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Cohen H.A. — A Course in Computational Algebraic Number Theory
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Название: A Course in Computational Algebraic Number TheoryАвтор: Cohen H.A. Аннотация: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1996Количество страниц: 563Добавлена в каталог: 28.05.2005Операции: Положить на полку |
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10 378 186 153 390 66 154 378 208 Abelian group 66 Addition chain 11 Addition theorem 370 Additive degeneracy 373 Adeles 188 Adjoint matrix 54 Adleman, L. 445 471 501 Affine subspace 486 Algebraic integer 153 Algebraic number 153 Algorithm 1 Ambiguous cycle 270 434 Ambiguous form 255 433 434 Antwerp IV 367 Approximation theorem 192 Archimedean valuations 187 Artinian rings 303 Atkin, O. 32 247 252 445 471 481 Axiom 2 507 Baby step giant step 240 Bach, E. 34 254 Bad reduction 373 Bareiss, E. 52 Berge, A.-M. 175 329 Berlekamp, E. 130 Bernardi, D. 305 382 394 Bignum 2 509 Binary quadratic form 225 Birch and Swinnerton — Dyer conjecture 393 Birch, B. 392 Birthday paradox 480 Bit operations 1 Bosma, W. 466 Brauer — Siegel theorem 216 Brent, R. 420 427 429 441 Buchmann, J. 288 303 315 352 Buhler, J. 501 Canfield, E. 481 Canonical height 411 Canonical height pairing 411 Cantor, D. 127 Carayol, H 390 Carmichael numbers 421 Cartan decomposition 107 Cartan, E. 107 CEILING 7 cfrac 477 CHARACTER 448 Characteristic polynomial 53 162 Chinese remainder theorem 19 Cholesky decomposition 104 Chudnovsky, D.and G. 490 Cl(K) 208 Class group 207 228 Class number 208 Codifferent 205 Coefficient explosion 5 112 114 Cohen, H. 168 254 288 296 352 445 Collins, G. 118 Column echelon form 59 Column vector 7 Comatrix 54 Compact representation of units 279 285 Complementary error function 238 Completely split prime 197 Complex multiplication 381 Compositeness test 419 composition 243 Conductor 224 Conductor-discriminant formula 168 Congruent number 376 Conjugate vector representation 161 Conjugates 154 Content 116 Continued fraction 21 265 271 426 478 Coordinate recurrence method 254 Coppersmith, D. 480 504 Coprime elements 116 Coprime ideals 182 Couveignes, J.-M. 405 Cremona, J. 394 417 Cycle of reduced forms 262 Cyclotomic field 446 Davenport, H. 462 de Bruijn, N. 481 de Weger, B. 93 Dedekind domain 185 Dedekind zeta function 214 Dedekind, R. 305 Dedekind’s eta-function 416 Deep insertion 91 Degenerate elliptic curve 373 Degree of a prime ideal 197 Deligne, P. 387 Denominator of a module 74 DERIVE 2 507 Determinant 52 Determinant of a lattice 80 Diaz y Diaz, F. 168 254 288 313 352 363 Different 205 Dirichlet character 448 Dirichlet, P. 211 Discriminant of a number field 166 Discriminant of a polynomial 119 Discriminant of a quadratic number 384 Discriminant of an n-element set 165 Distance function 279 Distinct degree factorization 126 Divisibility (in a ring) 114 Division polynomials 405 Double large prime variation 494 Doubly periodic function 368 Dual isogeny 380 Duke, W 298 Dullmann, S. 252 254 Dwork, B. 388 Early abort strategy 259 480 ECM 487 Eichler, M. 236 Eisenstein polynomial 315 Elementary divisor 76 Elementary operations 48 Elimination 48 Elkies, N. 405 471 Elliptic curve over K 369 Elliptic function 368 Elliptic integral 367 397 Elliptic logarithm 398 Enlarging procedure 304 Equivalence of quadratic forms 225 Erdos, P. 481 