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Isogenies on twisted Hessian curves

Fouazou Lontouo Perez Broon, Thinh Dang, Emmanuel Fouotsa, Dustin Moody
2021 Journal of Mathematical Cryptology  
Continuing this line of work, this paper derives explicit formulas for isogenies between elliptic curves in (twisted) Hessian form.  ...  In comparison with other isogeny formulas, we note that our formulas for twisted Hessian curves have the lowest costs for processing the kernel and our X-affine formula has the lowest cost for processing  ...  We now turn to formulas for 3-isogenies of twisted Hessian curves.  ... 
doi:10.1515/jmc-2020-0037 pmid:34322179 pmcid:PMC8314185 fatcat:5a43fqnas5dzne4hi6e7igrdui

Isogeny formulas for Jacobi intersection and twisted hessian curves

João Paulo da Silva, ,Institute of Computing, University of Campinas, Av. Albert Einstein 1251, Cidade Universitária "Zeferino Vaz", 13083-852, Campinas, SP, Brazil, Julio López, Ricardo Dahab
2019 Advances in Mathematics of Communications  
Shumow [17], we derived maps for elliptic curves represented in Jacobi Intersection and Twisted Hessian models.  ...  Finally, we present a comparison of computational cost to generate maps for isogenies of degree l, where l = 2k + 1.  ...  Here, we just present the maps used as isogeny evaluation for Twisted Hessian curves.  ... 
doi:10.3934/amc.2020048 fatcat:7k7lmpxbybanld43g5y4hxm6ry

Families of elliptic curves with rational 3-torsion

Dustin Moody, Hongfeng Wu
2012 Journal of Mathematical Cryptology  
These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves.  ...  We find the number of F q -isogeny classes of each family, as well as the number of F q -isomorphism classes of the generalized DIK curves.  ...  So there is likewise always a 3-isogeny to another twisted Hessian curve, resulting in pairs for each isogeny class.  ... 
doi:10.1515/jmc-2011-0013 fatcat:ff36jr4wkrddnmdob43uvjmgwu

A Survey Report On Elliptic Curve Cryptography

Samta Gajbhiye, Monisha Sharma, Samir Dashputre
2011 International Journal of Electrical and Computer Engineering (IJECE)  
The curve and its twists have the same j-invariant and is shown in [17] . Twisted Hessian curve [18] represents a generalization of Hessian curve.  ...  In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists.  ... 
doi:10.11591/ijece.v1i2.86 fatcat:etikame46ve2nkrio52rprs7ky

Twisted Hessian Curves [chapter]

Daniel J. Bernstein, Chitchanok Chuengsatiansup, David Kohel, Tanja Lange
2015 Lecture Notes in Computer Science  
., switching from Weierstrass to twisted Hessian saves yM. We reduced the number of dots plotted in this figure to avoid excessive PDF file sizes and display times, but a full plot is similar.  ...  triplings; the line has a positive slope, i.e., Hessian is faster.  ...  Let H be the twisted Hessian curve aX 3  ... 
doi:10.1007/978-3-319-22174-8_15 fatcat:3ei2izlv5zfpnaoaatz6d262t4

The geometry of efficient arithmetic on elliptic curves [article]

David Kohel
2016 arXiv   pre-print
means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies  ...  relative comparison of complexities of [ℓ] and addition ⊕ on twisted Hessians ([ℓ] = [3]) and on twisted Edwards models and Jacobi quartics ([ℓ]  ...  In the case of the twisted Hessian, the dual isogeny ψ =φ is given by (X : Y : Z) → (f 0 : f 1 : f 2 ), where f 0 = X 3 + Y 3 + Z 3 − 3XY Z, f 1 = X 2 Y + Y 2 Z + XZ 2 − 3XY Z, f 2 = XY 2 + Y Z 2 + X 2  ... 
arXiv:1601.03665v1 fatcat:3bl45fh765gvhb3s3o4itzn6x4

Twists of Hessian Elliptic Curves and Cubic Fields

Katsuya Miyake
2009 Annales mathématiques Blaise Pascal  
The Twist H µ,t ofH(µ, t) over K t In this section we give a twist H µ,t ofH(µ, t) over K t , and also show that H µ,t is a quadratic twist of the Hessian curve H µ .  ...  The 3-Isogeny of H µ,t By Proposition 2.2 in Section 2, we gave a simple affine model A µ of the 3-isogeny of the Hessian elliptic curve H µ for µ = 1; namely, A µ : y 2 + 3µxy + y = x 3 .  ... 
doi:10.5802/ambp.251 fatcat:prfbmbz6fbgonejjhwg7ki34ve

Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field [article]

Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver, Ari Shnidman
2017 arXiv   pre-print
of 3-isogenies over F.  ...  In dimension one, we deduce that if E/F is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded.  ...  For each s ∈ F * , we have the quadratic twists A s and A ′ s and a 3-isogeny φ s : A s → A ′ s .  ... 
arXiv:1709.09790v2 fatcat:qk3iue57cbf3tgml7d5wto7sga

