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Koblitz N. Ч A course in number theory and cryptography
Koblitz N. Ч A course in number theory and cryptography

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Ќазвание: A course in number theory and cryptography

јвтор: Koblitz N.

јннотаци€:

The purpose of this book is to introduce the reader to arithmetic topics, both ancient and modern, that have been at the center of interest in applications of number theory, particularly in cryptography. No background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasizing estimates of the efficiency of the techniques that arise from the theory. A special feature is the inclusion of recent application of the theory of elliptic curves. Extensive exercises and careful answers have been included in all of the chapters. Because number theory and cryptography are fast-moving fields, this new edition contains substantial revisions and updated references.


язык: en

–убрика: ћатематика/“еори€ чисел/¬ычислительна€ теори€ чисел/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

»здание: second edition

√од издани€: 1994

 оличество страниц: 235

ƒобавлена в каталог: 04.06.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
Abelian group      33
Abelian group, type of      174
Adleman Ч Huang primality test      190
Adleman Ч Pomerance Ч Rumely primality test      134Ч135
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      102Ч106
Algorithm, index-calculus      103Ч106
Algorithm, probabilistic      86 95 127
Algorithm, Schoof      179 183
Algorithm, Silver Ч Pohlig Ч Hellman      102Ч103 183
Alphabet      54
Alphabet, Cyrillic      63 78
Arms control      90Ч91 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      7Ч8
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      127Ч128 136
Casanova      84Ч85
Characteristic of a field      33
Chinese remainder theorem      21
Chor Ч Rivest knapsack      115
Ciphertext      54
Classical cryptosystem      88
Classical cryptosystem, Cohen Ч Lenstra primality test      134Ч135
coin toss      91 96Ч97 215
Coloring map or graph      118
complex numbers      17
Complex numbers, Gaussian integers      17 37 42Ч43 171
Composite number      12
Composition of cryptosystems      64 79
Congruence      19 193
conjugate      32
Continued fraction      155
Continued fraction, factorization method      158Ч159
Convergent      155
Crypt analysis      56
Cryptography      54
cryptography, public key      85
Cryptosystem      54Ч55 83
Cryptosystem, classical      88
Cryptosystem, composition      64 79
Cryptosystem, Diffie Ч Hellman      98Ч99 181Ч182
Cryptosystem, ElGamal      100Ч101 109 182
Cryptosystem, elliptic curve      181Ч182
Cryptosystem, knapsack      113Ч115
Cryptosystem, Massey Ч Omura      100 109 182 216
Cryptosystem, Merkle Ч Hellman      113Ч114
Cryptosystem, private key      88
Cryptosystem, product      64 78Ч79
Cryptosystem, public key      85
Cryptosystem, RSA      22 92Ч93 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 Ч Hellman assumption      99 121
Diffie Ч Hellman assumption, key exchange      98Ч99 181Ч182
Digital Signature Standard      101Ч102
DIGITS      1
Digits, binary (bit)      3
Digits, number of      3
Digraph      54 59
Digraph, transformation      59
Dirichlet L-series      134
Discrete log      97Ч98
Discrete log, algorithms for      102Ч106
Discrete log, on elliptic curve      180
Divisibility      12
Divisibility, exact      12
Division points      173
Divisor      12
Divisor, nontrivial      12
Divisor, proper      12
Elgamal cryptosystem      100Ч101 109 182
ElGamal cryptosystem, signature      109Ч110
Elliptic curve      167Ч168
Elliptic curve, addition law      168Ч170
Elliptic curve, complex points      171
Elliptic curve, cryptosystem      181Ч182
Elliptic curve, factorization      191Ч192 195Ч198
Elliptic curve, global      183
Elliptic curve, nonsupersingular      181
Elliptic curve, over finite field      174
Elliptic curve, primality test      188Ч190
Elliptic curve, rank      173
Elliptic curve, real points      176Ч177 227
Elliptic curve, reduction      184 193Ч194
Elliptic curve, super singular      181
Elliptic curve, torsion subgroup      173 185
Elliptic curve, Weil pairing      180Ч181
Elliptic curve, zero element      169
Elliptic curve, zeta-function      175
Elliptic function      173
Enciphering      54
Enciphering, key      56 83
Enciphering, matrix      71Ч72
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 21Ч22
Euler phi-function, pseudoprime      129
exponentiation      23 97
Factor base      145
Factor base, algorithm      103 148
Factoring      27Ч29 92
Factoring, continued fraction method      158Ч159
Factoring, Fermat factorization      15 96 143Ч144
Factoring, Monte Ч Carlo method      138Ч140
Factoring, Pollard p - 1 method      192Ч193
Factoring, quadratic sieve      160Ч162
Factoring, rho method      138Ч142
Factoring, trial division      126 138
Factoring, with elliptic curves      191Ч192 195 198
Fermat factorization      15 96 143Ч144
Fermat factorization, prime      29 51 109 190
FermatТs Little Theorem      20 126
Fibonacci numbers      16Ч17 77Ч78 159 211Ч212 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      35Ч36
Finite fields, generator      34
Finite fields, irreducible polynomials over      38Ч39 104 110
Finite fields, roots of unity in      42
Finite fields, square roots in      42 48 52 96 179Ч180
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 42Ч43 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
HasseТs theorem      174
Hexadecimal      10
Imbedding plaintexts      179
Index-calculus algorithm      103Ч106
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      113Ч115
Knapsack cryptosystem, problem      112
Knapsack cryptosystem, superincreasing      112
LagrangeТs Theorem      157
Lattice      171
Least absolute residue      145
Least absolute residue, common multiple      13
Legendre symbol      43 174
Lenstra elliptic curve factorization      191Ч192 195Ч198
Lifting      52 80
Line at infinity      171
Linear algebra      58 66Ч68
Linear algebra, modulo 2      146Ч147
Linear algebra, modulo N      68Ч70 105
Linear map      57 67 68 70
Massey Ч Omura cryptosystem      100 109 182 216
Matrices      66Ч67 68
Matrices, inverses      67 69
Matrices, Merkle Ч Hellman cryptosystem      113Ч114
Mersenne prime      28 29 51 125 191 207
Mersenne prime in the Gaussian integers      228
Message unit      54
Miller Ч Rabin primality test      130Ч131
Miller Ч Rabin primality test, time estimate for      136Ч137
Modular exponentiation      23Ч24 97
Modulus      19
Monic polynomial      17 32
Monte Ч Carlo factorization      138Ч142
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      152Ч153 164Ч165
Numerical equivalents      55
Oblivious transfer      120Ч123
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      187Ч188
Pohlig Ч Silver Ч Hellman algorithm      102Ч103 183
Point at infinity      168 171
Pollard p - 1 method      192Ч193
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 Ч Huang      190
Primality test, Adleman Ч Pomerance Ч Rumely      134Ч135
Primality test, Atkin      187 190
Primality test, Cohen Ч Lenstra      134Ч135
Primality test, elliptic curve      188Ч190
Primality test, Miller Ч Rabin      130Ч131
Primality test, Pepin      190
Primality test, Pocklington      187Ч188
Primality test, Solovay Ч 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 78Ч79
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      160Ч162
Random      92
Random, walk      174
Rank of an elliptic curve      173
Reduction of an elliptic curve      184 193Ч194
Relatively prime      14
Repeated squaring method      23 97 104
Repeating expansion of fraction      10 200 222
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