Factorization of n = 87463 with the Quadratic Sieve To find a factor base, we need to find primes such that n is a square modulo p and for each such p determine the solutions of x2 ≡ n (mod p). Testing small primes up to 37 we find that only for p ∈ {2, 3, 13, 17, 19, 29} there are solutions to x2 ≡ n (mod p), namely: p 2 3 13 17 19 29 x 1 1, 2 5, 8 7, 10 5, 14 12, 17 We now √ start sifting, using an interval of length 2 · 30 around b nc = 295. (See table on last page.) For the values of x for which x2 − n splits completely, the exponent vector modulo 2 is: x −1 2 3 13 17 19 29 265 1 1 1 0 1 0 0 278 1 0 1 1 0 0 1 296 0 0 0 0 1 0 0 299 0 1 1 0 1 1 0 307 0 1 0 1 0 0 1 316 0 0 0 0 1 0 0 We now solve (the matrix is transposed as we solve Av = 0 and not vA = 0): 1 1 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 ·v =0 1 0 0 modulo 2. One solution is v = (1, 1, 1, 0, 1, 0) We thus take the 1st, 2nd, 3rd and the 4th x-value and get x = 265 · 278 · 296 · 307 = 6694540240 ≡ 34757 (mod n) p y = (2652 − n) · (2782 − n) · (2962 − n) · (3072 − n) = 2 · 34 · 132 · 17 · 29 = 13497354 ≡ 28052 (mod n) This yields the gcds: gcd(x − y, n) = 149, which gives a factorization. gcd(x + y, n) = 587 x 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 2 3 13 17 19 29 x2 − n splits X X X X XX −2 · 3 · 132 · 17 XX X X X X X X XX XX X X X XX X X XX X X X X X X X XX X X XX X X X XX XX X X X X XX X XX X −33 · 13 · 29 x 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 2 3 13 17 19 29 x2 − n splits X X 32 · 17 X X XX X X 2 · 3 · 17 · 19 XX X X X X XX XX X X X X XX X X X XX X X X X X XX X 36 · 17 X XX X X X X XX XX X X X XX X X X 2 · 32 · 13 · 29