An algorithm to compute volcanoes of 2-isogenies of elliptic curves over finite fields

Miret, Josep M.; Moreno, Ramiro; Sadornil, Daniel; Tena, Juan Guillermo Lohmann Luca de; Valls, Magda

doi:10.1016/j.amc.2005.10.020

Cited by 13 publications

(11 citation statements)

References 4 publications

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“…Let E be a curve on the isogeny volcano such that v ℓ (N ) < v ℓ (M ). As explained in [17] (in the case ℓ = 2, but the result is general), a is such that v ℓ (a − 1) ≥ min {v ℓ (g), v ℓ (#E(F q ))/2} .…”

Section: 2mentioning

confidence: 98%

“…These algorithms need to walk on the crater, to descend from the crater to the floor or to ascend from the floor to the crater. In many cases, the structure of the ℓ-Sylow subgroup of the elliptic curve, allows one, after taking a step on the volcano, to decide whether this step is ascending, descending or horizontal (see [17,18]). Note that, since a large fraction of isogenies are descending, finding one of them is quite easy.…”

Section: Introductionmentioning

confidence: 99%

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Pairing the Volcano

Ionica

Joux

2010

Lecture Notes in Computer Science

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Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are ℓ-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel's and Fouquet-Morain algorithms, many steps are taken before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trialand-error approach. In this paper, we propose an alternative method that efficiently finds all points P of order ℓ such that the subgroup generated by P is the kernel of an horizontal or an ascending isogeny. In many cases, our method is faster than previous methods. This is an extended version of a paper published in the proceedings of ANTS 2010. In addition, we treat the case of 2-isogeny volcanoes and we derive from the group structure of the curve and the pairing a new invariant of the endomorphism class of an elliptic curve. Our benchmarks show that the resulting algorithm for endomorphism ring computation is faster than Kohel's method for computing the ℓ-adic valuation of the conductor of the endomorphism ring for small ℓ.

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Section: 2mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Pairing the Volcano

Ionica

Joux

2010

Lecture Notes in Computer Science

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“…These are (multi)graphs whose vertices are elliptic curves up to isomorphism, and whose edges are isogenies between them (again up to isomorphism). The use of isogeny graphs for algorithmic applications goes back to Mestre and Oesterlé [49], followed notably by Kohel [41], and has been continued by many authors [29,26,31,50,37].…”

Section: Isogeny Graphsmentioning

confidence: 99%

Towards Practical Key Exchange from Ordinary Isogeny Graphs

Feo

Kieffer

Smith

2018

Lecture Notes in Computer Science

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We revisit the ordinary isogeny-graph based cryptosystems of Couveignes and Rostovtsev-Stolbunov, long dismissed as impractical. We give algorithmic improvements that accelerate key exchange in this framework, and explore the problem of generating suitable system parameters for contemporary pre-and post-quantum security that take advantage of these new algorithms. We also prove the session-key security of this key exchange in the Canetti-Krawczyk model, and the IND-CPA security of the related public-key encryption scheme, under reasonable assumptions on the hardness of computing isogeny walks. Our systems admit efficient key-validation techniques that yield CCA-secure encryption, thus providing an important step towards efficient post-quantum non-interactive key exchange (NIKE).

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“…Finally, according to Certicom [17] , the elliptic curve should be selected with a cardinal of the form #E(F 2 137 ) = h · p, where p is a big prime and h 4 is the cofactor [14,15] . A point of prime order p should be chosen as generator.…”

Section: Selection Of Parametersmentioning

confidence: 99%

A Secure Elliptic Curve-Based RFID Protocol

Martínez

Valls

Roig

et al. 2009

J. Comput. Sci. Technol.

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Nowadays, the use of Radio Frequency Identification (RFID) systems in industry and stores has increased. Nevertheless, some of these systems present privacy problems that may discourage potential users. Hence, high confidence and efficient privacy protocols are urgently needed. Previous studies in the literature proposed schemes that are proven to be secure, but they have scalability problems. A feasible and scalable protocol to guarantee privacy is presented in this paper. The proposed protocol uses elliptic curve cryptography combined with a zero knowledge-based authentication scheme. An analysis to prove the system secure, and even forward secure is also provided.

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An algorithm to compute volcanoes of 2-isogenies of elliptic curves over finite fields

Cited by 13 publications

References 4 publications

Pairing the Volcano

Pairing the Volcano

Towards Practical Key Exchange from Ordinary Isogeny Graphs

A Secure Elliptic Curve-Based RFID Protocol

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