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Jan Braun
vor 6 Jahren
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6 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,...

9 ( ). N > 1 N = p e 1 1 pe pe l l = l k=1 p e k k, p i P p i e i N 1 i l N = = = l = 3 p 1 = 2 e 1 = 2 p 2 = 3 e 2 = 1 p 3 = 5 e 3 = 1 p a p b p a b a > b p, a, b N p a k 1 N a = k 1 p p b k 2 N b = k 2 p a b = (k 1 p) (k 2 p) = (k 1 k 2 ) p p a b P P = {2, 3, 5,..., p} N = ( p) + 1 N P P N q N ( p) q N ( p) = 1 P

10 x 2 a ( m) x, a Z x a m N\{0} m ggt (a, m) = 1 a m ±x a m a m (±3) 2 2 ( 7), x Z (±x) 2 2 ( 5). m p a p a p ±x ( a p) (a/p) a Z p P\{2} p ( ) a 0, a p = 1, a p p 1, a p ( ) 2 = 1 7 ( ) 2 = 1 5 ( ) 9 = 0 3

11 ( ). p, q P\{2} p q p q ( )( ) { p q 1, ( ), = ( 1) (p 1)(q 1)/4 = q p 1,. (. ). p P\{2} ( ) { 2 1, ( ) ( ), = ( 1) (p2 1)/8 = p 1,. a, b Z ( ) a p = ( ) a p p ( ) ab p = ( )( ) a p b p ( ) 1 p = 1 ( ) ( )( ) 4 (2.6) 2 2 (2.4) = = ( 1) (52 1)/8 ( 1) (52 1)/8 = p q a b ggt (a, b) = 1

12 p a p p p P a Z p a a p 1 1 ( p) p a p a b p, a, b N b > 0 p a k N a = k p b a b = (k p) b = k b p b = (k b p b 1 ) p p a b a 0 ( p) a b 0 ( p) p, a, b N b > 0 (). a Z p P\{2} ( ) a a (p 1)/2 ( p). p p a [ g ] = {b b g ( p)} p p P\{2} b Z g Z [ g ] Z p = {[ g ] 1, [ g ] 2,..., [ g ] p 1 } [ g ] p a Z [ a ] Z p α {1, 2,..., p 1} g α a ( p) a p α α = 2β (g β ) 2 a ( p) a (p 1)/2 (g 2β ) (p 1)/2 (g β ) (p 1) (g (p 1) ) β (1) β 1 ( p). a p α α = 2β + 1 a (p 1)/2 (g (2β+1) ) (p 1)/2 (g 2β g) (p 1)/2 (g 2β ) (p 1)/2 (g) (p 1)/2 ( p). (g 2β ) (p 1)/2 1 ( p) (g 2β ) (p 1)/2 (g) (p 1)/2 (g) (p 1)/2 (g (p 1) ) 1/2 (1) 1/2 (1) 1 ( p).

13 (1) ±1 ( p) 1 (g) (p 1)/2 1 ( p) g (p 1)/2 p 1 p a 2 p U (p 1) (p 1) N\U (p 1)/2 N 0 3 (3 1)/2 = 1 p a a 0 ( p) a b 0 ( p) b N\{0} b = (p 1)/2 a a b a (p 1)/2 0 ( p). a Z, p P\{2} a a p t 1 a 0 2 a a a/2 p 8 {3, 5} t t a, p p, a a p 3 ( 4) t t a a p p = 1

14 n O O(f(n)) = {g(n) c N\{0} n 0 N n n 0 : g(n) c f(n)} g(n) O(f(n)) n 0 c f(n) f(n) O(n k ) k N k O(2 p(n) ) p(n) L α R 0 α 1 N N c R L N [α, c] = e (c+o(1))( N)α ( N) 1 α, N α = 0 L N [0, c] = e (c+o(1)) N = ( N) c+o(1) N α = 1 L N [1, c] = e (c+o(1)) N N α L N [0, c] L N [1/2, c] L N [1, c]

16 N N N N N N N N N d d 2 > N d > N N N N = p q p q N p q > N N = p q p > N q < N N P N/2 N N\{0, 1} p N p N p P N d d N N d

17 N = = = = = 49 > 41 N = 492 = N N\{0, 1} F [ ] N N 2 N N N /2 F F 2 d 3 d 2 N d N N N /d F F d d d + 2 N 1 F F N

18 N N = x 2 y 2 = (x + y) (x y) U N = a b N, a, b U x, y Z x y N N = x 2 y 2 = (x + y) (x y) N, a, b U N = a b a = x + y b = x y a + b = (x + y) + (x y) = 2x a b = (x + y) (x y) = 2y x = (a + b)/2 Z y = (a b)/2 Z ( ) a + b 2 ( ) a b 2 x 2 y 2 = = 2 2 ( a 2 + 2ab + b 2) ( a 2 2ab + b 2) 2 2 = 4ab = a b = N, 4 N N = a b N, a, b U x = N y 2 = x 2 N y 2 = y y Z x 1 y 2 = (x + 1) 2 N x y y Z a b N a = x + y b = x y N N N p N = 3p N x = (3 + p)/2 y = (3 p)/2 ( ) 3 + p 2 ( ) 3 p 2 N =. 2 2

