Introduction The MQ Problem Polynomial Equivalence Problems

Charles Bouillaguet

Description

Introduction The MQ Problem Polynomial Equivalence Problems Supposedly Hard Problems In Multivariate Cryptography Charles Bouillaguet Université de Versailles Saint-Quentin Versailles, France Séminaire CARAMEL 20 janvier 2012

• mq problem


• quadratic encryption


• problem underlying


• polynomials over


• supposedly hard


• hard over


• solving mq


• multivariate quadratic

Sujets

• Versailles

Informations

• Publié par: pefav
• Publié le: 01 janvier 2012
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

IntroductionTheQMrPboelPmlonymolEiaivquenalPrceelbosm

Charles Bouillaguet

Séminaire CARAMEL 20 janvier 2012

Université de Versailles Saint-Quentin Versailles, France

Supposedly Hard Problems In Multivariate Cryptography

Rationale SolvingMQPolynomial Systems isNP-hardover any field

The Hard Problem Underlying Multivariate Cryptography

I RSA Encryption:

y=x e modN,withx,y∈Z/NZ

y 1 =x 1 2 +x 1 x 3 +x 2 x 3 +x 2 x 4 +x 3 2 +x 3 x 4 +1 y 2 =x 1 2 +x 1 x 2 +x 1 x 3 +x 2 2 +x 2 x 4 +x 3 2 +x 4 2 +1 y 3 =x 1 x 2 +x 1 x 4 +x 2 x 3 +x 2 x 4 +x 3 2 +x 3 x 4 +x 4 2 y 4 =x 1 x 2 +x 1 x 3 +x 2 2 +x 2 x 3 +x 3 x 4 withx,y∈  F q  n

I Multivariate Quadratic Encryption:

ortnIobPrMQhenTioctdulaviecneborPsmelmPleynoliaomqulE

ontiucodtrInyloPmelborPQMehTonimlaqEiuavelcneProblems

A Common Construction:Obfuscation 1 Non-linear functionψ:  F q  n →  F q  n I easily invertible, sometimespublic(as in SFLASH) 2 Express it as multivariate polynomials over  F q  n 3 Obfuscateψ: compose withsecret matricesSandT 4 PK=T◦ψ◦S(theobfuscated representationofψ)

ψ

Multivariate Quadratic Trapdoor One-Way Functions

Atrapdoormust be embedded in the equations

Multivariate Quadratic Trapdoor One-Way Functions

Atrapdoormust be embedded in the equations

A Common Construction:Obfuscation 1 Non-linear functionψ:  F q  n →  F q  n I easily invertible, sometimespublic(as in SFLASH) 2 Express it as multivariate polynomials over  F q  n 3 Obfuscateψ: compose withsecret matricesSandT 4 PK=T◦ψ◦S(theobfuscated representationofψ)

ψ

◦

◦

S

T

nyloPmeluqElaimoceenalivmsleobPrtnorIionTductProbheMQ

robleMQPonThuctirtdonIsemblroePnclevaiuqElaimonyloPme

Multivariate Quadratic Trapdoor One-Way Functions

Atrapdoormust be embedded in the equations

A Common Construction:Obfuscation 1 Non-linear functionψ:  F q  n →  F q  n I easily invertible, sometimespublic(as in SFLASH) 2 Express it as multivariate polynomials over  F q  n 3 Obfuscateψ: compose withsecret matricesSandT 4 PK=T◦ψ◦S(theobfuscated representationofψ)

PK=