1. Advanced Information Security 2 SCALAR MULTIPLICATION Dr. Turki F. Al-Somani 2015

2. Module Outlines 2  Scalar Multiplications (SM) in ECC  Generic-Point SM Algorithms double-and-add (or binary), w-ary, addition-subtraction, signed w-ary, and width-w addition-subtraction.  Fixed Point SM Algorithms Fixed-Point Windowing Fixed-Point Comb  New Generic Point Parallel SM Algorithms Postcomputations Precomputations

3. Module Outlines 3  Complexities Comparisons  Time  Area  Summary

4. Scalar Multiplication 4  Scalar multiplication is the basic operation for ECCs.  Scalar multiplication in the group of points of an elliptic curve is the analogous of exponentiation in the multiplicative group of integers modulo a fixed integer m.  The scalar multiplication operation, denoted as kP, where k is an integer and P is a point on the elliptic curve, represents the addition of k copies of point P.

5. Scalar Multiplication (contd.) 5  Scalar multiplication is computed by a series of point doubling and point addition operations of the point P depending upon the bit sequence representing the scalar multiplier k.

6. 6 Scalar Multiplication (contd.)  K can be expanded to binary representation. INPUTK, P OUTPUT KP 1.Initialize Q[0] = , Q[1] = P 2for i = 0 to m-1 3 if k[i] = 1 then 4 Q[0] = ADD(Q[0],Q[1]) 5 end if 6 Q[1] = DBL(Q[1]) 7end for 8return Q[0]

7. 7 Scalar Multiplication (contd.)  most-to-least (left-to-right) version INPUTK, P OUTPUT KP 1.Initialize Q[0] = P 2for i = m-2 downto 0 3 Q[0] = DBL(Q[0]) 4 if k[i] = 1 then 5 Q[0] = ADD(Q[0],P) 6 end if 7end for 8return Q[0]

8. Scalar Multiplication Algorithms 8  Generic Scalar Multiplication Algorithms:  can be used to compute an arbitrary point multiplication  fixed-point multiplication is a common operation in elliptic curve cryptographic algorithms.  Fixed-point Scalar Multiplication Algorithms:  can be used to compute point multiplications involving known points.  point multiplication with a known point can be computed much more efficiently than for arbitrary points.

9. Generic-Point SM Algorithms 9  Algorithms:  double-and-add (or binary),  w-ary,  addition-subtraction,  signed w-ary,  and width-w addition-subtraction.

10. double-and-add (or binary) 10

11. w-ary 11

12. addition-subtraction 12

13. signed w-ary 13

14. width-w addition-subtraction 14

15. Fixed-Point SM Algorithms 15  Algorithms:  Fixed-Point Windowing  Fixed-Point Comb

16. Fixed-Point Windowing 16

17. Fixed-Point Comb 17  Arrangement of k:

18. Fixed-Point Comb (contd.) 18  The precomputation table:

19. Fixed-Point Comb (contd.) 19

20. Time & Area Complexities 20

21. New Generic Point Parallel SM Algorithms 21

22. Postcompuations Methods (2009) 22

23. Postcompuations Methods 23

24. Postcompuations Methods 24

25. Postcompuations Methods (2014) 25

26. Postcompuations Methods 26

27. Precomputations Methods (2015) 27

28. Precomputations Methods (2015) 28

29. Summary 29  Scalar Multiplication is the sole of ECC.  The selection the scalar multiplication algorithm depends on your design objectives:  Time  Area  Power  Security !!

30. THANKS & GOOD LUCK NEXT IS: 3 PROJECTIVE COORDINATES Dr. Turki F. Al-Somani 2015