Euchner, M. 91 Euclidean Domain 114 Euclid’s algorithm 12 Euler product 250 Expected running time 2 Exponential time 2 Factor base 260 478 Fast multiplication methods 3 Fermat number 424 495 Fermat’s Last Theorem 151 208 392 459 Fermat’s Little Theorem 421 439 450 Fermigier, S. 394 Field membership problem 179 Fincke, U. 105 FLOOR 7 Floyd, R. 427 FLT 151 208 Fractional ideal 183 Frobenius homomorphism 309 Functional equation for elliptic curves 390 Functional equation for number fields 215 Functional equation for quadratic fields 238 266 267 Functional equation, sign of 391 Fundamental discriminant 224 Fundamental domain 368 Fundamental units 210 Galois closure 157 Galois group 157 322 Gap 508 Gauss sum 448 Gauss, K.F. 52 Gaussian elimination 48 Gaussian pivoting 48 Gaussian reduction 23 Gauss’s Lemma 116 GCD 7 115 Generalized Riemann hypothesis 34 Genus field 474 Germain, S. 151 GMP 2 509 Goldfeld, D. 216 234 Goldwasser, S. 445 Gram matrix 80 Gram — Schmidt orthogonalization 82 Greatest common divisor 7 12 115 GRH 34 Gross, B. 216 234 385 394 Group ring 446 h(K) 208 H(n) 234 Hadamard’s inequality 51 82 Hafner, J. 69 70 77 252 Hashing 299 Hasse — Weil L-function 389 Hasse, H. 373 462 Height 411 Hensel’s Lemma 137 Hermite normal form 67 Hermite normal form of a 67 Hermite normal form of a matrix 67 Hermite, C. 198 Hermite’s constant 334 Hessenberg form 55 Hilbert class field 384 416 Hilbert class polynomial 415 HNF 67 HNF-basis 189 Huang, M.D. 445 471 Hurwitz class number 234 Ideal 182 Ideal, class 208 Ideal, equivalence 207 Ideal, intersection 207 219 Ideal, inversion 204 Ideal, product 190 Ideal, representation 188 190 Ideal, two-element representation 192 Ideal, valuation 201 Idele class group 209 Ideles 188 Image of a matrix 58 INDEX 167 Inert prime 197 Inessential discriminantal divisor 199 364 Infinite prime 198 Infrastructure method 279 Integral basis 166 Integral domain 114 Integral ideal 183 Integral kernel 74 98 Integrally closed 185 Intelligent Gaussian elimination 480 Intelligent Hermite reduction 254 Inverse image 60 Irreducible element 114 Irreducible polynomial 124 Isogeny 379 Isomorphism problem 179 Iwasawa decomposition 83 j(E) 377 Jacobi sum 448 Jacobi symbol 28 Kant 508 Karatsuba, A. 3 Kernel of a matrix 57 Kilian, J. 445 Knuth, D. 298 Kodaira type 407 Kolyvagin, V. 394 Kouya, T. 394 Kraitchik, M. 478 Kronecker symbol 28 Kronecker, L. 211 L(p) 109 L(X) 254 L-function 266 388 389 L-series 237 LaMacchia, B. 89 254 Large prime variation 258 480 Laska, M. 409 Lattice 23 80 Lattice, determinant of 80 Legendre symbol 27 Legendre symbol, generalized 219 Legendre, A. 478 Lehman, S. 425 Lehmer, D.H. 13 423 443 478 Lenstra, A.K. 84 141 494 495 Lenstra, H.W. 84 141 184 201 296 298 303 315 320 419 442 445 466 481 484 503 Leopoldt’s conjecture 216 lg 7 Lisp 2 LLL algorithm 87 LLL algorithm, integral 94 LLL-reduced basis 85 Logarithmic embedding 210 Louboutin, S. 301 Lovasz, L. 84 141 Lucas — Lehmer test 443 Lucas, E. 443 LUP form of a matrix 50 Macsyma 2 507 Magma 2 508 Manasse, M. 494 495 Manin, Y. 392 Manin’s constant 392 Maple 2 507 Martinet, J. 217 329 Mathematica 2 507 Matrix representation 160 Maximal ideal 184 Maximal order 186 303 Mazur, B. 375 McCurley, K. 69 70 77 252 288 Mersenne number 424 443 495 Mestre, J.-F. 394 418 Mignotte, M. 134 Miller, G. 421 Minimal polynomial 153 Minkowski, H. 198
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