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Index

(k, n)-threshold scherne, 349 R(x),82 EXP, 147 NP, 148 NP-complete, 152 NP-hard, 152 P,147 A(n), 71 >..(n),62 p,(n), 63 rj;(n),61 7r, 141 7r(x),65 7r2(x),86 7rM(x),29 1/J(x),72 a-(n),51 T(n), 51 8(x),69 ((s),75 b-sequence, 173 kth (higher) power nonresidue, 122 kth (higher) power residue, 121 kth power nonresidue, 106 kth power residue, 106 nth prime, 84 s(n), 51 NP-SPACE, 149 P-SPACE, 149 Li(x),73

Abel, N. H., 14 absolute least residue, 108 adder, 272 additive group, 15 additive identity, 17 additive inverse, 17 additivity, 6 Adlernan, L., 313 Advanced Encryption Standard (AES),

302

affine transformation, 292 Al-Khwarizmi, 142 algebraic computation law, 129 algebraic equation, 42 algebraic numbers, 13 algorithm, 142 algorithmic number theory, 139 aliquot k-cycle, 54 alphabet, 148 amicable k-tuple, 54 amicable pair, 54, 141, 249 amicable tripIe, 54 APR test, 191 APRCL test, 191 arithmetic function, 50 arithmetic mean, 53 arithmetic progression of primes, 89,

141 associativity, 14 asymmetrie cryptosystems, 289 asymmetrie key cryptosystem, 304 Atkin, A. O. L., 188 authentication, 288

base-2 pseudoprimality test, 172 basis vector, 231 Bernoulli's number, 75 binary computers, 269 binary Goldbach conjecture, 7, 252 binary operation, 13 bit operation, 156 block cipher, 294 Brent, R. P., 212 Brun's constant, 87

Caesar, J., 291 Caesar cipher, 291 Carmichael, R. D., 62 Carmichael number, 171 Carmichael's >"-function, 62, 98, 314 Carmichael's theorem, 98

376

CESG,309 CFRAC factoring algorithm, 203 CFRAC method, 201 character cipher, 291 Chebyshev, P. L., 69 Chebyshev's function, 69 check digit, 278 Chen, J. R., 8, 88 Chinese Remainder Theorem (CRT),

101, 345, 349 Chinese test, 172 Church, A., 145 Church-Turing thesis, 145 ciphertext space, 289 closure, 14 Cocks, C. C., 307 coin-tossing states, 144 combined test, 182 common multiple, 26 Communication-Electronics Security

Group (CESG), 306 commutative group, 14 commutative ring, 16 commutativity, 14 complement, 148 complete quotients, 39 complete system of residues, 93 completely multiplicative function, 51 complex numbers, 13 complex zeros, 77 complexity classe, 148 composite, 20 composite Fermat numbers, 30 computable, 142 computation, 145 computationally intractable (or

infeasible), 150 computationally tractable (or feasible),

150 congruence, 90 congruence classes, 91 congruent, 91 continued fraction, 36 Continued FRACtion (CFRAC)

method,194 convergent, 37 convergents, 44 Converse of Fermat's little theorem, 98 Converse of Wilson's theorem, 99 Cook, S., 151 Cook-Karp Thesis, 151 covered writing, 356 cryptanalysis, 288

cryptography, 288 cryptology, 288 cubic Diophantine equation, 125 cyclic group, 15

Index

Data Encryption Standard (DES), 299, 319

database decryption, 346 database encryption, 345 database security, 344 De la Vallee-Poussin, C. J., 71 decidable, 142 decision problems, 148 decryption key, 289 decryption process (algorithm), 289 deterministic encryption, 326 Deutsch, D., 230 Dickson, L. E., 259 Diffie, W., 303 Diffie-Hellman key-exchange, 310 Digital Signature Algorithm (DSA),

342 Digital Signature Standard (DSS), 342 digital signatures, 336 Diophantine equation, 41 Diophantus, 41 Dirac, P. A. M., 231 Dirichlet, J. P. G. L., 81 Dirichlet characters, 81 Dirichlet L-functions, 81 Dirichlet series, 81 discrete exponential bit generator, 287 discrete exponential generator, 286 discrete logarithm, 120 discrete logarithm problem, 218, 305 Disquisitiones Arithmeticae, 90 dividend, 32 divisibility, 20 division algorithm, 32 division ring, 16 divisor, 20 domain, 50 double encryption, 301 double hashing, 275

