1. Montgomery multiplication Algorithm Mohammad Farmani Under supervision of : Dr. S. Bayat-sarmadi 2 nd. Semister,1392-93 Sharif University of Technology 1

2. 2 Main Topic  Montgomery modular multiplication algorithm Main Article: “Montgomery Multiplication in GF(2 k )” Written by: Cetin K. KOC and Tolga Acar,1998 Copyright © 2014 Hardware Security and Trust

3. Outline Introduction Montgomery modular multiplication of integers Montgomery modular multiplication in GF(2 k ) Conclusion Montgomery multiplication algorithm Sharif University of Technology 3 Copyright © 2014 Hardware Security and Trust

4. The importance and applications of the arithmetic operations in the Galois field GF(2 k ) in : Coding theory Computer algebra Cryptography …. Importance of the exponentiation Using a series of multiplication for The exponentiation Montgomery multiplication algorithm Sharif University of Technology 4 Copyright © 2014 Hardware Security and Trust

5. Cryptographic applications require fast arithmetic operations Proposed an effective modular multiplication of integers by P.L. Montgomery 1985 Conversion to the Montgomery domain : a : an intger M : modulus r : Radix Montgomery multiplication algorithm Sharif University of Technology 5 Copyright © 2014 Hardware Security and Trust

6. Example: M = 11, r = 2 4 = 16 There is a one-to-one correspondence between integers and Montgomery residues for 0 < a < M-1 Montgomery multiplication algorithm Sharif University of Technology 6 Copyright © 2014 Hardware Security and Trust

7. Outline Introduction Montgomery modular multiplication of integers Montgomery modular multiplication in GF(2 k ) Conclusion Montgomery multiplication algorithm Sharif University of Technology 7 Copyright © 2014 Hardware Security and Trust

8. Define: r -1 is the inverse of r mod M: r -1 r = 1 (mod M) Montgomery multiplication algorithm Sharif University of Technology 8 Copyright © 2014 Hardware Security and Trust

9. Example : Montgomery multiplication algorithm Sharif University of Technology 9 Copyright © 2014 Hardware Security and Trust

10. Montgomery multiplication algorithm Sharif University of Technology 10 Copyright © 2014 Hardware Security and Trust

11. Example : Z initially 0 Z = (0 + 5 + 11) / 2 = 8 Z = (8 + 5 + 11) / 2 = 12 Z = (12 + 5 + 11) / 2 = 14 Z = (14 + 0) / 2 = 7 (final result) Montgomery multiplication algorithm Sharif University of Technology 11 Copyright © 2014 Hardware Security and Trust X = 7 = 0111 Y = 5 = 0101 M = 11 = 1011 Z = 0 for i = 0 to n-1 Z = Z + xiY if Z is odd then Z = Z + M Z = Z/2 if Z ≥ M then Z = Z – M

12. Conversion using MM Conversion of integers to/from Montgomery residues with one MM operation Montgomery multiplication algorithm Sharif University of Technology 12 Copyright © 2014 Hardware Security and Trust

13. Montgomery multiplication algorithm Sharif University of Technology 13 Copyright © 2014 Hardware Security and Trust MM x r2r2 X’X’ X’ 1 X

14. Outline Introduction Montgomery modular multiplication of integers Montgomery modular multiplication in GF(2 k ) Conclusion Montgomery multiplication algorithm Sharif University of Technology 14 Copyright © 2014 Hardware Security and Trust

15. Montgomery multiplication algorithm Sharif University of Technology 15 Copyright © 2014 Hardware Security and Trust

16. Montgomery multiplication algorithm Sharif University of Technology 16 Copyright © 2014 Hardware Security and Trust

17. Montgomery multiplication algorithm Sharif University of Technology 17 Copyright © 2014 Hardware Security and Trust

18. Montgomery multiplication algorithm Sharif University of Technology 18 Copyright © 2014 Hardware Security and Trust

19. Montgomery multiplication algorithm Sharif University of Technology 19 Copyright © 2014 Hardware Security and Trust

20. Montgomery multiplication algorithm Sharif University of Technology 20 Copyright © 2014 Hardware Security and Trust

21. Montgomery multiplication algorithm Sharif University of Technology 21 Copyright © 2014 Hardware Security and Trust

22. Montgomery multiplication algorithm Sharif University of Technology 22 Copyright © 2014 Hardware Security and Trust

23. Montgomery multiplication algorithm Sharif University of Technology 23 Copyright © 2014 Hardware Security and Trust

24. Montgomery multiplication algorithm Sharif University of Technology 24 Copyright © 2014 Hardware Security and Trust

25. Outline Introduction Montgomery modular multiplication of integers Montgomery modular multiplication in GF(2 k ) Conclusion Montgomery multiplication algorithm Sharif University of Technology 25 Copyright © 2014 Hardware Security and Trust

26. Montgomery multiplication algorithm Sharif University of Technology 26 Copyright © 2014 Hardware Security and Trust

27. End of presentation, Any question?