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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
CHAPTER 3.ELLIPTIC CURVE CRYPTOGRAPHIC PROCESSOR3.2.1 Montgomery algorithm for scalar multiplicationThe binary method for scalar multiplication on elliptic curves can be implemented as follows:set k ← (k l−1 ...k 1 k 0 ) 2set P 1 ← P, P 2 ← 2Pfor i = l − 2 to 0 doif k i = 1 thenelseset P 1 ← P 1 + P 2 , P 2 = 2 · P 2 .set P 2 ← P 1 + P 2 , P 1 = 2 · P 1 .end ifend forOne fast algorithm for fast scalar multiplication on elliptic curves is the ladder algorithmrst proposed by Montgomery [12]. It is based on the observation that in the binarymethod, the dierence between P 1 and P 2 is always equal to P . This makes it possible toimplement scalar multiplication with fewer registers.3.3 Specications of the cryptographic processorThe ECC core that was chosen as an example implementation for this project is designedfor elliptic curve calculations over the eld F 2 163. It provides a hardware implementation ofthe add-and-double operation that is at the heart of Montgomery's ladder multiplicationalgorithm.The ALU and the register le are controlled by a Finite State Machine (FSM), which isalso part of the hardware implementation. An external controller implements the physicalinterface for encryption or authentication and repeatedly calls the ECC core to performthe operations on two curve points.The main specications are given in table 3.1. The core is designed to operate at a verylow clock frequency of 100 kHz.technologytarget clock frequencypower dissipationUMC 90nm100 kHzmin.Table 3.1: Specications of the ECC core3.4 ArchitectureShown in Figure 3.1, the architecture of the ECC core consists of a core containing theArithmetic and Logic Unit (ALU) on the left, and a 6-word register le on the right.19
Page 4 and 5: Contents1 Introduction 61.1 Low-pow
Page 7 and 8: CHAPTER 1.INTRODUCTIONI DS ≈ I 0
Page 9 and 10: CHAPTER 2.SUBTHRESHOLD SOURCE-COUPL
Page 17: 3 Elliptic curve cryptographic proc
Page 21 and 22: CHAPTER 3.ELLIPTIC CURVE CRYPTOGRAP
Page 23 and 24: 4 Results4.1 Design and simulation
Page 25 and 26: CHAPTER 4.RESULTS`wasting' current
Page 27 and 28: CHAPTER 4.RESULTS32.521.5Iss [mA]10
Page 29 and 30: 5 Outlook5.1 Possible improvements
Page 31: 6 ConclusionThis work successfully
Page 34 and 35: AppendixVHDL code for the nite eld
Page 36 and 37: 16 process ( c l k )17 begin18 i f
Page 38 and 39: 4 use IEEE .STD_LOGIC_UNSIGNED.ALL;
Page 40: 7172 v = axis ( ) ;7374 text ( v (

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