Let $G$ be a connected commutative algebraic group over $\mathbb{F}_q$. If $\text{Fr}_q : G \to G$ denotes the $q$-Frobenius morphism, we define the Lang isogeny $L_q$ to be the endomorphism of $G$ given by $g \mapsto \text{Fr}_q(g)g^{-1}$. I have two questions about this important map.

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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 called the Lang isogeny. Lang’s theorem has very useful consequences. We record the most basic one here: Corollary 1.5.

Lang isogeny

9 Feb 2020 The authors began this project during the semester program “Computational aspects of the Lang- lands program” held at ICERM in fall 2015. 19 Mar 2021 Lang's isogeny. L_. XI. Abel-Jacobi // PicX. L−1 ⊗ τ L. (xi )i∈I. ↦→ OX×S(∑ wi xi ).

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We achieve this by constructing some modular compactifications of (PGL r × PGL r × PGL r)/PGL r and of the Lang isogeny in PGL r. The map , induced by , is Lang isogeny which is sujective and separable. Hence induces an isomorphism on tangent spaces as wanted..

I For every degree ‘-isogeny ’: E !E0there exists a unique degree ‘-isogeny (called the dual) ’_: E0!E such that ’_ ’= [‘]. De nition An isogeny graph is a graph where a vertex represents the j-invariant of an elliptic curve over F q and an undirected edge represents a degree ‘isogeny de ned over F q and its dual.

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

The Lang isogeny of Gdeﬁned as the morphism L G(x) = ˙(x)x-1 is a ﬁnite, étale homomorphism of groups whose kernel is the discrete subgroup G(k). We have an exact sequence: 0 !G(k) !G!LG G!0. Every ‘-adic representation ˚: G(k) !GL(V) gives rise to a ‘-adic sheaf F ˚ on G, by means of the Lang isogeny. Its trace function theoretic shadow can be An isogeny $ f: G \rightarrow G _ {1} $ is said to be separable if $ \mathop{\rm ker} ( f ) $ is an étale group scheme over $ k $.
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0and The converse is trickier; it uses the Lang isogeny L G: G !G deﬁned by g 7!Frob(g)g1. This is an abelian étale cover of G with Galois group G(F q). This construction gives an N 2Loc 1(G) for any ˜: G(F q) !Z ‘. Exercise 1.5.

As the morphism is a quotient morphism of smooth stacks, it is smooth. We are left to prove the smoothness of . Let be the push forward of the inclusion via the group homomorphism .
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whether the isogeny class is a base change of an isogeny class de ned over a smaller eld (i.e., whether the isogeny class is primitive), and if it is not primitive, the isogeny classes for which it is a base change; the twists of the isogeny class: the isogeny classes to which it becomes isogenous after a base change.

2007-01-25 · We propose the first quantum-resistant password-authenticated key exchange scheme based on supersingular elliptic curve isogenies. The scheme is built upon supersingular isogeny Diffie-Hellman [15], and uses the password to generate permutations which obscure the auxiliary points. 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.

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

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.

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. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. supersingular isogeny graph 2010Childs-Jao-Soukharev: Apply Kuperberg’s (and Regev’s) hidden shift subexponential quantum algorithm to CRS 2011Jao-De Feo: Build Difﬁe-Hellman style key exchange from supersingular isogeny graph (SIDH) 2018De Feo-Kieffer-Smith: Apply new ideas to speed up CRS 2018Castryck-Lange-Martindale-Panny-Renes: Apply 2018-11-18 · 4 W.Castryck,T.Lange,C.Martindale,L.Panny,andJ.Renes mentationisabouttentimesfasterthanourproof-of-conceptCimplementation, butevenat80ms,CSIDHispractical. 2020-09-21 · Isogeny of complex tori, rather than isomorphism, will turn out to be the appropriate equivalence relation in the context of modular forms. Usage notes [ edit ] In some contexts, (e.g., universal algebra ), an epimorphism may be defined as a surjective homomorphism , and the definition of isogeny may change accordingly.