EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th, 2007 Algorithms for public-key cryptology Montgomery Arithmetic.

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Presentation transcript:

EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th, 2007 Algorithms for public-key cryptology Montgomery Arithmetic

RSA: ElGamal: Motivation Need of efficient algorithms for modular multiplication Most of the time computing modular multiplications

Radix representation Notation Multiple-precision integer arithmetic depending on the processor (normalized)

General overview Ordinary RepresentationMontgomery Representation Sequential multiplications performed in Montgomery representation Montgomery Multiplication

Isomorphic Ordinary RepresentationMontgomery Representation Montgomery radix Montgomery Multiplication

Definition Definition:

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute? Algorithm

How to compute?

Subtraction-less Montgomery multiplication Algorithm

Subtraction-less Montgomery multiplication Algorithm

Subtraction-less Montgomery multiplication Algorithm

Ordinary RepresentationMontgomery Representation Conversion back and forth from ordinary representation and Montgomery representation

How to compute R 2 mod m ? Ordinary RepresentationMontgomery Representation Montgomery Bootstrapping

What about modular inversion? Ordinary RepresentationMontgomery Representation Montgomery Bootstrapping

How to compute m 0 -1 mod B? Montgomery Bootstrapping

Montgomery Squaring

RSA pseudorandom bit generator