Advanced Information Security 4 FIELD ARITHMETIC

Dr. Turki F. Al-Somani2015

2

Module Outlines

Finite Field Arithmetic 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.

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].

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.

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.

THANKS & GOOD LUCK NEXT IS: 5 ECC CRYPTOGRAPHY

Dr. Turki F. Al-Somani2015