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Degenerate Curve Attacks [chapter]

Samuel Neves, Mehdi Tibouchi
2016 Lecture Notes in Computer Science  
In this paper, we dispel that belief and present the first attack of this nature against (twisted) Edwards curves, Jacobi quartics, Jacobi intersections and more.  ...  Invalid curve attacks are a well-known class of attacks against implementations of elliptic curve cryptosystems, in which an adversary tricks the cryptographic device into carrying out scalar multiplication  ...  Extended Jacobi Quartics. Let E a,b : y 2 = dx 4 + 2ax 2 + 1 be an extended Jacobi quartic curve over F p , and consider the set G of points in F 2 p of the form (0, y), y = 0.  ... 
doi:10.1007/978-3-662-49387-8_2 fatcat:momfsae73jhmbivqc34gyhokwy

Degenerate curve attacks: extending invalid curve attacks to Edwards curves and other models

Samuel Neves, Mehdi Tibouchi
2018 IET Information Security  
In this paper, we dispel that belief and present the first attack of this nature against (twisted) Edwards curves, Jacobi quartics, Jacobi intersections and more.  ...  Invalid curve attacks are a well-known class of attacks against implementations of elliptic curve cryptosystems, in which an adversary tricks the cryptographic device into carrying out scalar multiplication  ...  Extended Jacobi quartics. Let E a,b : y 2 = dx 4 + 2ax 2 + 1 be an extended Jacobi quartic curve over F p , and consider the set G of points in F 2 p of the form (0, y), y = 0.  ... 
doi:10.1049/iet-ifs.2017.0075 fatcat:ci32wljdzbcejjeiarxz3ps6kq

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.  ...  In affine coordinates, our formulas require 46.8% less computation than the Huff model and 48% less computation than the formulas given for the Extended Jacobi Quartic model when computing isogenies of  ...  Jacobi Quartic [24] ---Weierstrass [17] ---Jacobi Intersection computation than the Huff model and 48% less computation than the given formula for the Extended Model Jacobi Quartic when computing isogenies  ... 
doi:10.3934/amc.2020048 fatcat:7k7lmpxbybanld43g5y4hxm6ry

Decaf: Eliminating Cofactors Through Point Compression [chapter]

Mike Hamburg
2015 Lecture Notes in Computer Science  
This allows cofactor-4 curves to efficiently implement prime-order groups.  ...  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  ...  Jacobi quartic curves. A Jacobi quartic curve has two parameter, called A and e, and is defined by J e,A := (s, t) ∈ P 2 (F) : t 2 = es 4 + 2As 2 + 1 with an identity point at (0, 1).  ... 
doi:10.1007/978-3-662-47989-6_34 fatcat:ubxc7wpqjfcmthvndho7np5yrq

Exponentiating in Pairing Groups [chapter]

Joppe W. Bos, Craig Costello, Michael Naehrig
2014 Lecture Notes in Computer Science  
study exponentiations in pairing groups for the most common security levels and show that, although the Weierstrass model is preferable for pairing computation, it can be worthwhile to map to alternative curve  ...  Endomorphisms on the Jacobi Quartic Model.  ...  The curve W is birationally equivalent to the (extended) Jacobi quartic curve J : (8) where τ ((−θ : 0: 1)) = (0: − 1: 1) ∈ J .  ... 
doi:10.1007/978-3-662-43414-7_22 fatcat:fohleebcbjfefitqex4mcugmly

New number-theoretic cryptographic primitives

Éric Brier, Houda Ferradi, Marc Joye, David Naccache
2019 Journal of Mathematical Cryptology  
The generalized signature schemes replace the Jacobi symbol with higher-power residue symbols.  ...  The signature is a bounded-size prime whose Jacobi symbols with respect to the {n_{i}} 's match the message digest.  ...  Definition 3 (Jacobi imprint).  ... 
doi:10.1515/jmc-2019-0035 fatcat:4qn7el77gnbh5kdfhm3bfpakji

A comparison between the secp256r1 and the koblitz secp256k1 bitcoin curves

Azine Houria, Bencherif Mohamed Abdelkader, Guessoum Abderezzak
2019 Indonesian Journal of Electrical Engineering and Computer Science  
<p><span>Bitcoin uses elliptic curve cryptography for its keys and signatures, but the specific secp256k1 curve used is rather unusual.  ...  Due to this characteristic, several questions come up concerning Satoshi's choice of this curve rather than that of the NIST standard secp256r1 curve.  ...  2 Extended Jacobi Quartic form 3 Generalized Hessien 4 Jacobi Quartic form or Edwards form e) Parameter b Table 4 . 4 NIST-Recommended Random Elliptic Curves Over Prime Fields [6] Parameters  ... 
doi:10.11591/ijeecs.v13.i3.pp910-918 fatcat:zwp5wriawjeu7c4mwftcbkhy7y

Hashing into Generalized Huff Curves [chapter]

Xiaoyang He, Wei Yu, Kunpeng Wang
2016 Lecture Notes in Computer Science  
Moreover, based on our deterministic encodings, we construct two hash functions from messages to generalized Huff curves indifferentiable from a random oracle.  ...  Huff curves are well known for efficient arithmetics to their group law. In this paper, we propose two deterministic encodings from Fq to generalized Huff curves.  ...  Conclusion We provide two constructions of deterministic encoding into generalized Huff curves over finite fields, namely, brief SWU encoding and cube root encoding.  ... 
doi:10.1007/978-3-319-38898-4_2 fatcat:ibwfa3pyijgn5hb26mkzbbm3x4

