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

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Presentation on theme: "EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th, 2007 Algorithms for public-key cryptology Montgomery Arithmetic."— Presentation transcript:

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

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

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

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

5 Isomorphic Ordinary RepresentationMontgomery Representation Montgomery radix Montgomery Multiplication

6 Definition Definition:

7 How to compute? Algorithm

8 How to compute? Algorithm

9 How to compute? Algorithm

10 How to compute? Algorithm

11 How to compute? Algorithm

12 How to compute? Algorithm

13 How to compute? Algorithm

14 How to compute? Algorithm

15 How to compute? Algorithm

16 How to compute? Algorithm

17 How to compute? Algorithm

18 How to compute? Algorithm

19 How to compute? Algorithm

20 How to compute? Algorithm

21 How to compute? Algorithm

22 How to compute? Algorithm

23 How to compute? Algorithm

24 How to compute? Algorithm

25 How to compute? Algorithm

26 How to compute? Algorithm

27 How to compute? Algorithm

28 How to compute? Algorithm

29 How to compute? Algorithm

30 How to compute? Algorithm

31 How to compute?

32

33 Subtraction-less Montgomery multiplication Algorithm

34 Subtraction-less Montgomery multiplication Algorithm

35 Subtraction-less Montgomery multiplication Algorithm

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

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

38 What about modular inversion? Ordinary RepresentationMontgomery Representation Montgomery Bootstrapping

39 How to compute m 0 -1 mod B? Montgomery Bootstrapping 0 011 0 001 0 011 0 010 1 001 00 011 0 011 1 001 0 011 0 011 1 000 0 001 010 1 011 0 100

40 0 011 0 001 0 011 0 010 1 001 00 011 0 011 1 001 0 011 0 011 1 000 0 001 010 1 011 0 100

41 Montgomery Squaring

42 RSA pseudorandom bit generator

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