Page 365 - Hardware Implementation of Finite-Field Arithmetic

Index 345

elliptic curve cryptography, 287 finite field (Cont.):
elliptic curve operations optimal extension field (OEF), 132
basic algorithms, 293 optimal extension field (OEF),
group law, 290 Type-I, 133
nonadjacent forms, 294 optimal extension field (OEF),
point addition, 290 Type-II, 133
point addition, GF(2 ), 305 order, 17
163
point multiplication, 292 properties, 17
point multiplication, F , 17
q
τ-ary representation, 301 Frobenius constants, 157
163
point multiplication, GF(2 ), Frobenius map, 299
304, 306
point multiplication, Montgomery G G
algorithm, 297 Galois field, 17, 20, 91, 139
projective coordinates, 294 Galois group, 21
elliptic curve operator gcd, 91, 100, 139, 140, 147. See also
163
point addition, GF(2 ), 305 greatest common divisor
point multiplication, GF(2 ), 308 generator, 7, 8
163
m
equivalence GF(2 ), 163
class, 5, 15 GF(p ), 117
m
relation, 5, 15 GF(q), 17
ESP, 213 greatest common divisor, 2, 3, 12,
Euclidean algorithm, 2, 91, 92, 139, 139, 207
140, 142, 207, 255 group, 8
extended, 3, 4 abelian, 8
extended, for polynomials, 14 commutative, 8
for polynomials, 12 cyclic, 7, 8, 17
Euler phi function, 6 multiplicative, 6
exponentiator
Montgomery exponentiation, H H
LSB-first, 85 Hamming weight, 213, 242
Montgomery exponentiation, Hasse theorem, 289
MSB-first, 83
extension (field), 10, 17, 18 I I
algebraic, 19 ideal, 9
degree, 19 identity element, 8
finite, 19 additive, 9, 10
optimal extension field (OEF), 132 multiplicative, 9, 10, 11
simple, 19 inverse, multiplicative, 5
Itoh-Tsujii algorithm, 257
F F
Fermat’s little theorem, 6, 17, 91, 110 K K
Fermat’s theorem, 207, 255, 275 Karatsuba-Ofman multiplication, 169
field, 10, 16 Koblitz curve, 299
characteristic, 10 Kronecker delta function, 270
extension, 10, 17, 18
isomorphic, 17 L L
prime, 10
finite field, 17 linear function, 269
bases, 20 López-Dahab projective coordinates,
GF(2 ), 22, 235 295
m
m
GF(p ), 117
OEF polynomial multiplication, M M
133 Massey-Omura multiplier, 235, 238
OEF reduction modulo an Mastrovito multiplication, 177
irreducible binomial, 133 matrix decomposition, 178