Evaluating software packages for attacks against elliptic curve cryptography by Moore, N

“…Until the coming of quantum computers and post-quantum cryptography, elliptic curve cryptography seems to be a prime candidate for modern encryption systems. …”

Unlikely intersections in products of families of elliptic curves and the multiplicative group by Barroero, F, Capuano, L

“…Let Eλ be the Legendre elliptic curve of equation Y 2 = X(X - 1)(X - λ). We recently proved that, given n linearly independent points P1(λ), ..., Pn(λ) on Eλ with coordinates in Q(λ), there are at most finitely many complex numbers λ0 such that the points P1(λ0), ..., Pn(λ0) satisfy two independent relations on Eλ0 . …”

Appendix and erratum to "Massey products for elliptic curves of rank 1" by Balakrishnan, J, Kedlaya, K, Kim, M

Key exchange in elliptic curve cryptography based on the decomposition problem by Hilyati Hanina Zazali, Wan Ainun Mior Othman

“…The security of our protocol was based on discrete logarithm problem, which was not infeasible and strictly difficult to retrieve in elliptic curve cryptography without any prior knowledge.…”
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An improved binary method for scalar multiplication in elliptic curve cryptography by Mohamed, Mohamad Afendee, Md. Said, Mohamad Rushdan, Mohd Atan, Kamel Ariffin, Ahmad Zulkarnain, Zuriati

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Signed decomposition method for scalar multiplication in elliptic curve cryptography by Md. Said, Mohamad Rushdan, Mohamed, M. A., Mohd Atan, Kamel Ariffin, Ahmad Zukarnain, Zuriati

“…Addition chain is the solution to computability constraint of the problematic large number arithmetic. In elliptic curve cryptography, a point arithmetic on elliptic curve can be reduced to repetitive addition and doubling operations. …”
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Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication by Park, Peter S., Swaminathan, Ashvin A., Chen, Evan

“…Let E/ℚ be an elliptic curve with complex multiplication (CM), and for each prime p of good reduction, let a[subscript E](p) = p + 1 − #E(𝔽[subscript p]) denote the trace of Frobenius. …”
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On L-functions of modular elliptic curves and certain K3 surfaces by Amir, Malik, Hong, Letong

“…We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number $$\ge 19$$ ≥ 19 . …”
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Elliptic curves and cryptography : a pseudorandom bit generator and other tools by Kaliski, Burton Stephen

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P-adic L-functions for elliptic curves over CM fields by Haran, Shai M. J. Haran.

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Speeding up secure web transactions using elliptic curve cryptography by Lim, Lai Koon.

“…Elliptic Curve Cryptography (ECC) is becoming an attractive alternative to traditional public-key cryptography systems like RSA, Digital Signature Algorithm (DSA) and Diffie-Hellman (DH) as ECC provides similar level of security with smaller key sizes resulting in faster computations, lower power consumption, as well as memory and bandwidth savings. …”
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On the low-lying zeros of Hasse–Weil L-functions for elliptic curves by Baier, Stephan, Zhao, Liangyi

“…In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. …”
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Descent Via Isogeny on Elliptic Curves with Large Rational Torsion Subgroups. by Flynn, E, Grattoni, C

“…We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. …”

Descent via isogeny on elliptic curves with large rational torsion subgroups. by Flynn, E, Grattoni, C

“…We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. …”

Natural boundaries for Euler products of Igusa zeta functions of elliptic curves by du Sautoy, M

“…We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.…”

On the Improvement of Addition Chain in Applications to Elliptic Curve Cryptosystem Status: Submitted by Mohamed, Mohamad Afendee

“…A hard problem most of the time can be broken into a sequence of simple tasks from which a solution to the original problem is obtainable. Originally, elliptic curve cryptography is based on a non-singular algebraic curve of genus 1. …”
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On the mathematical complexity and the time implementation of proposed variants of elliptic curves cryptosystems by Al-Saffar, Najlae F. Hameed, Md. Said, Mohamad Rushdan

“…The group of the elliptic curve points forms an abelian group, which is considered as a suitable choice for constructing a problem similar to the Discrete Logarithm Problem. …”
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Integer Sub-Decomposition (Isd) Method For Elliptic Curve Scalar Multiplication by Ajeena, Ruma Kareem K.

“…Oleh yang demikian formula perkalian kP scalar ISD boleh dinyatakan seperti berikut: kP = k11P+k12ψ1(P)+k21P+k22ψ2(P): In this study, a new method called integer sub-decomposition (ISD) based on the Gallant, Lambert, and Vanstone (GLV) method to compute the scalar multiplication kP of the elliptic curve E over prime finite field Fp that have efficient computable endomorphisms ψj for j = 1; 2, resulting in pre-computed values of λ jP, where λ j ∈ [1;n−1] has been proposed. …”
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New Representation Method For Integers And Its Application On Elliptic Curve Cryptography by Eghdamian, Arash

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Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves by Bhargava, Manjul, Kane, Daniel M., Lenstra, Hendrik W., Poonen, Bjorn, Rains, Eric

“…Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.…”
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