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Efficient Arithmetic on Hessian Curves [chapter]

Reza R. Farashahi, Marc Joye
2010 Lecture Notes in Computer Science  
This paper provides efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen parameters.  ...  In particular, the group of F-rational points on a generalized Hessian curve has complete addition formulas if and only if c is not a cube in F.  ...  Several forms of elliptic curves over finite fields with several coordinate systems have been studied to improve the efficiency and the speed of the arithmetic on the group law (mainly addition and doubling  ... 
doi:10.1007/978-3-642-13013-7_15 fatcat:hh45ydg65ncrjgc4zp7cwwdd5i

Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three [chapter]

Reza R. Farashahi, Hongfeng Wu, Chang-An Zhao
2013 Lecture Notes in Computer Science  
This paper presents new explicit formulae for the point doubling, tripling and addition for ordinary Weierstraß elliptic curves with a point of order 3 and their equivalent Hessian curves over finite fields  ...  The cost of basic point operations is lower than that of all previously proposed ones.  ...  For example, Koblitz implemented the digital signature algorithm on a special family of supersingular elliptic curves in characteristic three with great efficiency [14] .  ... 
doi:10.1007/978-3-642-35999-6_10 fatcat:y2frex6zr5bn3azkgqusxs5mmu

Pairing-Friendly Twisted Hessian Curves [chapter]

Chitchanok Chuengsatiansup, Chloe Martindale
2018 CSR und Social Enterprise  
This paper presents efficient formulas to compute Miller doubling and Miller addition utilizing degree-3 twists on curves with j-invariant 0 written in Hessian form.  ...  We give the formulas for both odd and even embedding degrees and for pairings on both G1 × G2 and G2 × G1.  ...  This implies that the point arithmetic has to be performed on curves in Weierstrass form -not all curves can be written in special forms such as Hessian or Edwards form.  ... 
doi:10.1007/978-3-030-05378-9_13 dblp:conf/indocrypt/Chuengsatiansup18 fatcat:darg3zlvrjgldf35j6reusqz7i

Efficient FPGA Implementations of Point Multiplication on Binary Edwards and Generalized Hessian Curves Using Gaussian Normal Basis

R. Azarderakhsh, A. Reyhani-Masoleh
2012 IEEE Transactions on Very Large Scale Integration (vlsi) Systems  
This paper presents the implementation results of an elliptic curve crypto-processor over binary fields (2 ) on binary Edwards and generalized Hessian curves using Gaussian normal basis (GNB).  ...  Efficient implementation of point multiplication is crucial for elliptic curve cryptographic systems.  ...  In Section II, preliminaries of Gaussian normal basis representation and arithmetics on the binary Edwards and generalized Hessian curves are presented.  ... 
doi:10.1109/tvlsi.2011.2158595 fatcat:aixceifoazexxhrxagsla2vrjy

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

Michał Wroński, Tomasz Kijko, Robert Dryło
2022 arXiv   pre-print
Hessian curves) with a point of order 3.  ...  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  ...  The role that symmetries on elliptic curves play in the efficiency of their arithmetics has been widely discussed e.g. in [16] and [17] .  ... 
arXiv:2111.04533v2 fatcat:z7f4t65hobd57c2rivnnjpbezy

Toric forms of elliptic curves and their arithmetic

Wouter Castryck, Frederik Vercauteren
2011 Journal of symbolic computation  
We scan a large class of one-parameter families of elliptic curves for efficient arithmetic.  ...  The generality of this algorithm is further illustrated by computing uniform addition formulas and formulas for generalized Montgomery arithmetic.  ...  For each elliptic curve, the first line of output contains its index (in our list S 6 ) and the curve equation. The second line is the base face τ b .  ... 
doi:10.1016/j.jsc.2011.02.003 fatcat:xl7orq4jmbcjfg42bebmk5hg3u

On the Implementation of Unified Arithmetic on Binary Huff Curves [chapter]

Santosh Ghosh, Amit Kumar, Amitabh Das, Ingrid Verbauwhede
2013 Lecture Notes in Computer Science  
 Introduction  Introduction Power analysis of UBHC • Simple power analysis on SASEBO Board • Pin-point the vulnerability of UBHC arithmetic  Countermeasure • Side-channel resistant arithmetic • Secure  ...  17-Sep-13 3 Elliptic Curve Against Side-channel Attacks  Double-and-add always, Montgomery Ladder  Atomic Operations  Unified Formula • Edwards Curve • Huff Curve SeCoE, Intel Corporation  ...  -Page 361, just before Conclusion: "In this respect, the Huff curve is one step ahead compared to its competitors Edwards [2] and Generalized Hessian curves [9] − on both of which the point addition is  ... 
doi:10.1007/978-3-642-40349-1_20 fatcat:g47ii7pqh5eztkiej6eyqwayma

