Factoring and Testing Primes in Small Space Viliam Geffert P.J.Šafárik University, Košice, Slovakia Dana Pardubská Comenius University, Bratislava, Slovakia.

Παρουσίαση με θέμα: "Factoring and Testing Primes in Small Space Viliam Geffert P.J.Šafárik University, Košice, Slovakia Dana Pardubská Comenius University, Bratislava, Slovakia."— Μεταγράφημα παρουσίασης:

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2 Factoring and Testing Primes in Small Space Viliam Geffert P.J.Šafárik University, Košice, Slovakia Dana Pardubská Comenius University, Bratislava, Slovakia

3 Factoring and Testing Primes in Small Space Viliam Geffert P.J.Šafárik University, Košice, Slovakia Dana Pardubská Comenius University, Bratislava, Slovakia

4 n is a prime

5 n is a prime if it cannot evenly be divided by any number other than 1 and itself

6 Euclid [280 B.C. ]: infinite number of primes

7 n is a prime if it cannot evenly be divided by any number other than 1 and itself Euclid [280 B.C. ]: infinite number of primes J.Hadamard, C.J.de la Valée Poussin [1896]:

8 n is a prime if it cannot evenly be divided by any number other than 1 and itself Euclid [280 B.C. ]: infinite number of primes J.Hadamard, C.J.de la Valée Poussin [1896]: Prime Number Theorem

9 n is a prime if it cannot evenly be divided by any number other than 1 and itself Euclid [280 B.C. ]: infinite number of primes J.Hadamard, C.J.de la Valée Poussin [1896]: Prime Number Theorem p m = (1 + o(1) ). m. ln m - p m denotes the m-th prime

10 Fundamental Theorem of Arithmetic - Factoring n = p i 1 k i 1. p i 2 k i 2. …. p i ℓ k i ℓ k i j > 0, integer

11 Fundamental Theorem of Arithmetic - Factoring n = p i 1 k i 1. p i 2 k i 2. …. p i ℓ k i ℓ k i j > 0, integer Factoring is computationally very hard

12 Fundamental Theorem of Arithmetic - Factoring n = p i 1 k i 1. p i 2 k i 2. …. p i ℓ k i ℓ k i j > 0, integer Factoring is computationally very hard - utilized to design secure cryptographic systems (data transmission over internet)

13 Primes= { n | n is a prime } - n binary coded

14 Primes= { n | n is a prime } - n binary coded un-Primes= {1 n | n is a prime }

15 Primes= { n | n is a prime } - n binary coded un-Primes= {1 n | n is a prime } Agrawal, Kayal, Saxena [2004]: Primes ϵ P

16 Primes= { n | n is a prime } - n binary coded un-Primes= {1 n | n is a prime } Agrawal, Kayal, Saxena [2004]: Primes ϵ P - no factorization, if n is composite

17 Primes= { n | n is a prime } - n binary coded un-Primes= {1 n | n is a prime } Agrawal, Kayal, Saxena [2004]: Primes ϵ P - no factorization, if n is composite P.Shor [1994]: polynomial time quantum algorithm for factoring

18 Primes= { n | n is a prime } - n binary coded un-Primes= {1 n | n is a prime } Agrawal, Kayal, Saxena [2004]: Primes ϵ P - no factorization, if n is composite P.Shor [1994]: polynomial time quantum algorithm for factoring How much space is sufficient?

19 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n)

20 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) n

21 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) n

22 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) n

23 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) n O(loglog n) bits

24 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) n O(loglog n) bits

25 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n)

26 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ? Is n a prime ? … 1 1 1 … YES: P PP

27 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ? Is n a prime ? … 1 1 1 … YES: … x1x1 1 … NO: x2x2 CC C P PP

28 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ? Is n a prime ? … 1 1 1 … YES: … x1x1 1 … NO: x2x2 - space below loglog n CC C P PP

29 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ? Is n a prime ? … 1 1 1 … YES: … x1x1 1 … NO: x2x2 … WRONG GUESS: - space above loglog n W CC C P PP

30 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n)

31 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) (2) un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n)

32 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) (2) un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) - (2) is not a trivial consequence of (1), since it is not known whether ASPACE(loglog n) is closed under complement

33 (1) un-Primes ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) (2) un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) - (2) is not a trivial consequence of (1), since it is not known whether ASPACE(loglog n) is closed under complement - additional bonus - factoring

34 Algorithm based on “elementary school” primality testing: for X = 2, 3,..., n-1 do if X divides n then “n is composite” end “n is prime”

