Isogeny formulas for Jacobi intersection and twisted hessian curves

João Paulo da Silva; Julio López; Ricardo Dahab

doi:10.3934/amc.2020048

Article Contents

2020, Volume 14, Issue 3: 507-523. Doi: 10.3934/amc.2020048

This issue Previous Article Challenge codes for physically unclonable functions with Gaussian delays: A maximum entropy problem Next Article The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes

Isogeny formulas for Jacobi intersection and twisted hessian curves

Institute of Computing, University of Campinas, Av. Albert Einstein 1251, Cidade Universitária "Zeferino Vaz", 13083-852, Campinas, SP, Brazil

Received: December 2018

Revised: December 2018

Early access: November 2019

Published: August 2020

The first author is supported by Intel/FAPESP grant 14/50704-7 under project "Secure Execution of Cryptographic Algorithms"

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Abstract

The security of public-key systems is based on the difficulty of solving certain mathematical problems. With the possible emergence of large-scale quantum computers several of these problems, such as factoring and computing discrete logarithms, would be efficiently solved. Research on quantum-resistant public-key cryptography, also called post-quantum cryptography (PQC), has been productive in recent years. Public-key cryptosystems based on the problem of computing isogenies between supersingular elliptic curves appear to be good candidates for the next generation of public-key cryptography standards in the PQC scenario. In this work, motivated by a previous work by D. Moody and D. Shumow [17], we derived maps for elliptic curves represented in Jacobi Intersection and Twisted Hessian models. Our derivation follows a multiplicative strategy that contrasts with the additive idea presented in the Vélu formula. Finally, we present a comparison of computational cost to generate maps for isogenies of degree $ l $, where $ l = 2k + 1 $. 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 degree $ 3 $. Considering higher degree isogenies as $ 101 $, our formulas require $ 23.4\% $ less computation than the Huff model and $ 24.7 \% $ less computation than the formula for the Extended Jacobi Quartic model.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. Operation Counting for Isogenies Evaluation in Alternative Models in affine coordinates

MODEL	OPERATION COST
Edwards [17]	$ (3k+1)M+2S+3kC+I $
Huff [17]	$ (4k-2)M+2S+2kC+2I $
Ext. Jacobi Quartic [24]	$ (4k+2)M+3S+(7k+4)C+2I $
Weierstrass [17]	$ (3+o(1))(2k+1)M+S+(3+o(1))(2k+1)C+I $ ¹
Intersec. de Jacobi	$ (4k+2)M+3S+(5k+1)C+I $
Twisted Hessian	$ (5k+2)M+3S+(9k)C+I $

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Table 2. Operations Counting for Evaluation of Isogenies in Alternative Models, given in affine coordinates, as a function of the number of multiplications in the base field

MODEL	OPERATION COST
	$I=100M$, $S=1M$, $*const=0M$	$I=100M$, $S=0, 8M$, $*const=0M$	$I=100M$ $S=0, 67M$, $*const=0M$
	Edwards [17]	$(3k+103)M$	$(3k+102.6)M$	$(3k+102.34)M$
Huff [17]	$(4k+200)M$	$(4k+199.6)M$	$(4k+199.34)M$
Ext. Jacobi Quartic [24]	$(4k+205)M$	$(4k+204.4)M$	$(4k+204.01)M$
Weierstrass [17]	-	-	-
Jacobi Intersection	$(4k+105)M$	$(4k+104.4)M$	$(4k+104.01)M$
Twisted Hessian	$(5k+105)M$	$(5k+104.4)M$	$(5k+104.01)M$

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Table 3. Operations Counting for Isogenies Evaluation in Alternative Models using Projective Coordinates

MODEL	OPERATION COST
Edwards [17]	$ (3k+3)M+4S+3kC $
Huff [17]	$ (4k+3)M+3S+4kC $
Ext. Jacobi Quartic [24]	-
Weierstrass [17]	-
Jacobi Intersection	$ (4k+7)M+5S+(6k+2)C $
Twisted Hessian	$ (5k+2)M+3S+(9k)C $

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Table 4. Operations Counting for Isogeny Evaluation on Alternative Models, using projective coordinates, as a function of the number of multiplications in the base field

MODEL	OPERATION COST
MODEL	$S=1M$, $*const=0M$	$S=0, 8M$, $*const=0M$	$S=0, 67M$, $*const=0M$
Edwards [17]	$(3k+7)M$	$(3k+6.2)M$	$(3k+5.68)M$
Huff [17]	$(4k+6)M$	$(4k+5.4)M$	$(4k+5.01)M$
Ext. Jacobi Quartic [24]	-	-	-
Weierstrass [17]	-	-	-
Jacobi Intersection	$(4k+12)M$	$(4k+11)M$	$(4k+10.35)M$
Twisted Hessian	$(5k+5)M$	$(5k+4.4)M$	$(5k+4, 01)M$

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