ECPP (Elliptic Curve Primality Proving), 188

ECPP Algorithm, 188 effective procedure, 142 efficient (good) algorithm, 150 electronic commerce, 355 EIGamal cryptosystem, 311 elliptic curve, 125, 332

Index

elliptic curve analogue of Diffie­Hellman, 334

elliptic curve analogue of ElGamal, 335 elliptic curve analogue of Massey-

Omura,334 elliptic curve analogue of RSA, 335 elliptic curve bit generator, 287 elliptic curve cryptography (ECC), 332 elli ptic curve test, 186 elliptic function, 127 elliptic integral, 127 Ellis, J. H., 307 embedded message, 357 embedding messages on elliptic curves,

333 encryption key, 289 encryption process (algorithm), 289 ENIGMA code, 288 equivalence classes, 91 equivalence relation, 91 Eratosthenes of Cyrene, 22 Erdös, P., 72 error detection and correction, 277 Euclid, 2, 21 Euclid's algorithm, 32, 34 Euclid's Elements, 35 Euclid-Euler Theorem, 55 Euler, L., 59 Euler probable prime, 179 Euler pseudoprime, 179 Euler's (totient) t/J-function, 61 Euler's criterion, 107 Euler's pseudoprimality test, 178 Euler's rule for amicable pairs, 59 Euler's theorem, 98 exclusive or (XOR), 301 exponential complexity, 158 exponentially bounded, 147 exponentially solvable, 147

factor, 20 factoring by trial divisions, 197 fast group operations, 164 fast modular exponentiations, 161 fast point additions, 164 Federal Information Processing

Standard, 299 Fermat, P., 11 Fermat numbers, 30, 140 Fermat probable prime, 171 Fermat pseudoprime, 171 Fermat's factoring algorithm, 198 Fermat's Last Theorem (FLT), 11

Fermat's little theorem, 97 Fermat's pseudoprimality test, 170 Feynman, R. P., 230 Fibonacci, L. P., 180 Fibonacci numbers, 180 field, 17 finite fields, 19 finite group, 15

377

finite order of a point on an elliptic curve, 133

finite simple continued fraction, 38 FIPS 186, 342 FIPS 46, 299 FIPS 46-2, 299 FIPS 46-3, 299 firewall; 352 fixed-base number systems, 261 fixed-point attack, 326 Fundamental Theorem of Arithmetic,

24, 261

Galileo spacecraft, 281 Galois, E, 19 Galois field, 19 Gauss, C. F., 67 Gauss's lemma, 108 Generalized Riemann Hypothesis, 82 generating function, 81 geometrie composition law, 129 geometrie mean, 52 Goldbach partition, 9 Goldbach's conjecture, 6, 141, 252 Goldwasser, S., 187 Government Communications

Headquarters (GCHQ), 306 greatest common divisor (gcd), 25 group, 14 group laws on elliptic curves, 133

Hadamard, J., 70 halting problem, 146 Hardy, G. H., 7 Hardy-Ramanujan taxi number, 9 harmonie mean, 53 hashing function, 274 Hasse, H, 134 height, 131 Hellman, M. E., 304 high-order congruence, 104 Hilbert space, 231 hybrid cryptosystem, 306

identity, 14

378

incongruent, 91 index caleulus algorithm, 226 index caleulus method, 226 index of a to the base g, 120 index of an integer modulo n, 120 inefficient (bad) algorithm), 150 infinite fields, 18 infinite group, 15 infinite order of a point on an elliptie

curve, 133 infinite simple continued fraction, 38 instantaneous description (ID), 145 integer, 12 integer factorization problem, 192, 305 integral domain, 16 International Standard Book Number

(ISBN), 278 Internet, 352 inverse, 14 invertible function, 305 irrational number, 13 irrational numbers, 38 isomorphie, 265 isomorphism, 265

Jacobi, C. G., 111 Jacobi symbol, 111

Karp, R., 151 key bundle, 301 key space, 289 Kilian, J., 187 Knuth, D. E., 193 Koblitz, N, 332 Kronecker, L., 12

