modular polynomials
Recently Published Documents


TOTAL DOCUMENTS

33
(FIVE YEARS 0)

H-INDEX

5
(FIVE YEARS 0)

2020 ◽  
Vol 15 (1) ◽  
pp. 31-44
Author(s):  
Yasushi Takahashi ◽  
Momonari Kudo ◽  
Ryoya Fukasaku ◽  
Yasuhiko Ikematsu ◽  
Masaya Yasuda ◽  
...  

AbstractRecently, supersingular isogeny cryptosystems have received attention as a candidate of post-quantum cryptography (PQC). Their security relies on the hardness of solving isogeny problems over supersingular elliptic curves. The meet-in-the-middle approach seems the most practical to solve isogeny problems with classical computers. In this paper, we propose two algebraic approaches for isogeny problems of prime power degrees. Our strategy is to reduce isogeny problems to a system of algebraic equations, and to solve it by Gröbner basis computation. The first one uses modular polynomials, and the second one uses kernel polynomials of isogenies. We report running times for solving isogeny problems of 3-power degrees on supersingular elliptic curves over 𝔽p2 with 503-bit prime p, extracted from the NIST PQC candidate SIKE. Our experiments show that our first approach is faster than the meet-in-the-middle approach for isogeny degrees up to 310.



2020 ◽  
Vol 216 ◽  
pp. 403-459
Author(s):  
Enea Milio ◽  
Damien Robert
Keyword(s):  


2020 ◽  
Vol 213 ◽  
pp. 464-498
Author(s):  
Chloe Martindale




2019 ◽  
Vol 15 (03) ◽  
pp. 569-584
Author(s):  
Fabien Pazuki

We provide explicit bounds on the difference of heights of the [Formula: see text]-invariants of isogenous elliptic curves defined over [Formula: see text]. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman’s inequality and isogeny estimates by Gaudron and Rémond. We give applications in the study of Vélu’s formulas and of modular polynomials.



2018 ◽  
Vol 188 ◽  
pp. 71-87 ◽  
Author(s):  
Adel Betina ◽  
Emmanuel Lecouturier
Keyword(s):  


2016 ◽  
Vol 165 ◽  
pp. 1-14
Author(s):  
Florian Breuer ◽  
Hans-Georg Rück


2015 ◽  
Vol 11 (02) ◽  
pp. 631-643 ◽  
Author(s):  
Srinath Baba ◽  
Håkan Granath

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.



2015 ◽  
Vol 18 (1) ◽  
pp. 603-632 ◽  
Author(s):  
Enea Milio

We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove heuristically that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.



Sign in / Sign up

Export Citation Format

Share Document