19 N = n = 562 k x k yk 2 x2 k N y k = 75 2 N = = ( ) (566 75) = N N = a b N, a, b U x N y 2 x x N y 2 Z y 2 y x + 1 x x + 1 [ x + y 2, x ] y 2 (x + 1) 2 N = (x 2 + 2x + 1) N = (x 2 N) + 2x + 1 yk 2 2x + 1

20 N x, y Z x y x 2 y 2 ( N) x 2 y 2 0 ( N) x ± y ( N) x ± y 0 ( N) N (x 2 y 2 ) N (x + y) (x y) N (x + y) N (x y) ggt (x + y, N) ggt (x y, N) N x q(x) = x 2 N q(x i ) 0 i k y 2 = q(x 0 ) q(x 1 )... q(x k ) = (x 2 0 N) (x 2 1 N)... (x 2 k N) y 2 x 2 0 x x 2 k = (x 0 x 1... x k ) 2 = x 2 ( N), x 2 i N x2 i ( N) x 2 y 2 N N = N 1 N = 1 N

21 F B B N 1 F B 1 q(x i ) x min x i x max 1 F B = {p P p B} { 1}. q(x i ) B q(x i ) B B B = e N N B x min x max q(x i ) B M I = N N M = B 2 = e ] [x min, x max N = [ ] N M, N + M N 2 2 B M F B p F B N p q(x i ) = x 2 i N x i I p q(x i ) x 2 i N 0 p x2 i N p N p (N/p) = 1 F B = F B \{p F B p > 2 (N/p) 1}.

22 S = {q(x i ) x i I} B q(x i ) B p q(x i ) = x 2 i N x min x i < x min+p p p p k N q(x i + kp) = (x i + kp) 2 N = x 2 i + 2x i kp + (kp) 2 N = (x 2 i N) + 2x i kp + (kp) 2 = q(x i ) + 2x i kp + (kp) 2 q(x i ) ( p). p p 1 F B S 1 p F B B S B F B 1 B S B q(x i ) v(q(x i )) 1 q(x i ) q(x i ) = x 2 i N = p e 1 1 pe pe l l, v(q(x i )) = v(x 2 i N) = (e 1, e 2,..., e l ). p p x 2 i N Z p p q(x i ) p p p 0

23 q(x i ) 0 i k q(x i ) q(x 0 ) q(x 1 )... q(x k ) = p 2m e 1 1 p 2n e p 2o e l l = (p m e 1 1 p n e p o e l l ) 2 = y 2, m, n, o N 2m, 2n, 2o N\U v(q(x i )) B B B q(x i ) y 2 x 2 N x 2 y 2 ( N) N N = ggt (x y, N) N ggt (x y, N), N = N 1 N = 1 N N B q(x i )

24 N = F B B N 1 N B B = 19 F B = { 1, 2, 3, 5, 7, 11, 13, 17, 19}. M = B 2 = 19 2 = 361 M = 40 [ I = N M 2, N + M 2 = [296 20, ] N ] N = {276, 277, 278,..., 296,..., 314, 315, 316}. p F B N p (N/p) 1 F B = F B \{5, 7, 11} = { 1, 2, 3, 13, 17, 19} S = {q(x i ) = x 2 i N x i I} B S = {q(276), q(277), q(278),..., q(296),..., q(314), q(315), q(316)} = { 11287, 10734, 10179,..., 153,..., 11133, 11762, 12393} 1 S = {11287, 10734, 10179,..., 153,..., 11133, 11762, 12393}. 2 2 q(277) = 10734

25 2 q(277 + k 2) k N\{0} 2 F B 3 3 q(277) = q(278) = q(277 + k 3) q(278 + k 3) 3 p q(x i ) S B S = {11287, 1789, 29,..., 1,..., 1237, 5881, 1} q(296) q(316) F B q(296) = 153 = = q(316) = = = q(299) = 1938 = B B N B q(x i ) v(q(x i )) v(q(296)) = (0, 0, 2, 0, 1, 0) v(q(299)) = (0, 1, 1, 0, 1, 1) v(q(316)) = (0, 0, 6, 0, 1, 0) M = w wm = 0 M w = (1, 0, 1) q(296) q(316) p F B

