# 2006-01-01

2018-12-15 · Isogenies on supersingular elliptic curves are a candidate for quantum-safe key exchange protocols because the best known classical and quantum algorithms for solving well-formed instances of the isogeny problem are exponential. We propose an implementation of supersingular isogeny Diffie-Hellman (SIDH) key exchange for complete Edwards curves.

0, up to isogeny one can always spread out Ato an abelian scheme Aas above. In Section 5, using the methods from Section 4, we prove Theorem 1.1: First we reduce to the situation considered above which allows us to de ne F. Then using Theorem 1.2 we show that Theorem 1.1 follows from a variant of the following formal Mordell-Lang result: 2018-11-18 Lang’s theorem Any smooth cubic curve Ede ned over a nite eld Ois called an isogeny. Therefore, an isogeny must be surjective and must have nite kernel. In fact, we noted that: An isogeny is a group homomorphism.

Lang calls L=K “of Albanese type” if its “geometric part” Lk=K¯ ¯k is obtained by pullback, via a canonical map ﬁ: V = VK! AK, from a separable isogeny B ! AK deﬁned over the algebraic closure ¯k of k. Such an extension is abelian if the isogeny and ﬁ are deﬁned over k and the kernel of Tate's isogeny theorem states that there is an isogeny from E 1 to E 2 which is defined over F p. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny. The algorithm proposed in this paper has exponential complexity in the worst case.

Qualification, Doctor of Philosophy. Awarding Institution.

## Objective The Journal of Mathematical Cryptology (JMC) is a forum for original research articles in the area of mathematical cryptology. JMC is a fully peer-reviewed, open access, electronic-only journal publishing works of wide significance, originality and relevance. Works in the theory of cryptology and articles linking mathematics with cryptology (including quantum cryptology) are welcome

In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the \(\ell \) -isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and Objective The Journal of Mathematical Cryptology (JMC) is a forum for original research articles in the area of mathematical cryptology. JMC is a fully peer-reviewed, open access, electronic-only journal publishing works of wide significance, originality and relevance. Works in the theory of cryptology and articles linking mathematics with cryptology (including quantum cryptology) are welcome Geometrization of the Local Langlands Program McGill May 6-10, 2019 Notes scribed by Tony Feng

### 2018-12-07

In the latter case, a uniform bound is obtained on the index of a prime term. isogeny class. In the present paper we generalize our isogeny estimates to abelian va-rieties of arbitrary dimension. We also prove the corresponding finiteness theorem, referred to by Lang [L] as Finiteness I; namely, if A is an abelian va-riety defined over a number field k, there are only finitely many k-isomorphism Isogeny-Based Cryptography Master’s Thesis Dimitrij Ray Department of Mathematics and Computer Science Coding Theory and Cryptology Group Supervisors: prof. dr. Tanja Lange dr.

Supervisors/Advisors. Supersingular Isogeny Key Encapsulation (SIKE) by Jao et al. Language. Processor. Scheme.

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It su ces to check that Lis etale on G k at any k-point g 0, with each ber L 1(L(g 0)) a right G(k)-coset inside G(k).

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Math 249C. Lang’s theorem and unirationality 1. Introduction This handout aims to prove two theorems. The rst theorem is very useful for solving problems with connected reductive groups over in nite elds, and the second is useful for bypassing the failure of the Zariski-density consequences of the rst theorem when working over nite elds.

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### (Lang-Tate. [11]). As will be shown in •˜2, we can generalize this reasoning to an arbitrary isogeny ƒة: G•¨H of group varieties, defined over a global field k which.

Chloe Martindale Lorenz Panny, MSc Eindhoven, July 2018 Hello! I love solving difficult security problems with cryptography. By day, I deploy secure cryptography in Texas Instrument’s IoT devices. By night, I research post-quantum cryptography. Recently, I have been actively investigating applications, security, and implementations of isogeny-based cryptography.