Abstract
We analyze the point decomposition problem (PDP) in binary elliptic curves. It is known that PDP in an elliptic curve group can be reduced to solving a particular system of multivariate non-linear equations derived from the so called Semaev summation polynomials. We modify the underlying system of equations by introducing some auxiliary variables. We argue that the trade-off between lowering the degree of Semaev polynomials and increasing the number of variables provides a significant speed-up.
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Notes
- 1.
We prefer to use point rather than element because elliptic curve group elements are commonly called points.
- 2.
This is roughly when the number of relations exceeds \(|\mathcal {B}|\).
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Acknowledgment
I would like to acknowledge two recent papers [9, 12]. Semaev [12] claims a new complexity bound \(2^{c(\sqrt{n\ln n}})\) for solving ECDLP(2, n) under the assumption that the degree of regularity in Groebner computations of particular polynomial systems is \(D_{\mathsf {reg}}\le 4\). Semaev also shows that ECDLP(2, n) can be solved in time \(2^{o(c\sqrt{n\ln n})}\) under a weaker assumption that \(D_{\mathsf {reg}} = o(\sqrt{n/\ln n})\) The techniques used in [12] and in this paper are similar. In [9], Kosters and Yeo provide experimental evidence that the degree of regularity of the underlying polynomial systems is likely to increase as a function of n, whence the conjecture \(D_{\mathsf {reg}} \approx D_{\mathsf {FirstFall}}\) may be false.
I would like to thank Michiel Kosters and Igor Semaev for their comments on the first version of this paper.
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Karabina, K. (2016). Point Decomposition Problem in Binary Elliptic Curves. In: Kwon, S., Yun, A. (eds) Information Security and Cryptology - ICISC 2015. ICISC 2015. Lecture Notes in Computer Science(), vol 9558. Springer, Cham. https://doi.org/10.1007/978-3-319-30840-1_10
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DOI: https://doi.org/10.1007/978-3-319-30840-1_10
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