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Quadratic Field Sieve

Quadratic Field Sieve. QFS Matt Spear Steven Guy.

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Quadratic Field Sieve

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  1. Quadratic Field Sieve QFS Matt Spear Steven Guy 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357

  2. Agenda • Introduction to sieves • Euclid’s GCD in base 2 • Definitions • Algorithms • RHO example • Factor Bases • QFS example • Introduction to MPQFS

  3. Prime Number Sieve Sieve of Eratosthenes • Start with all numbers greater than 1 • Divide all by the first number • Repeat until no numbers are left to divide by, i.e. the last number is all left. • What remains are the prime numbers.

  4. Prime Number Sieve 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 Initial Sieve Space

  5. Prime Number Sieve 2 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | 47 | 49 | 51 | 53 | 55 | 57 | 59 | 61 | 63 | 65 | 67 | 69 | 71 | 73 | 75 | 77 | 79 | 81 | 83 | 85 | 87 | 89 | 91 | 93 | 95 | 97 | 99 | 101 After Divide by two

  6. Prime Number Sieve 2 3 | 5 | 7 | | | 11 | 13 | | | 17 | 19 | | | 23 | 25 | | | 29 | 31 | | | 35 | 37 | | | 41 | 43 | | | 47 | 49 | | | 53 | 55 | | | 59 | 61 | | | 65 | 67 | | | 71 | 73 | | | 77 | 79 | | | 83 | 85 | | | 89 | 91 | | | 95 | 97 | | | 101 After Divide by three

  7. Prime Number Sieve 2 3 | 5 | 7 | | | 11 | 13 | | | 17 | 19 | | | 23 | | | | | 29 | 31 | | | | | 37 | | | 41 | 43 | | | 47 | 49 | | | 53 | | | | | 59 | 61 | | | | | 67 | | | 71 | 73 | | | 77 | 79 | | | 83 | | | | | 89 | 91 | | | | | 97 | | | 101 After Divide by five

  8. Prime Number Sieve 2 3 | 5 | 7 | | | 11 | 13 | | | 17 | 19 | | | 23 | | | | | 29 | 31 | | | | | 37 | | | 41 | 43 | | | 47 | | | | | 53 | | | | | 59 | 61 | | | | | 67 | | | 71 | 73 | | | | | 79 | | | 83 | | | | | 89 | | | | | | | 97 | | | 101 After all possible divisions

  9. Prime Number Less Than 1602 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601

  10. Euclid's GCD Algorithm (Binary) g := 1 while u is even &&v is even u := u/2 v := v/2 g := 2*g EndWhile // now u or v (or both) are odd while u > 0 if u is even, then u := u/2 elseif v is even, then v := v/2 elsethen t := |u-v|/2 if u < v, then v := t else u := t EndIf EndIf EndWhile return g*v

  11. Groups • An algebraic structure (G, Δ) with one associative composition (operation) (Δ) • Contains a neutral element for the Δ and every element is invertible over Δ • Is Abelian If the Δ is also commutative For Example: (Nn,+) (addition modulo n) is an abelian group with neutral element e = 0 and inverse of x = n - x

  12. Rings • An algebraic structure (A,+,·) with (A,+) being an abelian group and (A,·) being associative composition distributed over + • Is commutative ring if · is commutative For Example: (Nn,+n,·n) is a commutative ring, called the ring of integers mod n.

  13. Fields • A commutative ring with every nonzero element possessing a · inverse (x·x* = 1) • Is Finite if the size of the field is non-infinite For Example: Z/pZ is a Finite Field when p is a prime integer, the field of integers modulo p (Fp). (if p is not prime all numbers will not have an inverse i.e. if p = 10 then 2 has no inverse over ·)

  14. Quadratic Residues • Solutions n to the equation x2 ≡ n (mod p) • If an element is not the square of a number it is a nonresidue. For Example: In F11 : the residues are {1, 4, 9, 5, 3} as 12 ≡ 1; 22 ≡ 4; 32 ≡ 9; 42 ≡ 5; 52 ≡ 3. The nonresidues are {2,6,7,8,10}.

  15. Legendre Symbol • Used to determine if a number is a quadratic residue. • Defined as:

  16. Legendre(a,p) if a ≡ 0 (mod p) then return 0 EndIf x := a, y := p, L := 1 while true x := (x mod y) if x > y/2 then x := y-x if y ≡ 3 (mod 4) then L := L · -1 EndIf EndIf if x = 0 then return –1 EndIf while x ≡ 0 (mod 4) x := x/4 EndWhile if x ≡ 0 (mod 2) then x := x/2 t := (y mod 8) if t = 5 or t = 3 then L := L · -1 EndIf EndIf if x = 1 then return L EndIf if x,y ≡ 3 (mod 4) then L := L · -1 EndIf t := x, x := y, y := t EndWhile