High-degree compression functions on alternative models of elliptic curves and their applications [article]

Michał Wroński, Tomasz Kijko, Robert Dryło
2022 arXiv   pre-print
We will study alternative models of elliptic curves with points of order 2 and 4, and specifically Huff's curves and the Hessian family of elliptic curves (like Hessian, twisted Hessian and generalized  ...  Hessian curves) with a point of order 3.  ...  , generalized Hessian and twisted Hessian curves 4.2.1.  ... 
arXiv:2111.04533v2 fatcat:z7f4t65hobd57c2rivnnjpbezy

Speeding up Huff form of elliptic curves

Neriman Gamze Orhon, Huseyin Hisil
2018 Designs, Codes and Cryptography  
The original Huff form was introduced as ax(y 2 − 1) = by(x 2 − 1) by Huff in [9] and a twisted version to cover more elliptic curves was given as ax(y 2 −d) = by(x 2 −d) by Joye, Tibouchi and Vergnaud  ...  E.g. ax 2 + y 2 = 1 + dx 2 y 2 (twisted Edwards form) [1] , y 2 = dx 4 + 2ax 2 + 1 (extended Jacobi quartic form) [4] , ax 3 +y 3 +1 = dxy (twisted Hessian form) [2] , y 2 = x 3 +ax+b (short Weierstrass  ...  Therefore, ϕ is an isogeny from H to G.  ... 
doi:10.1007/s10623-018-0475-4 fatcat:cavy6xkjwjgkjdfk3fd76phmz4

Decaf: Eliminating Cofactors Through Point Compression [chapter]

Mike Hamburg
2015 Lecture Notes in Computer Science  
We propose a new unified point compression format for Edwards, Twisted Edwards and Montgomery curves over large-characteristic fields, which effectively divides the curve's cofactor by 4 at very little  ...  While such laws exist for prime-order curves [6, 11] , they are faster and much simpler for other elliptic curves such as (twisted) Edwards curves [14, 5, 4] , Hessian curves [15] , Jacobi quartics  ...  The cofactor h is always divisible by 3 for Hessian curves, and by 4 for the other models.  ... 
doi:10.1007/978-3-662-47989-6_34 fatcat:ubxc7wpqjfcmthvndho7np5yrq

Mass deformations of four-dimensional, rank 1, N=2 superconformal field theories [article]

Philip C. Argyres, John Wittig
2010 arXiv   pre-print
The map between the Jacobi and Legendre forms is a 2-isogeny, while that between the Hessian and Legendre forms is a 3-isogeny. 2-isogenies We can map the Jacobi form to the Legendre form by x = x 1  ...  In particular, there are three traditional presentations of elliptic curves which are related by simple isogenies: Legendre: y 2 = x 3 + f x + g, Jacobi: y 2 = x 4 + α x 2 + β, (4.1) Hessian: γ = y 3 +  ... 
arXiv:1007.5026v1 fatcat:dwlmou6x4feqznbqdwpfj5cxgi

Analogues of Vélu's formulas for isogenies on alternate models of elliptic curves

Dustin Moody, Daniel Shumow
2015 Mathematics of Computation  
Isogenies of elliptic curves have been well-studied, in part because there are several cryptographic applications. Using Vélu's formula, isogenies can be evaluated explicitly given their kernel.  ...  In this paper we show how to similarly evaluate isogenies on Edwards curves and Huff curves.  ...  Another research topic is to find similar isogeny formulas for other models of curves, such as Hessian curves, Jacobi quartics, or Jacobi intersections.  ... 
doi:10.1090/mcom/3036 fatcat:wpk2bldlwzguvp3f6snilkbaii

Computing Isogenies Between Montgomery Curves Using the Action of (0, 0) [chapter]

Joost Renes
2018 Lecture Notes in Computer Science  
As a particular case, we provide efficient formulas for 2-isogenies between Montgomery curves and show that these formulas can be used in isogeny-based cryptosystems without expensive square root computations  ...  This generalization removes the restriction of a cyclic kernel and allows for any separable isogeny whose kernel does not contain (0, 0).  ...  This curve is known as a triangular curve [BCKL15] and is isomorphic to the twisted Hessian curve [BCKL15, Theorem 5.3] (a 3 − 27)x 3 + y 3 + 1 = 3axy . SIDH.  ... 
doi:10.1007/978-3-319-79063-3_11 fatcat:muxrwtdndbc73elopkucjqfkwm

Page 6078 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
The author also considers the Hessian family of elliptic curves E*: y?>+axy+y =x’, all of which have a point of order 3 at (0,0).  ...  But then Frey’s 3-descent shows that the 3-Selmer group of the twisted curve £¢ is trivial.  ... 
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