An empirical study to demonstrate that EdDSA can be used as a performance improvement alternative to ECDSA in Blockchain and IoT

Guruprakash J, Srinivas Koppu
2022 Informatica (Ljubljana, Tiskana izd.)  
Then, we performed an empirical comparison of ECDSA with the Edwards curve digital signature algorithm (EdDSA).  ...  This paper investigates the widely used elliptic curve digital signature algorithm (ECDSA) and its application to blockchain and IoT.  ...  Curve ADD reADD mADD DBL UNI Edwards 10M +1S+1D 10M+1S+1D 9M+1S+1D 3M+4S 10M+1S+1D Hessian 12M 12M 10M 7M+1S 12M Jacobi intersection 13M+2S+1D 11M+2S+1D 11M+2S+1D 3M+4S 13M+2S+1D Jacobi quartic 10M+3S+  ... 
doi:10.31449/inf.v46i2.3807 fatcat:dbn2dwzxkrbavg2c6d5oql7mqe

The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds [chapter]

Andreas-Stephan Elsenhans, Jörg Jahnel
2006 Lecture Notes in Computer Science  
We decided to store the values of z e +v e +w e into the hash table. Afterwards, we have to look up the values of ax e − by e .  ...  Its fibers are plane quartics which split into two conics as (v + w) 4 + v 4 + w 4 = 2(v 2 + vw + w 2 ) 2 . After resolution of singularities, the two conics become disjoint.  ... 
doi:10.1007/11792086_23 fatcat:omjybcmwvzaj7l5rdljm5kbhwy

Report on Pairing-based Cryptography

Dustin Moody, Rene Peralta, Ray Perlner, Andrew Regenscheid, Allen Roginsky, Lily Chen
2015 Journal of Research of the National Institute of Standards and Technology  
The first panel discussed where and how IBE could fit into the NIST cryptographic toolkit, along with barriers to adoption.  ...  Mathematical Background Since 1985, elliptic curves have been used in cryptography. Elliptic curve cryptosystems have some advantages over other systems.  ...  These include: i) Huff curves: x(ay 2 -1) = y(bx 2 -1) ii) Jacobi quartics: y 2 = ex 4 -2dx 2 + 1 iii) (twisted) Edwards curves: (a)x 2 + y 2 = 1 + dx 2 y 2 We include a M,S are operations in F q k , while  ... 
doi:10.6028/jres.120.002 pmid:26958435 pmcid:PMC4730686 fatcat:zxzb76skivd5zhfg5hmiohichq

Hyperelliptic Curves with Compact Parameters

Ezra Brown, Bruce T. Myers, Jerome A. Solinas
2005 Designs, Codes and Cryptography  
We present a family of hyperelliptic curves whose Jacobians are suitable for cryptographic use, and whose parameters can be specified in a highly efficient way.  ...  (System Parameters)Input: the hash value c of the user's ID Output: p, r, d 1.  ...  We suggested that this random start value be determined by the hash output of an ID string chosen by the user.  ... 
doi:10.1007/s10623-004-1718-0 fatcat:kvmx3ihpg5e5vbrlx4clt7bnki

Protecting ECC Against Fault Attacks: The Ring Extension Method Revisited

Marc Joye
2020 Journal of Mathematical Cryptology  
This paper revisits the ring extension method and its adaptation to the elliptic curve setting.  ...  AbstractDue to its shorter key size, elliptic curve cryptography (ECC) is gaining more and more popularity.  ...  A Further Models A.1 Jacobi Quartic Model The (extended) Jacobi quartic model is presented in [9] (see also [16, 22, 29, 31] ).  ... 
doi:10.1515/jmc-2019-0030 fatcat:ymsslt5qtngwlggbem7aknci5m

Analogue of Vélu's Formulas for Computing Isogenies over Hessian Model of Elliptic Curves [article]

Fouazou Lontouo Perez Broon, Emmanuel Fouotsa
2019 IACR Cryptology ePrint Archive  
Vélu's formulas for computing isogenies over Weierstrass model of elliptic curves has been extended to other models of elliptic curves such as the Huff model, the Edwards model and the Jacobi model of  ...  elliptic curves.  ...  The Section 5 will be devoted to a comparison of the computational cost in term of basic fields operations of isogenies over Edward, Huff, Jacobi quartic and Hessian models of elliptic curves.  ... 
dblp:journals/iacr/BroonF19 fatcat:btx36lp6ffbtppiazz3rnrr3cy

Space-Efficient Identity Based EncryptionWithout Pairings

Dan Boneh, Craig Gentry, Michael Hamburg
2007 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)  
Identity Based Encryption (IBE) systems are often constructed using bilinear maps (a.k.a. pairings) on elliptic curves.  ...  Consequently, we can hash (ID, i) into [1, 8N 2/3 ] without hurting the simulation. This can potentially speed up prime number generation.  ...  The proof is by direct substitution into (3) .  ... 
doi:10.1109/focs.2007.50 dblp:conf/focs/Marx07a fatcat:uiv4n2oatneqpk5zmnz74gwaxu
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