Number-Theoretic Algorithms

Transcription

1 Number-Theoretic Algorithms Hengfeng Wei March 31 April 6, 2017 Hengfeng Wei Number-Theoretic Algorithms March 31 April 6, / 36

2 Number-Theoretic Algorithms 1 Modular Arithmetic 2 Euclid s Algorithm 3 Pairwise Relatively Prime 4 Chinese Remainder Theorem

3 Modular Arithmetic Cancellation in modular arithmetic (TC ) ad bd (mod n) = a b (mod n) ad bd (mod n), d n = a b (mod n) (mod 4) 3 5 (mod 4) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

4 Modular Arithmetic Changing the modulus (mod 4) 3 5 (mod 4) 3 5 (mod 2) ad bd (mod nd) a b (mod n) (d 0) (a mod n)d = ad mod nd (distributive law) ad bd (mod n) a b (mod n (d, n) ) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

5 Modular Arithmetic Changing the modulus n = n 1 n 2 n k a b (mod n) = a b (mod n i ) a b (mod 100) = a b (mod 20) = a b (mod 5) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

6 Modular Arithmetic Changing the modulus n = n 1 n 2 n k a b (mod n 1 ), a b (mod n 2 ) a b (mod lcm(n 1, n 2 )) a b (mod n 1 ), a b (mod n 2 ) a b (mod n 1 n 2 ), if n 1 n 2 1 i k, a b (mod n i ) a b (mod n), if n i n j Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

7 Number-Theoretic Algorithms 1 Modular Arithmetic 2 Euclid s Algorithm 3 Pairwise Relatively Prime 4 Chinese Remainder Theorem

8 Euclid s Algorithm Worst-case analysis of Euclid s algorithm (TC ) 1. If a > b 0, Euclid(a, b) makes 1 + log ϕ b recursive calls. Lamé s theorem: a > b 1, b < F k+1 = r < k. k = 2 + log ϕ b To prove b < F 3+logϕ b. F k = ϕk ˆϕ k 5 > ϕk 1 5 Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

9 Euclid s Algorithm Worst-case analysis of Euclid s algorithm (TC ) 2. Improve this bound to 1 + log ϕ ( b (a,b) ). (16, 12) a (a, b) = (a, b) ( (a, b), b (a, b) ) = (12, 4) = (4, 0) (4, 3) = (3, 1) = (1, 0) = 4 = 1 a Euclid(a, b) Euclid( (a, b), b (a, b) ) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

10 Euclid s Algorithm Worst-case analysis of Euclid s algorithm (TC ) 2. Improve this bound to 1 + log ϕ ( b (a,b) ). a Euclid(a, b) Euclid( (a, b), b (a, b) ) Euclid(b, a mod b) Euclid(? b (a, b), a (a, b) mod b Euclid(b, a mod b) Euclid( (a, b), a mod b (a, b) ) a (a, b) mod b (a, b) = a mod b (a, b) b (a, b) ) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

11 Euclid s Algorithm Worst-case analysis of Euclid s algorithm (TC ) 2. Improve this bound to 1 + log ϕ ( b (a,b) ). Lemma (Generalization of Lemma 31.10) If a > b 1, d = (a, b) and Euclid(a, b) performs k 1 recursive calls, then a df k+2 and b df k+1. Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

12 Euclid s Algorithm Average-case analysis of Euclid s algorithm T (m, 0) = 0; T (m, n) = 1 + T (n, m mod n) n 1 When m is chosen at random: T n = 1 T (k, n) n 0 k<n Assume that, for 0 k < n, (n mod k) is random : T n n (T 0 + T T n 1 ) = n = H n ln n + O(1) Reference The Art of Computer Programming, Vol 2: Seminumerical Algorithms (Section 4.5.3) by Donald E. Knuth, 3rd edition. Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

13 Number-Theoretic Algorithms 1 Modular Arithmetic 2 Euclid s Algorithm 3 Pairwise Relatively Prime 4 Chinese Remainder Theorem

14 Pairwise Relatively Prime Pairwise relatively prime (TC ) n 1, n 2, n 3, n 4 are pairwise relatively prime gcd(n 1 n 2, n 3 n 4 ) = gcd(n 1 n 3, n 2 n 4 ) = 1 Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