Lagarias, J. C., 244 Landau, E., 7 language, 148 least (nonnegative) residue of x modulo

n,92 least common multiple (lern), 26 Legendre, A. M., 68, 107 Legendre symbol, 107 Legendre's congruence, 199 Lehman's method, 194 Lehmer, D. H., 183 Lenstra, H. W. Jr., 215 Lenstra's Elliptie Curve Method

(ECM), 195, 215 linear congruence, 96 linear Congruential generator, 283 linear Diophantine equation, 42 Littlewood, J. E., 7

logarithm, 154 logarithmic integral, 73 Lucas, F. E., 180 Lucas numbers, 180 Lucas probable prime, 181 Lucas pseudoprimality test, 182 Lucas pseudoprime, 181 Lucas sequences, 180 Lucas test, 181 Lucas theorem, 181 Lucas-Lehmer test, 183 Lucas-Lehmer theorem, 183 Lucas-Lehmer test for Mersenne

primes, 185

Möbius, A. F., 63 Möbius f.L-function, 63 Möbius inversion formula, 64 magnitude, 270 Massey-Omura cryptosystem, 312 Meissel, D. F. E., 244 Mersenne, M., 27 Mersenne number, 27 Mersenne primes, 27, 140 Mertens's conjecture, 60 message concealing, 357 message digest, 343 message extracting, 357 message space, 289 middle-square method, 282 Miller, G., 174 Miller-Rabin test, 174

Index

minimal perfect hashing function, 276 minimal collision-free hashing function,

276 mixed-base number systems, 261 modular arithmetic in 7l,/n7l" 94 modular exponentiation, 160 modular inverse, 95 monographic cipher, 291 monoid,14 Morain, F., 188 Mordell, L. J., 135 multiple, 20 multiple encryption, 301 Multiple Polynomial Quadratie Sieve

(MPQS),194 multiplieative function, 50 multiplieative generator, 284 multiplicative group, 15 multiplieative identity, 18 multiplicative inverse, 18, 95 multiplicativity, 5

Index

National Institute of Standards and Technology (NIST), 299

natural numbers, 12 non-secret cover-message, 357 non-secret encryption, 307 non-zero field element, 18 noncomputable, 143 nonnegative integers, 12 nonpositional number systems, 261 nontrivial divisor, 20 nontrivial square root of 1, 173 nontrivial zeros, 77 nonwitness, 177 Number Field Sieve (NFS), 5, 194, 206,

227 number systems, 261

odd perfect numbers, 140 Odlyzko, A. M., 60, 244 one's complement representation, 272 one-way function, 304 order of a modulo n, 115 order of a field, 19 order of a group, 15 order of a point on an elliptic curve,

133

packer filter, 353 parity, 2 parity check, 3, 277 parity check bit, 277 partial quotients, 36 Pell's equation, 45 perfect hashing function, 276 perfect number, 54 period, 38 periodic simple continued fraction, 38 Pocklington's theorem, 186, 316 point at infinity, 126 polarization, 358 Pollard, J. M.,208 Pollard's p factoring algorithm, 213 Pollard's p-method, 194, 208 Pollard's "p - I" factoring algorithm,

214 Pollard's "p - I" method, 214 polygraphic cipher, 294 polynomial bounded, 147, 148 polynomial complexity, 158 polynomial congruence, 104 polynomial congruential equation, 104 polynomial security, 327 polynomially solvable, 147

Pomerance, C., 204 positional number systems, 261 positive integers, 12 power generator, 285

379

practically feasible computation, 147 practically tractable computation, 147 primality, 4 primality testing problem, 167 prime, 20 prime counting function, 65 prime distribution function, 65 prime factor, 24 prime factorization, 5, 24 prime Fermat numbers, 30 Prime Number Theorem, 67 prime numbers, 65 prime power, 19 prime tripies, 4 primitive root of n, 116 principle of superposition, 232 privacy, 288 private key, 304 probabilistic encryption, 327, 328 probabilistic Turing machine (PTM),