35 O(loglog n) space due to

36 - Modular representation (based on Chinese Remainder Theorem)

37 O(loglog n) space due to - Modular representation (based on Chinese Remainder Theorem) M = p 1. p 2. …. p m n = (z 1, …, z m ), z i = n mod p i

38 O(loglog n) space due to - Modular representation (based on Chinese Remainder Theorem) M = p 1. p 2. …. p m n = (z 1, …, z m ), z i = n mod p i memory space: log p m  O(loglog n)

39 O(loglog n) space due to - Modular representation (based on Chinese Remainder Theorem) M = p 1. p 2. …. p m n = (z 1, …, z m ), z i = n mod p i memory space: log p m  O(loglog n) - Scalar representation

40 O(loglog n) space due to - Modular representation (based on Chinese Remainder Theorem) M = p 1. p 2. …. p m n = (z 1, …, z m ), z i = n mod p i memory space: log p m  O(loglog n) - Scalar representation n = α. M, α ϵ  0,1)

41 O(loglog n) space due to - Modular representation (based on Chinese Remainder Theorem) M = p 1. p 2. …. p m n = (z 1, …, z m ), z i = n mod p i memory space: log p m  O(loglog n) - Scalar representation n = α. M, α ϵ  0,1) -- truncated to 3. loglog n bits

42 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m M 0 Z

43 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m M 0 Z In reality

44 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m M 0 Z Z In reality

45 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m M 0 Z

46 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m m  O(log Z) M 0 Z

47 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m m  O(log Z) p m  O(m. log m)  O(log Z. loglog Z) M 0 Z

48 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m m  O(log Z) p m  O(m. log m)  O(log Z. loglog Z) log p m  O(loglog Z) M 0 Z

49 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m ? Is Z a prime ? M 0 Z

50 __ Z = (z 1, …, z m ), Z < √MM = p 1. p 2. …. p m ? Is Z a prime ? if  z i = 0 then “ Z is a composite ” M 0 Z

51 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 ? Is Z a prime ? M 0 Z

52 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 Z is a prime iff  X = (x 1, …, x m ), X ϵ {2, …, Z-1} X does not divide Z M 0 Z X

53 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 Z is a prime iff  X = (x 1, …, x m ), X ϵ {2, …, Z-1} X does not divide Z Z

54 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 Z is a prime iff  X = (x 1, …, x m ), X ϵ {2, …, Z-1} X does not divide Z Z branching universally at each X ϵ {2, …, Z-1}

55 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 Z is a prime iff  X = (x 1, …, x m ), X ϵ {2, …, Z-1} X does not divide Z  Z branching universally at each X ϵ {2, …, Z-1}      

56 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 Z is a prime iff  X = (x 1, …, x m ), X ϵ {2, …, Z-1} X does not divide Z M 0 Z X Z  X

57 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 Z is a prime iff  X = (x 1, …, x m ), X ϵ {2, …, Z-1} X does not divide Z M 0 Z X

58 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? M 0 Z X

59 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? if  x i = 0 then “ X does not divide Z ” M 0 Z X

60 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z X

61 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z x i required, for some i X

62 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z  log p m space log p m  O(loglog Z) x i required, for some i compute p i X

63 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z  log p m space log p m  O(loglog Z) x i required, for some i compute p i existentially guess x i ϵ {0, …, p i -1} X xixi

64 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z split universally compute p i existentially guess x i ϵ {0, …, p i -1} X xixi xixi

65 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z split universally - one parallel process verifies the guessed value x i X xixi xixi pipi pipi pipi xixi

66 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z X xixi - one parallel process executes the main program, assuming the guessed value x i is correct

67 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z z i required, for some i X

68 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z  log p m space log p m  O(loglog Z) z i required, for some i compute p i X

69 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z  log p m space log p m  O(loglog Z) z i required, for some i compute p i existentially guess z i ϵ {0, …, p i -1} X zizi

70 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z split universally compute p i existentially guess z i ϵ {0, …, p i -1} X zizi zizi

71 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z split universally - one parallel process verifies the guessed value z i X zizi zizi

72 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z split universally - one parallel process verifies the guessed value z i X zizi pipi pipi zizi zizi

73 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z X zizi - one parallel process executes the main program, assuming the guessed value z i is correct

74 M 0 Z X __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? Z X log p m  O(loglog Z) z 1, z 2, …, z m x 1, x 2, …, x m

75 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? if  x i = 0 then “ X does not divide Z ” M 0 Z X

76 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} ? X divides Z ? M 0 Z X