15 Pairwise Relatively Prime Pairwise relatively prime (TC ) n 1, n 2,..., n k are pairwise relatively prime a set of lg k pairs of numbers derived from the n i are relatively prime. ( ) k = Θ(k 2 ) (complete graph) 2 gcd( 1 L, 1 R ) = gcd( 2 L, 2 R ) = = gcd( lg k L, lg k R ) = 1 k = 2 : gcd(n 1, n 2 ) = 1 k = 3 : gcd(n 1, n 2 n 3 ) = gcd(n 2, n 3 ) = 1 Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

16 Pairwise Relatively Prime Pairwise relatively prime: divide-and-conquer n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 1 n 2 n 3 n 4 n 5 n 6 { T (1) = 0 T (k) = T ( k 2 ) + T ( k 2 ) + 1 = T (k) = k 1 = Θ(k) T k = k 1 : (n i, n i+1 n i+2 n k ) 1 i < k Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

17 Pairwise Relatively Prime Pairwise relatively prime: smarter combination n 1 n 2 n 3 n 4 n 1 n 2 n 3 n 4 n 1 n 2 n 3 n 4 (n 1 n 2, n 3 n 4 ) = 1 (n 1, n 2 ) = 1, (n 3, n 4 ) = 1 { T (1) = 0 T (k) = T ( k 2 ) + 1 (n 1 n 2, n 3 n 4 ) = 1 (n 1 n 3, n 2 n 4 ) = 1 = T (k) = lg k Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

18 Pairwise Relatively Prime Pairwise relatively prime: the dividing pattern k = 7 : n 0, n 1, n 2,..., n 6 n 0 n 1 n 2 n 3 n 4 n 5 n 6 n 0 n 1 n 2 n 3 n 4 n 5 n 6 n 0 n 1 n 2 n 3 n 4 n 5 n 0 n 1 n 2 n 3 n 4 n 5 n 6 0 : : : : : : : 110 T (k) = lg k Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

19 Pairwise Relatively Prime Can we do even better? T (k) lg k Prove by (strong) mathematical induction. T (k) 1 + T ( k 2 ) 1 + lg k 2 = lg k Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

20 Pairwise Relatively Prime Biclique covering Covering a complete graph with few complete bipartite subgraphs. Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

21 Pairwise Relatively Prime Biclique covering: rethinking the first divide-and-conquer Reference for T (k) k 1 T (k) = k 1 edge-disjoint biclique partition On the Addressing Problem for Loop Switching by Graham and Pollak, Reference for weighted biclique partition Covering a Graph by Complete Bipartite Graphs by P. Erdős and L. Pyber, Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

22 Number-Theoretic Algorithms 1 Modular Arithmetic 2 Euclid s Algorithm 3 Pairwise Relatively Prime 4 Chinese Remainder Theorem

23 Chinese Remainder Theorem Chinese Remainder Theorem (CRT) Theorem (CRT) n 1,..., n k ; a 1,..., a k n i n j i j, n = n 1 n 2 n k!a (0 a < n) : a a i (mod n i ). Proof for uniqueness. a (a 1, a 2,..., a k ) a a (mod n i ) = n a a. Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

24 Chinese Remainder Theorem History of CRT Hengfeng Wei Number-Theoretic Algorithms March 31 April 6, / 36

25 Chinese Remainder Theorem History of CRT Hengfeng Wei Number-Theoretic Algorithms March 31 April 6, / 36

26 Chinese Remainder Theorem Proof of CRT (1) Nonconstructive proof. f : [0, n) [0, a i ) 1 i k f : a ( a mod n 1,..., a mod n k ) f is one-to-one. f is onto. a : f(a) = (a 1,..., a k ). Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

27 Chinese Remainder Theorem Proof of CRT (2) Constructive proof by induction. a a 1 (mod n 1 ) (1) a a 2 (mod n 2 ) (2) (1) = a = a 1 + n 1 y x = a 1 + n 1 n 1 1 (a 2 a 1 ) (mod n 1 n 2 ) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

28 Chinese Remainder Theorem Proof of CRT (3) Constructive proof by induction. a a 1 (mod n 1 ) (3) a a 2 (mod n 2 ) (4) n 1 n 2 = n 1 n 1 + n 2 n 2 = 1 x = a 1 n 1 n 1 + a 2 n 2 n 2 (mod n 1 n 2 ) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

