AM_Ple_LegConsolidated

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Mathematics and Computer Science. Supervisors/Advisors. Supersingular Isogeny Key Encapsulation (SIKE) by Jao et al. Language.

Lang isogeny

CSIDH restricts the isogeny graph under consideration to supersingular elliptic curves and isogenies de ned over F 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. We introduce a new constant-time variable-degree isogeny algorithm, a new application of the Elligator map, new ways to handle failures in isogeny computations, new combinations of the components of these computations, new speeds for integer multiplication, and more.

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isogeny-based cryptography makes use of isogenies between elliptic curves. An isogeny overF q as˚: E!E0asanon-constantrationalmapfrom E(F q) to E0(F q) thatisalsoagrouphomomorphism.Theisogeny’sdegree isitsdegreeas analgebraicmap.Sincethecomplexityofcomputinganisogenyscaleslinearly withthedegree,itispracticalonlytocomputeisogeniesofasmallbasedegree.

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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).

The kernel is evidently G(k), so we have a short exact sequence 0 !G(k) !G!
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Check that N is in fact a character local system, and that these constructions are inverse. 2 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. isogeny-based cryptography makes use of isogenies between elliptic curves. An isogeny overF q as˚: E!E0asanon-constantrationalmapfrom E(F q) to E0(F q) thatisalsoagrouphomomorphism.Theisogeny’sdegree isitsdegreeas analgebraicmap.Sincethecomplexityofcomputinganisogenyscaleslinearly withthedegree,itispracticalonlytocomputeisogeniesofasmallbasedegree.

Isogeny computation: velusqrt-asm includes new software for the new isogeny-evaluation algorithm and for the relevant polynomial arithmetic, and automatically tunes the parameter choices in the new algorithm.
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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. Objective The Journal of Mathematical Cryptology (JMC) is a forum for original research articles in the area of mathematical cryptology.

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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.

2 Lang map L G: G! L G G deﬁned by L G(g) = ˙(g)g1.