77 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z M 0 Z X

78 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z M= p 1. p 2. …. p m M 0 Z X

79 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z M= p 1. p 2. …. p m X [M] * Y= (X * Y) mod M = (x 1 [p 1 ] * y 1, …, x m [p m ] * y m ) M 0 Z X

80 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z M= p 1. p 2. …. p m X [M] * Y= (X * Y) mod M = (x 1 [p 1 ] * y 1, …, x m [p m ] * y m ) X [M] -1 : X [M] * X [M] -1 = 1 X [M] -1 = (x 1 [ p 1] -1, …, x m [ p m] -1 ),  i: x i ≠ 0 M 0 Z X

81 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z M 0 Z X

82 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z Y = (y 1, …, y m ) M 0 Z X Y

83 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y

84 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y Z X

85 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y Z X z 1, z 2, …, z m x 1, x 2, …, x m

86 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y Z X y i required, for some i z 1, z 2, …, z m x 1, x 2, …, x m

87 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y Z X y i required, for some i compute x i z 1, z 2, …, z m x 1, x 2, …, x m

88 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y Z X y i required, for some i compute x i compute z i z 1, z 2, …, z m x 1, x 2, …, x m

89 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y Z X y i required, for some i compute x i compute z i y i := x i [ p i] -1 [p i ] * z i z 1, z 2, …, z m x 1, x 2, …, x m

90 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z if Y  Z then “ X divides Z ” else “ X does not divide Z ” Y = (y 1, …, y m ) M 0 Z X Y

91 __ Z = (z 1, …, z m ), Z < √M,  i : z i ≠ 0, x i ≠ 0 X = (x 1, …, x m ), X ϵ {2, …, Z-1} X divides Z iff X [M] -1 [M] * Z  Z - to solve the problem we only need an algorithm deciding whether Y  Z - for any given Y Y = (y 1, …, y m ) M 0 Z X Y

92 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? M 0 Z Y

93 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Thm: Let Z = (z 1, …, z m ) and Y = (y 1, …, y m ) be two numbers in the residual representation, Z  M/2. M 0 Z Y

94 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Thm: Let Z = (z 1, …, z m ) and Y = (y 1, …, y m ) be two numbers in the residual representation, Z  M/2. If the values z i and y i can effectively be computed in O(log p m ) space, for each given i  {1, …, m}, M 0 Z Y

95 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Thm: Let Z = (z 1, …, z m ) and Y = (y 1, …, y m ) be two numbers in the residual representation, Z  M/2. If the values z i and y i can effectively be computed in O(log p m ) space, for each given i  {1, …, m}, then the question of whether Y  Z can also be decided in O(log p m ) space. M 0 Z Y

96 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Idea: convert Y and Z into scalar representation: M 0 Z Y

97 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Idea: convert Y and Z into scalar representation: Y = α y. M α y, α z ϵ  0,1) Z = α z. M M 0 Z Y

98 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Idea: convert Y and Z into scalar representation: Y = α y. M α y, α z ϵ  0,1) Z = α z. M M 0 Z Y

99 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Idea: convert Y and Z into scalar representation: Y = α y. M α y, α z ϵ  0,1) Z = α z. M Y  Z iff α y  α z M 0 Z Y

100 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? Idea: convert Y and Z into scalar representation: Y = α y. M α y, α z ϵ  0,1) Z = α z. M Y  Z iff α y  α z -- trunctated to 3. loglog n bits M 0 Z Y

101 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? M 0 Z Y

102 __ Z = (z 1, …, z m ), 0  Z < √M Y = (y 1, …, y m ), 0  Y < M ? Y  Z ? if Y  M / 2 then “ Y > Z ” M 0 Z Y

103 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 ? Y  Z ? M 0 Z Y

104 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 M 0 Z Y

105 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 W = (w 1, …, w m ) M 0 Z Y W

106 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W

107 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W X Z

108 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W Z X z 1, z 2, …, z m x 1, x 2, …, x m y 1, y 2, …, y m

109 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W Z X w i required, for some i z 1, z 2, …, z m x 1, x 2, …, x m y 1, y 2, …, y m

110 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W Z X w i required, for some i compute z i z 1, z 2, …, z m x 1, x 2, …, x m y 1, y 2, …, y m

111 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W Z X z 1, z 2, …, z m x 1, x 2, …, x m y 1, y 2, …, y m w i required, for some i compute z i compute y i

112 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W Z X w i required, for some i compute z i compute y i w i := z i [p i ] - y i z 1, z 2, …, z m x 1, x 2, …, x m y 1, y 2, …, y m

