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EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th , 2007

Algorithms for public-key cryptology Montgomery Arithmetic. EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th , 2007. Motivation. RSA:. ElGamal:. Most of the time computing modular multiplications. Need of efficient algorithms for modular multiplication . Notation.

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EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th , 2007

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  1. Algorithmsforpublic-keycryptologyMontgomeryArithmetic EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27th, 2007

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

  3. Notation Multiple-precision integer arithmetic Radix representation depending on theprocessor (normalized)

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

  5. Montgomery Multiplication Montgomery radix Ordinary Representation Montgomery Representation Isomorphic

  6. Definition Montgomery Multiplication 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. How to compute?

  33. Subtraction-less Montgomery multiplication Algorithm

  34. Subtraction-less Montgomery multiplication Algorithm

  35. Subtraction-less Montgomery multiplication Algorithm

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

  37. Montgomery Bootstrapping How to compute R2 mod m ? Ordinary Representation Montgomery Representation

  38. Montgomery Bootstrapping What about modular inversion? Ordinary Representation Montgomery Representation

  39. Montgomery Bootstrapping How to compute m0-1 mod B? 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0

  40. 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0

  41. Montgomery Squaring

  42. RSA pseudorandom bit generator

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