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. Chloe Martindale Lorenz Panny, MSc Eindhoven, July 2018

27 Jan 2013 The Lang isogeny Let G be a connected commutative algebraic group over Fq. If Frq:G→G denotes the q-Frobenius morphism, we define the 

For the etale property of Lat g 0, it is equivalent to prove the etale property of g7!L(g 0g) at the identity. But L(g 0g) = g [q] 0 (g [q]g 1)g 1 0 = g [q] 0 L(g) g 0 : An important example of an isogeny is the multiplication [n] X: X → X by an integer n != 0. We write X[n] := Ker([n] X) ⊂ X. (5.9) Proposition. For n !=0 , the morphism [n] X is an isogeny.

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For the etale property of Lat g 0, it is equivalent to prove the etale property of g7!L(g 0g) at the identity. But L(g 0g) = g [q] 0 (g [q]g 1)g 1 0 = g [q] 0 L(g) g 0 : An important example of an isogeny is the multiplication [n] X: X → X by an integer n != 0. We write X[n] := Ker([n] X) ⊂ X. (5.9) Proposition. For n !=0 , the morphism [n] X is an isogeny. If g =dim(X),wehave deg([n] X)=n2g.If(char(k),n)=1then [n] X is separable. Proof.

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. 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).

Lang map L G: G! L G G defined by L G(g) = ˙(g)g1. Since G is commutative, this is a homomorphism of groups, which is even an étale isogeny (since ˙has vanishing di erential). The kernel is evidently G(k), so we have a short exact sequence 0 !G(k) !G! L G G !0: Example 1.4. If G = G m then L G(x) = xq1, the Kummer isogeny, and we obtain the

Advances in Mathematics of Communications , 2020, 14 (3) : 507-523. doi: 10.3934/amc.2020048 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 Geometrization of the Local Langlands Program McGill May 6-10, 2019 Notes scribed by Tony Feng 1973-12-01 Lattices, elliptic curves over the complex numbers and isogeny graphs Marios Magioladitis University of Oldenburg July 2011 searching for Isogeny 21 found (60 total) alternate case: isogeny.

Lang calls L/K“of Albanese type” if its “geometric part” Lk/K¯ ¯k is obtained by pullback, via a canonical map α: 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 αare defined over kand the kernel of the isogeny consists of k-rational points.

We are left to prove the smoothness of . Let be the push forward of the inclusion via the group homomorphism . For the CSIDH-1024 prime, 2018 Castryck–Lange–Martindale–Panny–Renes included portable software, and velusqrt-asm includes asm software. 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. 2018-03-31 equations for evaluating an isogeny with kernel F at point P given by V elu’s formulas: ˚(P) = 0 @x P + X Q2Fnf1g (x P+Q x Q);y p + X Q2Fnf1g (y P+Q y Q) 1 A Isogeny formulas equivalent to V elu’s for Edwards curves were found by Moody and Shumow (2011). They presented new formulas for odd isogenies, and composite formulas for even isogenies (with kernel The following is the coding required for this isogeny : A sample run [ here ] is given next, and where the mapping of (1120,1391) on E2 is seen to map to (565,302) on E4: Let E 1 and E 2 be ordinary elliptic curves over a finite field F p such that # E 1 (F p) = # E 2 (F p).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. isogeny class.

In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex 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. Chloe Martindale Lorenz Panny, MSc Eindhoven, July 2018 Compactification de l'isogénie de Lang et dégénérescence des structures de niveau simple des chtoucas de Drinfeld Compactification of the Lang isogeny and degeneration of simple level structures of Drinfeld's shtukas. Note présentée par Pierre Deligne. Author links open overlay panel Laurent Lafforgue.
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An isogeny $ f: G \rightarrow G _ {1} $ is said to be separable if $ \mathop{\rm ker} ( f ) $ is an étale group scheme over $ k $. This is equivalent to the fact that $ f $ is a finite étale covering. An example of a separable isogeny is the homomorphism $ n _ {G} $, where $ ( n, p) = 1 $.

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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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. Check that N is in fact a character local system, and that these constructions are inverse. 2

The Lang isogeny induces a map ˇ 1(G;e) !G(k) (viewing it as a fibration over G with fiber G(k)). Thus, given a character ˜: G(k) !Q ‘ we can compose to get a character of the fundamental group. We denote the corresponding local system by L ˜. ˇ 1(G;e) / $ G(k) Q ‘ Example 1.5. See a book of Katz for a reference. If G = G m, and ˜: k !Q ‘ then L 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).

e) tunna trådar (whiskers), antingen mono- eller polykristallina av valfri längd, f) aromatisk SIKE (Supersingular Isogeny Key. Encapsulation). 3. Avkodning av 

doi: 10.3934/amc.2020048 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 Geometrization of the Local Langlands Program McGill May 6-10, 2019 Notes scribed by Tony Feng 1973-12-01 Lattices, elliptic curves over the complex numbers and isogeny graphs Marios Magioladitis University of Oldenburg July 2011 searching for Isogeny 21 found (60 total) alternate case: isogeny. Jacobian variety (713 words) exact match in snippet view article find links to article Honda–Tate theorem – classifies abelian varieties over finite fields upto isogeny David, Mumford; Nori, Madhav; Previato 2006-01-01 Posted by Akhil Mathew under algebraic geometry, number theory | Tags: crazy ideas, Fourier-Deligne transform, l-adic cohomology, Lang isogeny, torsors | Leave a Comment The topic of this post is a curious functor, discovered by Deligne, on the category of sheaves over the affine line, which is a “sheafification” of the Fourier transform for functions.

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. 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.