Ultra-Low-Power Digital Circuit Design - Microelectronic Systems ...

CHAPTER 3.ELLIPTIC CURVE CRYPTOGRAPHIC PROCESSORAddition a + b = r, where r is the remainder of the division of a + b by p.Multiplication a × b = s, where s is the remainder of the division of a × b by p.The nite eld F 2 m The binary nite eld, F 2 m, is a vector space of dimension m overthe prime eld F 2 . A polynomial basis of F 2 m can be introduced as follows: Let f(x) be anirreducible polynomial of degree m over F 2 called the reduction polynomial. Each elementa of F 2 m can now be written as a binary polynomial of degree m − 1 or less:a = a m−1 x m−1 + ... + a 1 x + a 0When a polynomial basis is specied, an element of F 2 mbit vector of length m.can therefore be written as aFinite eld operationscan be dened:Using a polynomial basis for F 2 m, the following eld operations• Addition: a + b = c = (c m−1 ...c 1 c 0 ), where c i = (a i + b i ) mod 2. Field addition isthe bitwise XOR of the bit vectors representing elements of F 2 m.• Multiplication: a · b = c = (c m−1 ...c 1 c 0 ), where c(x) = ∑ m−1i=0 c ix i is the remainder ofthe division of the polynomial ( ∑ m−1i=0 a ix i )( ∑ m−1i=0 b ix i ) by f(x). Field multiplicationcan be performed by the shift-and-add method, where one bit of b is considered ata time, starting at the MSB. If the bit is equal to one, a is added (using XOR) to arunning sum c. After each step, c is left-shifted by one bit and reduced modulo f(x).Elliptic curves over F 2 m The elliptic curve E(F 2 m) over F 2 m for the parameters a, b ∈F 2 m, b≠0 is dened to be the set of points P = (x, y) for x, y ∈ F 2 mthe equationthat are solution toy 2 + xy = x 3 + ax 2 + btogether with the special point O called the point at innity.Addition on elliptic curvesAddition of two points P = (x 1 , y 1 ), Q = (x 2 , y 2 ) ∈ E(F 2 m),P ≠ ±Q, results in a new point P + Q = (x 3 , y 3 ) ∈ E(F 2 m), where(x 3 = λ 2 + λ + x 1 + x 2 + a, y 3 = λ(x 1 + x 2 ) + x 3 + y 1 λ = y )2 + y 1.x 2 + x 1A similar expression can be found for the double of a point P .Elliptic scalar multiplicationScalar multiplication of a point P on an elliptic curve bythe integer k is dened to be the result of adding P to itself k times. This operation isthe underlying principle of all elliptic curve cryptographic schemes.There are ecientalgorithms for calculating kP . On the other hand, it is very hard to nd k if only P andkP are known. This is the so-called discrete logarithm problem.18