26 q(296) q(316) = (153) (12393) = (3 2 17) (3 6 17) = ( ) = (3 4 17) 2 = y 2 x 2 = ( ) 2 N y 2 = q(296) q(316) = (3 4 17) 2 ( ) 2 = x 2 ( 87463) x = x 2 = y = y 2 = ggt (x y, N) = 587 N N N = = ggt (x y, N) ggt (x y, N) = = N N B e N N F B {p P p B} { 1} M [ B 2 I ] N M, N + M N 2 2 F B F B\{p F B p > 2 (N/p) 1} S {q(x i ) = x 2 i N x i I} S 1 S B B q(x i ) = p e 1 1 pe pe l l v(q(x i )) = (e 1, e 2,..., e l ) v(q(x 0 )) + v(q(x 1 )) v(q(x k )) = 0 x x 0 x 1... x k y q(x 0 ) q(x 1 )... q(x k ) [ ggt (x y, N), N ggt (x y,n) ]

28 sieveinterval(...) ±x x 2 N ( p) N number p prime (N/p) = 1 ±x squareroots[0] squareroots[1] p q(x i ) = x 2 i N x min x i < x min+p p x i ±x ( p) p p generatematrix(...) M exponentvectors v(q(x i )) M w wm = 0

29 sieveinterval(...) logarithms

30 range(start, stop, step) xrange(start, stop, step) islice(...) count(...) S = {q(x i ) = x 2 i N x i I} x i I q(x i ) x 2 i S 2 (x 2 i N) = 2 (p e 1 1 pe pe l l ) = 2 (pe 1 1 ) + 2 (pe 2 2 ) (pe l l ), p e 1 1 pe pe l l x 2 i N S logarithms sievingarraylength 0

31 p q(x i ) = x 2 i N x min x i < x min+p p index logarithms 2 (p) index + k p k N\{0} logarithms 2 (p) 0 logarithms x i I q(x i ) S logarithms q(x i ) = x 2 i N B B B 1 B 1 startpoint x min endpoint x max 5 3 startpoint x i x i q(x i ) 30 ) threshold = 2 ( ( x min (x max x min )) 2 N 30, N number

32 F 7 = = = , F 7 quadraticsieve.pyx setup.py main.py legendresymbol.py F 7 F 7 =

33 F B + 10 F B F B F B F B +10 F B N N = N 1 N = 1 N I I 0, I 1,..., I 10 q(x) = x 2 N x N B B I F B + 10 B I 5, I 6, I 4, I 7, I 3, I 8, I 2, I 9, I 1, I 10 I 0

34 q(x) = x 2 N B x N B f(x) = ax 2 + 2bx + c a, b, c Z b 2 ac = N B Q(x) = a f(x) = a 2 x 2 + 2abx + ac = a 2 x 2 + 2abx + b 2 b 2 + ac = (ax + b) 2 (b 2 ac) = (ax + b) 2 N, Q(x i ) 0 i k y 2 = Q(x 0 ) Q(x 1 )... Q(x k ) = ((ax 0 + b) 2 N) ((ax 1 + b) 2 N)... ((ax k + b) 2 N) y 2 (ax 0 + b) 2 (ax 1 + b) 2... (ax k + b) 2 = ((ax 0 + b) (ax 1 + b)... (ax k + b)) 2 = x 2 ( N), (ax i + b) 2 N (ax i + b) 2 ( N) x 2 y 2 N f(x) = ax 2 +2bx+c a b a b a b

35 N N = {0, 1, 2, 3, 4,...} Z Z = {..., 2, 1, 0, 1, 2,...} Q Q = { v w v, w Z, w 0} I I = { 2, π, e,... } R R = Q I P P = {2, 3, 5, 7, 11, 13,...} U U = {2u + 1 u N} p N p N p N p P N N k N N = k p p N p P N N k N N k p ggt (a, m) a Z m Z ggt (a, m) = 1 a m x ( a p) min{y Z y x} x R x O(f(n)) O L N [α, c] L N\{0} {1, 2, 3, 4,...} N + Q

36 = = = = = = = = = = = = = =

37 L v R α 1/2 v L N [1/2, c] = v e (c+o(1))( N)1/2 ( N) 1/2

38 findfit(data, model) ( v e (c+o(1))( N)1/2 ( N) 1/2) = v + e (c+o(1))( N)1/2 ( N) 1/2 = v + (c + o(1))( N) 1/2 ( N) 1/2. o(1) N (v L N [1/2, c]) v + c ( N) 1/2 ( N) 1/2 = v + c N N. c v findfit(data, model) N N N n N N = e N = 2 N 2 e = 1 2 e 2 N = e 2 2 N = 2 2 N 2 n. c 1, 08 v 1, v L N [1/2, c] v e c ( N)1/2 ( N) 1/2 = v e c N N

39 Zeit in min , e 1,08 ln N ln ln N Faktorisierungen Bits B

40 N = p q p, q P\{2} p q a Z p P\{2} p U\P

41 N N\{0, 1} N = a b N, a, b U

45 N = a b N, a, b U N F B factorbase

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