FPGA based Hardware Accleration for Elliptic Curve Cryptography ...

2.2.2 Polynomial Rings over·7¸À¹4»
2.2. FINITE FIELD ARITHMETIC 9
Algorithm 3 Double-And-Add
Input:
6 =­ Y@\\\S@AYZ@AP®S3 and
Ë /
m ¯ _ c ¯
end while
I
then
Vwm
if¯:´
m ¯ _ c ¯
6 ¯ m°( _ c
Output:V
6²I and¯³ I
do
while±
while¯³ I
do
/ } _ €Hµ r>[xtPuV…
if ± 6 c then
Vwm
/ } _ v-{P{ uV§@g…
end if
Vwm
m ¯ _ c ¯
end while
else
end if
VwmoT
2.2.1 The Field·§¸º¹»
Finite
The smallest imaginable finite is- 6½¼ †¦ field , which has two elements only: The additive and the
multiplicative elementsI
andc neutral respectively. Its addition and multiplication tables resemble the truth
tables of the binary (¾ XOR ) and the binary (¿ AND ) operation respectively. The elements can directly
represented by a single bit.
returnV
sethÁ5à6 J¦ÄÆÅ ±ÈÇÉ® ;P±K5 ± L`;P± E -AR The of polynomials with in: coefficient together with
the additive elementI
5 ®
neutral , the multiplicative elementch5
®
neutral , and polynomial addition as well as
multiplication operations constitutes a over ring . Since the degree of a coefficient is given by it’s bit
position, an ofhÁ5. element can effectively be represented by it’s coefficients stored a bit vector.
2.2.3 Fields-·‰¸º¹ »
Finite
Given an irreducible E hÁ5. polynomial of ( degree , finite fields of extension ( degree are constructed
by modular arithmetic out of the previously defined polynomial rings as follows:
6 hÁ5.ÂK†k\ (2.3)
The set, which is underlying the Galois field, is thus the finite set of residue classes of polynomials modulo
the prime polynomial . The canonical representative of a polynomialv
’s residue class is the remainder of