144 probable prime, 171 pseudoprime, 171 pseudorandom numbers, 282 public key, 304 public-key cryptography, 289 public-key cryptosystem, 306 purely periodic simple continued

fraction, 38 Pythagoras, 58

Qin Jiushao, 101 quadratic congruence, 105 quadratic Diophantine equation, 45 quadratic irrational, 38 quadratic nonresidue, 106 quadratic nonresidue modulo n, 306 Quadratic reciprocity law, 109 quadratic residue, 106 quadratic residue modulo n, 305 quadratic residues generator, 286 Quadratic Residuosity Problem (QRP),

305, 327 Quadratic Sieve (QS), 204 quantum algorithm for discrete

logarithms, 241 quantum algorithm for integer

factorization, 238 quantum bit, 231

380

quantum computer, 231, 233 quantum cryptography, 358 quantum operation, 234 quantum register, 233, 238 quantum state, 231 quantum Turing machine (QTM), 235 qubit, 235 quotient, 32

Rabin, M. 0., 174 Rabin's modified bit generator, 287 Ramanujan, S., 9 random number generation, 282 random numbers, 282 randomized encryption, 327 rational numbers, 13, 38 read-keys, 345 real numbers, 13 real zeros, 77 real-valued function, 50 realbase logarithm, 120 rectilinear polarization, 358 reduced system of residues modulo n,

94 reflexive, 91 relatively prime, 25 remainder, 32 repeated doubling and addition, 164 repeated doubling method, 334 repeated multiplication, 160 repeated squaring, 160 repeated squaring and multiplication,

160 residue arithmetie in (Z/nZ)*, 267 residue dass, 92 residue dasses, 91 residue dasses modulo n, 13 residue computers, 269 residue number systems, 261, 269 residue of x modulo n, 92 residue representation of a number, 262 Riemann, G. F. B, 70, 74 Riemann (-function, 74, 75 Riemann function, 82 Riemann Hypothesis (RH), 77, 141 ring, 15 ring with identity, 16 Rivest, R. L., 313 root finding problem, 228, 325 RSA Assumption, 313 RSA bit generator, 286 RSA cryptosystem, 313 RSA generator, 285

running time, 147

secret key, 289, 304 secret sharing, 348 secret-key cryptography, 288 secret-key cryptosystem, 306 Selberg's estimate, 71 Selberg, A., 71 Selfridge, J. L., 175 semantic security, 327 semigroup, 14 seminumerical method, 249 Shamir, A., 313 Shanks, D., 219

Index

Shanks' baby-step giant-step method for discrete logarithms, 220

Shanks' dass group method, 194 Shanks' SQUFOF method, 194 Shannon, C. E., 155 Shannon bits, 231 shift transformation, 292 Shor, P., 238 sieve of Eratosthenes, 22 sign bit, 270 signature generation, 342 signature verification, 342 signed-magnitude representation, 270 Silver-Pohlig-Hellman algorithm, 222 simple continued fraction, 36 sociable group, 54 Solovay, R., 190 Solovay-Strassen test, 179 solvable, 142 square generator, 286 square root method, 222 steganographic system, 357 steganography, 356 stego-key, 357 stego-message, 357 Strassen, V., 190 strong probable prime, 175 strong pseudoprimality test, 173, 174 strong pseudoprime, 175 strong test, 173 subexponential complexity, 195 subgroup, 15 substitution cipher, 291 summatory function of 4(n), 72 Sun Zi, 101 superposition, 234 symmetrie,91 symmetrie cryptosystems, 288

Taylor, R., 11

Index

te Riele's rule, 60 te Riele, H. J. J., 60 ternary Goldbach conjecture, 7, 252 Thabit ibn Qurra, 56 Thabit's rule for amicable pairs, 57 time complexity function, 147 transcendental numbers, 13 transitive, 91 trapdoor, 305 trapdoor one-way function, 305 trial division, 194 Tripie DES (TDES), 301 trivial divisor, 20 trivial zeros, 77 Turing, A. M., 143 Turing machine, 143 Twin Prime Conjecture, 88 twin primes, 4, 65 two's complement representation, 272

381

U.S. National Institute of Standards and Technology (NIST), 342

undecidable, 143 unsolvable, 143

Vinogradov, I. M., 8 von Mangoldt function, 71 von Neumann, J., 282

Waring's problem, 141 Wiener's attack, 326 Wiles, A. J., 10, 11 Williamson, M, 308 Wilson, J., 99 Wilson's theorem, 99 witness, 177 words, 269 write-keys, 345