Arithmetic Operations in the Polynomial Modular Number System
Arithmetic Operations in the Polynomial Modular Number System
Arithmetic Operations in the Polynomial Modular Number System
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LIRMM<br />
<strong>Arithmetic</strong> needs for public key cryptography<br />
<strong>Modular</strong> <strong>Arithmetic</strong><br />
Introduction<br />
New <strong>Number</strong> <strong>System</strong><br />
<strong>Number</strong> system<br />
Adapted <strong>Modular</strong> <strong>Number</strong> <strong>System</strong><br />
Fundamental Theorem<br />
<strong>Arithmetic</strong> on PMNS<br />
<strong>Modular</strong> Multiplication<br />
Coefficient Reduction<br />
The RED Algorithm<br />
Conclusions<br />
Diffie-Hellman<br />
An exponentiation over <strong>the</strong> prime field F p<br />
Needs : Multiplication modulo p (prime)<br />
Length : 1024, 2048, . . . bits<br />
RSA<br />
An exponentiation on <strong>the</strong> r<strong>in</strong>g Z/nZ<br />
Needs : Multiplication modulo n (composite, n = p.q)<br />
Length : 1024, 2048, . . . bits<br />
ECC<br />
Elliptic curve po<strong>in</strong>t multiplication<br />
Needs : <strong>Arithmetic</strong> operations (+,-,*,/) over <strong>the</strong> f<strong>in</strong>ite field F q, where<br />
q is a power of a prime p<br />
Length : 160, 192, . . . bits<br />
Jean-Claude Bajard, Laurent Imbert, Thomas Plantard, 28 june 2005 4/18
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