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Recent progress on the elliptic curve discrete logarithm problem


We survey recent work on the elliptic curve discrete logarithm problem. In particular we review index calculus algorithms using summation polynomials, and claims about their complexity.

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  1. This is sometimes called the “non-uniform” model, but we do not discuss such interpretations in this paper. Note that an algorithm that stores a table of all discrete logs does not fit the model since the program length is \(O( r \log (r) )\) bits.

  2. It is not necessary that V be a subfield. If V is a one-dimensional subspace that is not a subfield then \(V^{(2)}\) is also a one-dimensional subspace, but \(V^{(2)} \ne V\).

  3. And more, including the first author and his Ph.D. student Shishay Gebregiyorgis.

  4. This is true only under genericity assumptions, and with appropriate monomial orderings.

  5. And one must be careful not to be fooled by the Strong law of small numbers [57].


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We thank Claus Diem, Michiel Kosters, Christophe Petit, Peter Wild and an anonymous referee for helpful comments on the draft of this article. The second author also thanks Maike Massierer, Pierre-Jean Spaenlehauer and Vanessa Vitse for various discussions on the topic.

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Correspondence to Steven D. Galbraith.

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This is one of several papers published in Designs, Codes and Cryptography comprising the 25th Anniversary Issue.

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Galbraith, S.D., Gaudry, P. Recent progress on the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 78, 51–72 (2016).

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