function field sieve algorithm for computing discrete logarithms in finite fields.  ...  ., Providence, RI, 1999. 2001¢:11136b 11Y16 68P25 94A60 Adleman, Leonard M. (1-SCA-C; Los Angeles, CA); Huang, Ming-Deh A. (1-SCA-C; Los Angeles, CA) Function field sieve method for discrete logarithms  ...

### Page 6869 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews
The author adapts the number field sieve factoring algorithm to the problem of finding discrete logarithms in a finite field with q = p” elements.  ...  6869 discrete logarithms in the group of rational points on the Jaco- bians of large genus hyperbolic curves over a field GF(q), logg < (2g + 1)'-* for each 0 <e <1, is proposed.  ...

### The Past, Evolving Present, and Future of the Discrete Logarithm [chapter]

Antoine Joux, Andrew Odlyzko, Cécile Pierrot
2014 Open Problems in Mathematics and Computational Science
In particular, this is the case for the multiplicative groups of finite fields with medium to large characteristic and for the additive group of a general elliptic curve.  ...  The first practical public key cryptosystem ever published, the Diffie-Hellman key exchange algorithm, relies for its security on the assumption that discrete logarithms are hard to compute.  ...  Discrete Logarithm in Finite Fields A Short History The earliest methods introduced to compute discrete logarithms are generic.  ...

### Page 3499 of Mathematical Reviews Vol. , Issue 87g [page]

1987 Mathematical Reviews
The method described in the article is developed di- rectly for finite fields in contrast to the known methods which are finite-field adaptations of the methods for solving systems over the reals. N.  ...  The authors present several related methods (of the Western- Miller family) of finding discrete logarithms in a field GF(p), p prime.  ...

### The Multiple Number Field Sieve with Conjugation and Generalized Joux-Lercier Methods [chapter]

Cécile Pierrot
2015 Lecture Notes in Computer Science
In this paper, we propose two variants of the Number Field Sieve (NFS) to compute discrete logarithms in medium characteristic finite fields.  ...  selections for the classical Number Field Sieve.  ...  Acknowledgment I would like to sincerely thank Taechan Kim for his careful proofreading.  ...

### Page 10 of Mathematical Reviews Vol. , Issue 2004a [page]

2004 Mathematical Reviews
The function field sieve of Adleman and Huang can be used to find discrete logarithms in the field F, with g elements, where ¢ = p" and p is small in the sense that p< n'/°.  ...  This results in a discrete logarithm algorithm whose conjectured complexity is the same as the function field sieve—L,[1/3, (32/9)'/* + o(1)] l—a where L,[a, c] = exp(c log“ (log(q)) log log’ ~“ )—which  ...

### Indiscreet logarithms in finite fields of small characteristic [article]

Robert Granger, Thorsten Kleinjung, Jens Zumbrägel
2016 arXiv   pre-print
the state-of-the-art algorithms for computing discrete logarithms in all finite fields.  ...  Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly  ...  Inspired by the number field sieve, the function field sieve was developed for computing discrete logarithms in small (fixed) characteristic [Adl94, AH99, JL02, JL06] .  ...

### Indiscreet logarithms in finite fields of small characteristic

Robert Granger, Thorsten Kleinjung, Jens Zumbrägel
2018 Advances in Mathematics of Communications
the state-of-the-art algorithms for computing discrete logarithms in all finite fields.  ...  Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly  ...  Inspired by the number field sieve, the function field sieve was developed for computing discrete logarithms in small (fixed) characteristic [Adl94, AH99, JL02, JL06] .  ...

### A unified polynomial selection method for the (tower) number field sieve algorithm

Palash Sarkar, Shashank Singh
2019 Advances in Mathematics of Communications
to the discrete logarithm problem over finite fields.  ...  This line of work has led to new asymptotic complexities for various cases of the discrete logarithm problem over finite fields.  ...  Acknowledgment We thank the reviewers for carefully reading the paper and providing comments which have helped in improving the work.  ...

### Page 1110 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews
., 877, Springer, Berlin, 1994; MR 96d:11135] for computing discrete logarithms in the multiplicative group of a finite field.  ...  In fact, the number field sieve has this complex- ity for small n relative to p, and the function field sieve has this complexity for large relative to p; roughly speaking, no algo- rithm of complexity  ...

### Page 7668 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews
The discrete logarithm algorithms mentioned include Shanks’ baby-step-giant-step method, the Silver-Pohlig-Hellman method, Adleman’s index calculus, Gordon’s number field sieve and the Xendi calculus for  ...  For a function f defined over Z/p!  ...
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