29 Chinese Remainder Theorem Proof of CRT (4) Constructive proof. 1. x 1 (mod n i ), x 0 (mod n j ) (i j) x = M i (M 1 i mod n i ) = x = M i M 1 i (mod n) 2. x a i (mod n i ), x 0 (mod n j ) (i j) 3. a a i (mod n i ), 1 i k a = x = a i M i M 1 i (mod n) 1 i k a i M i M 1 i (mod n) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

30 Chinese Remainder Theorem Proof of CRT (5) More efficient constructive proof. Reference The Residue Number System by Garner, Reference The Art of Computer Programming, Vol 2: Seminumerical Algorithms (Section 4.3.2) by Donald E. Knuth, 3rd edition. Hengfeng Wei Number-Theoretic Algorithms March 31 April 6, / 36

31 Chinese Remainder Theorem Operations over CRT a (a 1, a 2,..., a n ) a ± b (a 1 ± b 1, a 2 ± b 2,..., a n ± b n ) a b (a 1 b 1, a 2 b 2,..., a n b n ) TC Proof. a (a 1, a 2,..., a n ), (a, n) = 1 = a 1 (a 1 1, a 1 2,..., a 1 n ) a 1 a 1 i (mod n i ) = { a ai (mod n i ) (a, n) = 1 Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

32 Chinese Remainder Theorem The ϕ function Theorem (The ϕ function) ϕ(p) = p 1 ϕ(p k ) = p k p k 1 r ϕ(n) = n (1 1 r k ) (n = p i i ) p i=1 i i=1 We shall not prove this formula here. CLRS (Section 31.3) Let us prove this formula now. m n = ϕ(mn) = ϕ(m)ϕ(n) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

33 Chinese Remainder Theorem The ϕ function Theorem (The ϕ function) m n = ϕ(mn) = ϕ(m)ϕ(n) Proof. U mn = {a mod mn, (a, mn) = 1} U m = {b mod m, (b, m) = 1} U n = {c mod n, (c, n) = 1} f : U mn U m U n f(a mod mn) = (a mod m, a mod n). Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

34 Chinese Remainder Theorem Secret sharing using the CRT Definition ((k, n)-threshold secret sharing scheme) (3, 3)-secret sharing: Reference How to Share a Secret by Maurice Mignotte, Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

35 Chinese Remainder Theorem Secret sharing using the CRT 1. Choose m i : m 1 < m 2 < < m n, m i m j, n k m i < m i i=n k+2 i=1 2. Choose the secret S: n k m i < S < m i i=n k+2 i=1 3. Compute the shares: s i = S mod m i Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

36 Chinese Remainder Theorem Solving simultaneous congruences (TC ) x 1 (mod 9) x 2 (mod 8) x 3 (mod 7) x 10 (mod 504) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

37 Chinese Remainder Theorem Solving simultannous congruences CRT with large modulus 19x 556 (mod 1155) x 1 (mod 3) 19x 556 (mod 3) x 4 (mod 5) 19x 556 (mod 5) x 2 (mod 7) 19x 556 (mod 7) x 9 (mod 11) 19x 556 (mod 11) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

38 Chinese Remainder Theorem Solving simultaneous congruences CRT with non-pairwisely co-prime moduli x 3 (mod 8) x 11 (mod 20) x 1 (mod 15) { x 3 (mod 2 3 ) { x 3 (mod 2 3 ) x 3 (mod 2 2 ) { x 3 (mod 2 2 ) x 1 (mod 5) { x 1 (mod 3) { x 1 (mod 3) x 1 (mod 5) { x 1 (mod 5) x 1 (mod 5) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

39 Chinese Remainder Theorem Solving simultaneous congruences Theorem (CRT with non-pairwisely coprime moduli) a i a j (mod (n i, n j )) 0 a < lcm(n 1, n 2,..., n k ) Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

40 Chinese Remainder Theorem Simultaneous incongruences?a, 1 i k : a a i (mod n i ) NP-complete Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, / 36

41 Chinese Remainder Theorem Hengfeng Wei Number-Theoretic Algorithms March 31 April 6, / 36