  17. Square Root Modulo p • Sometimes it is useful to find an x such that x2≡ n (mod p), there are two methods for finding such an x: • Iterate over the subset 0 < x < (p - 1)/2 • Use the Shanks-Tonelli algorithm: Shanks-Tonelli(a,p) Choose random n untillegendre(n.p) = -1 Find e,q such that p – 1 = 2e · q and q is odd y := (nqmod p), r := e, x := a(q – 1)/2(mod p), b := a · x2 (mod p), x := a · x While b ≠ 1 (mod p) Find smallest m such that b2m ≡ 1 (mod p) t := y2(r-m-1) (mod p), y := t2 (mod p), r := m, x := x · t (mod p), b := b · y (mod p) EndWhile Return x

  18. RHO Derivation • Use proof any odd n є N+ > 2 can be represented by x2 - y2; therefore any composite n = x2 - y2 = p · q. Try to find x such that x2≡ y2 (mod n). This follows simply from definition of mod: n = x2 - y2 x2 = n + y2 and as mod returns r such that r = y2 – a · n (here a = 1)  x2≡ y2 (mod n).

  19. RHO Algorithm • Basis for most algorithms (including both QFS and NFS). • Set ƒi+1(x) = a · x2 + b · x + c with a,b,c є N+ • Set ƒ0(x) = 1,2 or some small integer • Compute ƒi(x) until gcd(ƒi+1 – ƒI, n) ≠ 1 • This number will be a factor of n.

  20. RHO Running Time • With a high probability RHO will find a factor in bit operations • Much faster than trial division

  21. Factor Bases • A set of prime integers one of the elements can be –1 (B = {p1,p2,…,pk}). • An integer is smooth over B iff all of its factors exist in B • The least absolute residue is (x2 mod n) in the interval (–n/2,n/2) • An integer is a B-number iff the least absolute residue (LAR) is smooth over B For Example: B = {-1,2,3,5}, n = 336, a = 8, b = 5, c = 9 LAR(a) = 64 = 26, LAR(b) = 25 = 52, LAR(C) = 81 = 34, these are B-numbers a = {0,6,0,0}, b = {0,0,0,2}, c = {0,0,4,0} therefore {b,c} is Linearly dependant over B mod 2 and gcd(b + c, n) = 14 a factor of n.

  22. QFS • Quadratic Field Sieve • A fast method for factoring large numbers less than 110-digits long. • Relies on algebraic number theory • Discovered by Pomerance in the early 1980’s. • Uses the ideas of RHO and Factor Bases • Uses a sieve similar to the prime number sieve shown earlier. • We shall denote floor(x) as [x] in the following

  23. QFS • Set P := • Set A := P3 • Make a matrix with row 1 all primes less than P such that legendre(n,pi) = 1 (if not discard pi) • Make column 1 be all t in the range ([√n] + 1,[√n] + A) • Make column 2 be t2 – n for all t. • For all the odd p (2 gets handled specially) solve the equation t2≡ n (mod pΘ) for Θ = 1,2,… until there is no solution in the range of column 1. • Let t1,t2 be the last pair of integers that satisfied the equation.

  24. QFS • For each element of column 2 if t differs from t1 by a multiple of p place a 1 in the row,column, repeat for p2, p3,…,pΘ except change the 1 to a 2,3,…,Θ. • Each time a 1 is placed or changed replace the t2 - n by (t2 – n)/p. • For p = 2 if n ≡ 1 (mod 8) treat 2 as above, otherwise simply place a 1 next to all odd t and replace the t2 - n by (t2 – n)/2 • Remove all rows where the t2 – n has not become 1. • As with Factor Bases find a linearly dependant subset of the rows (mod 2) we shall denote this as {t1,t2,…,tk} and the corresponding prime factors for each ti as {p1B1,p2B2,…,phΘh} where Θi is the number in the row,column specified by ti, pj.

  25. QFS • For this subset check that Where BΨ is the sum of the Θi in the Base vectors divided by 2. • Once a set has been found verify that • If so then will be a non-trivial factor of n

  26. QFS Example (n = 2279)

  27. QFS Example (n = 2279)

  28. QFS Example (n = 2279)

  29. QFS Example (n = 2279)

  30. QFS Example (n = 2279)

  31. QFS Example (n = 2279) • On 2 so check 2279≡ 7 (mod 8) good it is easier

  32. QFS Example (n = 2279) • Looking at the table it is obvious that rows 48,50,52,54 are linearly dependant mod 2. • (48 · 50 · 52 · 54)2≡ (52 · 7 · 13 · 17)2 (mod 2279) • Therefore gcd((48 · 50 · 52 · 54) – (52 · 7 · 13 · 17), 2279) is a factor, namely 53 • gcd((48 · 50 · 52 · 54) + (52 · 7 · 13 · 17), 2279) is the other factor namely 43. • It never hurts to double check so 53 · 43 = 2279 YAY We Factored 2279!!

  33. QFS Running Time • Runs in time • Requires approximately an equivalent amount of space • Faster than RHO as the function is between polynomial in log(n) and polynomial in n.

  34. MPQFS • Multiple Polynomial QFS • Allows for parallel processing of the QFS simply • Same algorithm except uses multiple polynomials of the form: • Q(x) := a · x2 + b · x + c • Where a is the square of an integer, b is in the interval [0,a) such that b2≡ a (mod n), c := b2/(4 · a). • By doing so reduces size of Factor Base and sieving interval for each Q(x) and can be run simultaneously

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