1. Advanced Information Security 4 Field Arithmetic
Dr. Turki F. Al-Somani
2015

2. Module Outlines Finite Field Arithmetic Summary GF(p) Arithmetic
GF(2m) Arithmetic
Polynomial basis
Normal basis
Addition/subtraction
Squaring
Multiplication
Inversion
Summary

3. Finite Field Arithmetic
In abstract algebra, a finite field is a field that contains only finitely many elements.
Finite fields are important in number theory, algebraic geometry, Galois theory, coding theory, and cryptography.

4. Finite Field Arithmetic (contd.)

5. Finite Field Arithmetic (contd.)

6. Finite Field Arithmetic (contd.)

7. Finite Field Arithmetic (contd.)

8. Finite Field Arithmetic (contd.)

9. GF(2m) Arithmetic
The finite GF(2m) field has particular importance in cryptography since it leads to particularly efficient hardware implementations.
Elements of the field are represented in terms of a basis.
Most implementations use either a Polynomial Basis or a Normal Basis.
Normal basis is more suitable for hardware implementations than polynomial basis since operations are mainly comprised of rotation, shifting and exclusive- OR operations which can be efficiently implemented in hardware.

10. Polynomial Basis

11. Polynomial Basis

12. Normal Basis

13. Normal Basis (contd.)

14. Normal Basis (contd.)

15. Optimal Normal Basis
An optimal normal basis (ONB) is one with the minimum number of terms, or equivalently, the minimum possible number of nonzero λij
This value is 2m-1, and since it allows multiplication with minimum complexity, such a basis would normally lead to more efficient hardware implementations.

16. Optimal Normal Basis (Contd.)
Note: Type 1 is circled.

17. Optimal Normal Basis Types
Now CN=2n-1
Type I:
Rule 2 means: for every i in the range [0, n-1], (2k mod n+1) must result in a unique integer in the range [1, n].

18. Cont.
Type II:
Rule 2a means that every 2k mod 2n+1, in the range [1 to 2n].
Therefore 2 is called the generator for all the possible locations in the 2n+1 field
Rule 2b means that even if 2k does not generate every element in the range [1, 2n], however, half of points in the field of form by rule 2a can be hit. It is because SQR(2k) can be taken.
The points generated by rule 2b are in the form of perfect squares.

19. ONB Type I & II (n ≤ 230)

20. Survey Paper (2006)

21. NB Multiplication
Multiplication is more complicated than addition and squaring operations in finite field arithmetic.
An efficient multiplier is highly needed and is the key for efficient finite field computations.
Finite filed multipliers using normal basis can be classified into two main categories:
𝜆-matrix based multipliers
Conversion based multipliers

22. 𝜆-matrix based multipliers
Massey and Omura Multiplier
Hasan et. al. Multiplier
Gao and Sobelman Multiplier
Reyhani-Masoleh and Hasan Multiplier

23. Example: Type I

24. Example: Type II

25. Massey and Omura Multiplier

26. Hasan et. al. Multiplier

27. Gao and Sobelman Multiplier

28. Reyhani-Masoleh and Hasan Multiplier

29. Comparisons

30. Conversion based multipliers
Sunar and Koc Multiplier
Wu et. al. Multiplier

31. Sunar and Koc Multiplier

32. Wu et. al. Multiplier

33. Comparisons

34. Normal Basis Inversion
Inversion algorithms:
Standard algorithms
Exponent Decomposing algorithms
Exponent Grouping inversion algorithms

35. Standard Algorithms

36. Exponent Decomposing Algorithms

37. Exponent Decomposing Algorithms (contd.)

38. Exponent Decomposing Algorithms (contd.)

39. Exponent Grouping inversion Algorithms

40. Exponent Grouping inversion Algorithms (contd.)

41. Exponent Grouping inversion Algorithms (contd.)

42. Comparisons

43. Pipelining Paper (2009)

44. Pipelining Paper (2009)

45. UQU Pipelining Paper (2010)

46. Systolic Arrays Paper (2011)

47. IEEE VLSI Systolic Arrays Paper (2014)

48. Summary
Efficient computations in finite fields and their architectures are important in many applications, including coding theory, computer algebra systems, and public-key cryptosystems (e.g., elliptic curve cryptosystems (ECC).
The most commonly used basis are polynomial basis and normal basis.
Normal basis is more suitable for hardware implementations than polynomial basis since operations in normal basis representation are mainly comprised of rotation, shifting and exclusive-ORing which can be efficiently implemented in hardware.

49. Thanks & Good Luck Next is: 5 ECC Cryptography
Dr. Turki F. Al-Somani
2015