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Isogenies on twisted Hessian curves
2021
Journal of Mathematical Cryptology
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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
2019
Advances in Mathematics of Communications
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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
2012
Journal of Mathematical Cryptology
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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
2011
International Journal of Electrical and Computer Engineering (IJECE)
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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]
2015
Lecture Notes in Computer Science
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., 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]
2016
arXiv
pre-print
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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
2009
Annales mathématiques Blaise Pascal
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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]
2017
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
2018
Designs, Codes and Cryptography
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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]
2015
Lecture Notes in Computer Science
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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]
2010
arXiv
pre-print
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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
2015
Mathematics of Computation
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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]
2018
Lecture Notes in Computer Science
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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
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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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