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| Koblitz N. — A course in number theory and cryptography |
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| Ïðåäìåòíûé óêàçàòåëü |
Abelian group 33
Abelian group, type of 174
Adleman — Huang primality test 190
Adleman — Pomerance — Rumely primality test 134—135
Affine map 57 59 68 75
Affine map, plane 171
algebraic 32
Algorithm 9
Algorithm, Berlekamp 104
Algorithm, deterministic 127
Algorithm, factor-base 103 148
Algorithm, for discrete log 102—106
Algorithm, index-calculus 103—106
Algorithm, probabilistic 86 95 127
Algorithm, Schoof 179 183
Algorithm, Silver — Pohlig — Hellman 102—103 183
Alphabet 54
Alphabet, Cyrillic 63 78
Arms control 90—91 214
Atkin primality test 187 190
Authentication 88 95
Automorphism 32 36
B-number 145 160
Base of number system 1
Base of number system, two 1 3
Big-O notation 7—8
Bit 3
Bit, operation 3
Bond, James 82 185 210 214
Breaking a code 56
Breaking a code, the knapsack 114
Caesar, Julius 56
Carmichael number 127—128 136
Casanova 84—85
Characteristic of a field 33
Chinese remainder theorem 21
Chor — Rivest knapsack 115
Ciphertext 54
Classical cryptosystem 88
Classical cryptosystem, Cohen — Lenstra primality test 134—135
coin toss 91 96—97 215
Coloring map or graph 118
complex numbers 17
Complex numbers, Gaussian integers 17 37 42—43 171
Composite number 12
Composition of cryptosystems 64 79
Congruence 19 193
conjugate 32
Continued fraction 155
Continued fraction, factorization method 158—159
Convergent 155
Crypt analysis 56
Cryptography 54
cryptography, public key 85
Cryptosystem 54—55 83
Cryptosystem, classical 88
Cryptosystem, composition 64 79
Cryptosystem, Diffie — Hellman 98—99 181—182
Cryptosystem, ElGamal 100—101 109 182
Cryptosystem, elliptic curve 181—182
Cryptosystem, knapsack 113—115
Cryptosystem, Massey — Omura 100 109 182 216
Cryptosystem, Merkle — Hellman 113—114
Cryptosystem, private key 88
Cryptosystem, product 64 78—79
Cryptosystem, public key 85
Cryptosystem, RSA 22 92—93 106 125 137 153
Cryptosystem, structure 56
Cryptosystem, symmetric 88
Cyclic group 34
Cyrillic 63 78
Data Encryption Standard 101
Deciphering 54
Deciphering, key 83
Deciphering, transformation 54
decryption 54
Determinant 67
Deterministic algorithm 127
Deterministic algorithm, encryption 89
Diffie — Hellman assumption 99 121
Diffie — Hellman assumption, key exchange 98—99 181—182
Digital Signature Standard 101—102
DIGITS 1
Digits, binary (bit) 3
Digits, number of 3
Digraph 54 59
Digraph, transformation 59
Dirichlet L-series 134
Discrete log 97—98
Discrete log, algorithms for 102—106
Discrete log, on elliptic curve 180
Divisibility 12
Divisibility, exact 12
Division points 173
Divisor 12
Divisor, nontrivial 12
Divisor, proper 12
Elgamal cryptosystem 100—101 109 182
ElGamal cryptosystem, signature 109—110
Elliptic curve 167—168
Elliptic curve, addition law 168—170
Elliptic curve, complex points 171
Elliptic curve, cryptosystem 181—182
Elliptic curve, factorization 191—192 195—198
Elliptic curve, global 183
Elliptic curve, nonsupersingular 181
Elliptic curve, over finite field 174
Elliptic curve, primality test 188—190
Elliptic curve, rank 173
Elliptic curve, real points 176—177 227
Elliptic curve, reduction 184 193—194
Elliptic curve, super singular 181
Elliptic curve, torsion subgroup 173 185
Elliptic curve, Weil pairing 180—181
Elliptic curve, zero element 169
Elliptic curve, zeta-function 175
Elliptic function 173
Enciphering 54
Enciphering, key 56 83
Enciphering, matrix 71—72
Enciphering, transformation 54
Encoding 179
Encryption 54
Euclidean algorithm 13
Euclidean algorithm for Gaussian integers 18
Euclidean algorithm for polynomials 17
Euler phi-function 15 21—22
Euler phi-function, pseudoprime 129
exponentiation 23 97
Factor base 145
Factor base, algorithm 103 148
Factoring 27—29 92
Factoring, continued fraction method 158—159
Factoring, Fermat factorization 15 96 143—144
Factoring, Monte — Carlo method 138—140
Factoring, Pollard p - 1 method 192—193
Factoring, quadratic sieve 160—162
Factoring, rho method 138—142
Factoring, trial division 126 138
Factoring, with elliptic curves 191—192 195 198
Fermat factorization 15 96 143—144
Fermat factorization, prime 29 51 109 190
Fermat’s Little Theorem 20 126
Fibonacci numbers 16—17 77—78 159 211—212 223
Fields 31
Fields, automorphism of 32 36
Fields, characteristic of 33