A Hybrid Cryptosystem to Enhance Security in IoT Health Care System

Kavitha. S, Department of Computer Applications, National Institute of Technology, Tiruchirapalli, TamilNadu, India, P.J. A.Alphonse
2019 International Journal of Wireless and Microwave Technologies  
Single cryptographic algorithm based solution suffered to provide efficient security as its high probability of attacks.  ...  Efficient arithmetic operation of DLP implements the hybrid algorithm on the hessian curve instead of elliptic curves, hence the ECC security attack is away from the IoT security [18] .  ...  To measurement of time taken to compute arithmetic operations on the curve with small parameters, and it was concluding hessian curve requires less time computation, than other curves.  ... 
doi:10.5815/ijwmt.2019.01.01 fatcat:gxq3ttkuu5chtgqtupxc5tl3ay

Families of elliptic curves with rational 3-torsion

Dustin Moody, Hongfeng Wu
2012 Journal of Mathematical Cryptology  
We also include some formulas for efficient computation on these curves, improving upon known results.  ...  These families include Hessian curves, twisted Hessian curves, and a new family we call generalized DIK curves.  ...  Arithmetic on generalized DIK curves In this section, we look at how to efficiently perform arithmetic on a general ized DIK curve.  ... 
doi:10.1515/jmc-2011-0013 fatcat:ff36jr4wkrddnmdob43uvjmgwu

The Hessian Form of an Elliptic Curve [chapter]

N. P. Smart
2001 Lecture Notes in Computer Science  
In this paper we use the Hessian form of an elliptic curve and show that it offers some performance advantages over the standard representation.  ...  In particular when a processor allows the evaluation of a number of field multiplications in parallel (either via separate ALU's, a SIMD type operation or a pipelined multiplication unit) one can obtain  ...  The Hessian Group Law The group law on curves in Hessian form has been known to be particularly simple for a long time, see [1, Formulary] , [2] and [5] .  ... 
doi:10.1007/3-540-44709-1_11 fatcat:7psfj4usljfoteiy4dfdwvcwva

Эффективная реализация схемы цифровой подписи ГОСТ Р 34.10 с помощью современных методов скалярного умножения на эллиптических кривых

Д М Дыгин, D M Dygin, С В Гребнев, S V Grebnev
2013 Mathematical Aspects of Cryptography  
Grebnev, Efficient implementation of the GOST R 34.10 digital signature scheme using modern approaches to elliptic curve scalar multiplication, Efficient implementation of the GOST R 34.10 digital signature  ...  Results of numerical experiments and recommendations on the selection of parameters of algorithms are presented.  ...  Among them, we have chosen for experiments the extended coordinates on Hessian curves [8] , projective and inverted coordinates on Edwards curves [6] and extended coordinates on twisted Edwards curves  ... 
doi:10.4213/mvk82 fatcat:lwonnypswjcebanxkjtyeynat4

A Review of Techniques for Implementing Elliptic Curve Point Multiplication on Hardware

Arielle Verri Lucca, Guilherme Augusto Mariano Sborz, Valderi Reis Quietinho Leithardt, Marko Beko, Cesar Albenes Zeferino, Wemerson Delcio Parreira
2020 Journal of Sensor and Actuator Networks  
Among various existent algorithms on asymmetric cryptography, we may cite Elliptic Curve Cryptography (ECC), which has been widely used due to its security level and reduced key sizes.  ...  Elliptic Curve Point Multiplication (ECPM) is the main function in ECC, and is the component with the highest hardware cost.  ...  curve based on other equations, such as binary Edwards or Hessian curves.  ... 
doi:10.3390/jsan10010003 fatcat:bjzznwtk7zhhvnxcc5yyf27mxq

Page 5692 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
Smart, The Hessian form of an elliptic curve (118-125).  ...  Katsuyuki Okeya and Kouichi Sakurai, Efficient elliptic curve cryptosystems from a scalar multiplication algorithm with recov- 93 SYSTEMS THEORY; CONTROL 5692 ery of the y-coordinate on a Montgomery-form  ... 

EFFICIENT MULTI-LEVEL ENCRYPTION PROCEDURE FOR CLOUD SECURITY

Sampath Kumar Tallapally
2020 JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES  
Cloud space to yourself is a one of the most considerable tentative issue in cloud computing as some of the clients are satisfied with existing policies or protocols where as rest of them are quite concerned  ...  Hessian curves are more efficient than ECC because it takes lesser memory and is faster because it does lesser arithmetic which is same to ECC in security. VI.  ...  Hessian curve is a plain curve which is an Elliptic Curve suggested for the applications used by ECC.  ... 
doi:10.26782/jmcms.2020.06.00031 fatcat:dzxllvaymvdallyjwlu2k4scuy

A Survey Report On Elliptic Curve Cryptography

Samta Gajbhiye, Monisha Sharma, Samir Dashputre
2011 International Journal of Electrical and Computer Engineering (IJECE)  
This paper also discuss the arithmetic involved in elliptic curve and how these curve operations is crucial in determining the performance of cryptographic systems.  ...  Performance of ECC and HEC is also discussed based on scalar multiplication and DLP.  ...  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.  ... 
doi:10.11591/ijece.v1i2.86 fatcat:etikame46ve2nkrio52rprs7ky
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