Discrete logarithms in finite fields and their cryptographic significance

A. M. Odlyzko

doi:10.1007/3-540-39757-4_20

Discrete logarithms in finite fields and their cryptographic significance

A. M. Odlyzko

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

220 Scopus citations

Abstract

Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ∈ GF(q) is that integer k, 1 ≤ k ≤ q−1, for which u = g ^k. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2ⁿ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2ⁿ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2ⁿ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2ⁿ) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.

Original language	English (US)
Title of host publication	Advances in Cryptology
Subtitle of host publication	Proceedings of EUROCRYPT 1984 - A Workshop on the Theory and Application of Cryptographic Techniques
Editors	Ingemar Ingemarsson, Norbert Cot, Thomas Beth
Publisher	Springer Verlag
Pages	224-314
Number of pages	91
ISBN (Print)	9783540160762
DOIs	https://doi.org/10.1007/3-540-39757-4_20
State	Published - 1985
Event	Workshop on the Theory and Application of Cryptographic Techniques, EUROCRYPT 1984 - Paris, France Duration: Apr 9 1984 → Apr 11 1984

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	209 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	Workshop on the Theory and Application of Cryptographic Techniques, EUROCRYPT 1984
Country/Territory	France
City	Paris
Period	4/9/84 → 4/11/84

Bibliographical note

Publisher Copyright:
© 1985, Springer-Verlag Berlin Heidelberg.

Publisher link

10.1007/3-540-39757-4_20

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Odlyzko, A. M. (1985). Discrete logarithms in finite fields and their cryptographic significance. In I. Ingemarsson, N. Cot, & T. Beth (Eds.), Advances in Cryptology: Proceedings of EUROCRYPT 1984 - A Workshop on the Theory and Application of Cryptographic Techniques (pp. 224-314). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 209 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-39757-4_20

Discrete logarithms in finite fields and their cryptographic significance. / Odlyzko, A. M.
Advances in Cryptology: Proceedings of EUROCRYPT 1984 - A Workshop on the Theory and Application of Cryptographic Techniques. ed. / Ingemar Ingemarsson; Norbert Cot; Thomas Beth. Springer Verlag, 1985. p. 224-314 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 209 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Odlyzko, AM 1985, Discrete logarithms in finite fields and their cryptographic significance. in I Ingemarsson, N Cot & T Beth (eds), Advances in Cryptology: Proceedings of EUROCRYPT 1984 - A Workshop on the Theory and Application of Cryptographic Techniques. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 209 LNCS, Springer Verlag, pp. 224-314, Workshop on the Theory and Application of Cryptographic Techniques, EUROCRYPT 1984, Paris, France, 4/9/84. https://doi.org/10.1007/3-540-39757-4_20

Odlyzko AM. Discrete logarithms in finite fields and their cryptographic significance. In Ingemarsson I, Cot N, Beth T, editors, Advances in Cryptology: Proceedings of EUROCRYPT 1984 - A Workshop on the Theory and Application of Cryptographic Techniques. Springer Verlag. 1985. p. 224-314. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-39757-4_20

Odlyzko, A. M. / Discrete logarithms in finite fields and their cryptographic significance. Advances in Cryptology: Proceedings of EUROCRYPT 1984 - A Workshop on the Theory and Application of Cryptographic Techniques. editor / Ingemar Ingemarsson ; Norbert Cot ; Thomas Beth. Springer Verlag, 1985. pp. 224-314 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ∈ GF(q) is that integer k, 1 ≤ k ≤ q−1, for which u = g k. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2n) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2n) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2n) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.",

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N2 - Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ∈ GF(q) is that integer k, 1 ≤ k ≤ q−1, for which u = g k. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2n) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2n) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2n) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.

AB - Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ∈ GF(q) is that integer k, 1 ≤ k ≤ q−1, for which u = g k. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2n) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2n) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2n) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.

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Discrete logarithms in finite fields and their cryptographic significance

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