Fields, finite 20 33
Fields, Galois extension 32
Fields, isomorphism 32
| Fields, of p elements 20 33
Fields, prime 33
Fields, splitting 33
Finite fields 20 33
Finite fields, automorphism of 36
Finite fields, existence and uniqueness 35—36
Finite fields, generator 34
Finite fields, irreducible polynomials over 38—39 104 110
Finite fields, roots of unity in 42
Finite fields, square roots in 42 48 52 96 179—180
Finite fields, subfields 38
Fixed digraph 81
Fixed digraph, message unit 62 64
Frequency analysis 56
Frobenius 183 229
Function, one-way 85
Function, one-way, trapdoor 85
Fundamental Theorem of Arithmetic 12 26
Galois field extension 32
Gauss sum 44 45 134
Gaussian integers 17 37 42—43 171
Generator of finite field 34
Germain, Sophie 207
Germain, Sophie, prime 207
Global elliptic curve 183
Graph 118
Greatest common divisor 12
Greatest common divisor of Gaussian integers 17
Greatest common divisor of polynomials 17 32
Group, abelian 33
Group, Abelian, cyclic 34
Hash function 89
Hasse’s theorem 174
Hexadecimal 10
Imbedding plaintexts 179
Index-calculus algorithm 103—106
Infinity, line at 171
Infinity, point at 168 171
Inverses, multiplicative 19
Irreducible polynomial 32 104 110
Isomorphism 32
Jacobi symbol 47
k-threshold scheme 27
Key 56
Key, deciphering 83
Key, enciphering 56 83
Key, exchange 89 98
Knapsack cryptosystem 113—115
Knapsack cryptosystem, problem 112
Knapsack cryptosystem, superincreasing 112
Lagrange’s Theorem 157
Lattice 171
Least absolute residue 145
Least absolute residue, common multiple 13
Legendre symbol 43 174
Lenstra elliptic curve factorization 191—192 195—198
Lifting 52 80
Line at infinity 171
Linear algebra 58 66—68
Linear algebra, modulo 2 146—147
Linear algebra, modulo N 68—70 105
Linear map 57 67 68 70
Massey — Omura cryptosystem 100 109 182 216
Matrices 66—67 68
Matrices, inverses 67 69
Matrices, Merkle — Hellman cryptosystem 113—114
Mersenne prime 28 29 51 125 191 207
Mersenne prime in the Gaussian integers 228
Message unit 54
Miller — Rabin primality test 130—131
Miller — Rabin primality test, time estimate for 136—137
Modular exponentiation 23—24 97
Modulus 19
Monic polynomial 17 32
Monte — Carlo factorization 138—142
Mordell theorem 173
Multiple of point 178
Multiplicity of root 32
Non-interaction 122
Non-singular 168
Nonresidue, quadratic 43
Nonsupersingular 181
NP-complete 112 118
Number field sieve 152—153 164—165
Numerical equivalents 55
Oblivious transfer 120—123
One-way function 85
Order of a point 173
Order of an element 33
Parameters 56 83
Pepin primality test 190
Plaintext 54
Pocklington primality test 187—188
Pohlig — Silver — Hellman algorithm 102—103 183
Point at infinity 168 171
Pollard p - 1 method 192—193
Polynomial time 10
Polynomials 17
Polynomials, derivative of 32
Polynomials, Euclidean algorithm for 17
Polynomials, g.c.d.of 17 32
Polynomials, irreducible 32
Polynomials, monic 17 32
Polynomials, multiple roots 17
Polynomials, primitive 38
Polynomials, ring of 31
Polynomials, unique factorization 32
Precomputation 104
Primality test 92 125
Primality test, Adleman — Huang 190
Primality test, Adleman — Pomerance — Rumely 134—135
Primality test, Atkin 187 190
Primality test, Cohen — Lenstra 134—135
Primality test, elliptic curve 188—190
Primality test, Miller — Rabin 130—131
Primality test, Pepin 190
Primality test, Pocklington 187—188
Primality test, Solovay — Strassen 129
Primality test, trial division 126
Prime field 33
Prime number 12
Prime number theorem 11 92
Prime number, Fermat 29 51 109 190
Prime number, in arithmetic progression 35
Prime number, Mersenne 28 29 51 125 191 207
Primitive polynomial 38
Primitive polynomial, root of unity 42
Private key cryptosystem 88
Probabilistic algorithm 86 95 127
Probabilistic algorithm, encryption 89
Product of cryptosystems 64 78—79
Projective equation 171
Projective equation, plane 171
Projective equation, point 171
Pseudoprime 126
Pseudoprime, Euler 129
Pseudoprime, public key 87 88
Pseudoprime, strong 130
Quadratic character 174
Quadratic character, nonresidue 43
Quadratic character, reciprocity 45 47
Quadratic character, residue 43
Quadratic character, sieve 160—162
Random 92
Random, walk 174
Rank of an elliptic curve 173
Reduction of an elliptic curve 184 193—194
Relatively prime 14
Repeated squaring method 23 97 104
Repeating expansion of fraction 10 200 222
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