113 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 if W  M / 2 then “ Y  Z ” else “ Y > Z ” W = (w 1, …, w m ) M 0 Z Y W

114 Z = (z 1, …, z m ), 0  Z < M / 2 Y = (y 1, …, y m ), 0  Y < M / 2 Y  Z iff Z [M] - Y  M / 2 - to decide whether Y  Z, we only need an algorithm deciding whether W  M / 2 - for any given W W = (w 1, …, w m ) M 0 Z Y W

115 W = (w 1, …, w m ), 0  W < M = p 1. …. p m ? W  M / 2 ? M 0 W

116 W = (w 1, …, w m ), 0  W < M = p 1. …. p m ? W  M / 2 ? - convert W to scalar representation: M 0 W

117 W = (w 1, …, w m ), 0  W < M = p 1. …. p m ? W  M / 2 ? - convert W to scalar representation: W = α. M α ϵ  0,1) M 0 W

118 W = (w 1, …, w m ), 0  W < M = p 1. …. p m ? W  M / 2 ? - convert W to scalar representation: W = α. M α ϵ  0,1) M 0 W

119 W = (w 1, …, w m ), 0  W < M = p 1. …. p m ? W  M / 2 ? - convert W to scalar representation: W = α. M α ϵ  0,1) W  M / 2 iff α  1 / 2 M 0 W

120 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i

121 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end

122 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end integer arithmetic modulo p i

123 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end integer arithmetic modulo p i O(log p m ) space

124 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end integer arithmetic modulo p i O(log p m ) space log p m  O(loglog Z)

125 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end real arithmetic

126 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end real arithmetic O(loglog Z) space

127 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end real arithmetic O(loglog Z) space truncated to ℓ  3. loglog(Z) bits

128 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end real arithmetic O(loglog Z) space truncated to ℓ  3. loglog(Z) bits numeric error 1 ε   2 ℓ 2p m

129 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i numeric error 1 ε   2 ℓ 2p m M 0 ½.M

130 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i numeric error 1 ε   2 ℓ 2p m M 0 ½.M α.M ε

131 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i numeric error 1 ε   2 ℓ 2p m M 0 ½.M α.M ε

132 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i numeric error 1 ε   2 ℓ 2p m M 0 ½.M α.M ε

133 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i numeric error 1 ε   2 ℓ 2p m M 0 ½.M α.M ε

134 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i M’= M / p m W’= (w 1, …, w m-1 ) W  ½. M iff W’  ½. M’ numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ M p m M p m M p m M p m α ’.M ’ ε

135 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i M’= M / p m W’= (w 1, …, w m-1 ) W  ½. M iff W’  ½. M’ re-run for m := m-1 numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ M p m M p m M p m M p m α ’.M ’ ε

136 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i m := m-1 numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ M p m M p m M p m M p m α ’.M ’ ε

137 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i m := m-1 numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ M p m M p m M p m M p m α ’.M ’ ε

138 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end m := m-1 numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ M p m M p m M p m M p m α ’.M ’ ε

139 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end m := m-1 numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ M p m M p m M p m M p m α ’.M ’ ε

140 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ α ’.M ’ ε

141 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ α ’.M ’ ε

142 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ α ’.M ’ ε

143 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ εα ’.M ’

144 ε W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ½.M ’ α ’.M ’

145 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 ε

146 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 m := m-1 ε

147 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 m := m-1 ε

148 W = (w 1, …, w m ), 0  W < M = p 1. …. p m m m α = [1] Σ ( [p i ] Π p j ) [ p i] -1 [p i ]. w i / p i  1 / 2 ? i=1 j=1, j  i α := 0.00 for i := 1, …, m do c := 1 for j := 1, …, m do if j ≠ i then c := c [p i ]. p j end c := c [ p i] -1 ; c := c [p i ]. w i φ := c / p i α := α [1] + φ end numeric error 1 ε   2 ℓ 2p m M 0 ε M’M’ 0 m := m-1 … repeated until the question W  M / 2 is solved ε

149 un-Primes, un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n)

150 un-Primes, un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ϵ accept-ASPACE x REVERSALS(loglog n)

151 un-Primes, un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ϵ accept-ASPACE x REVERSALS(loglog n) --optimal, cannot be improved

152 un-Primes, un-Composites ϵ accept-ASPACE(loglog n) ϵ pebble-DSPACE(loglog n) ϵ accept-ASPACE x REVERSALS(loglog n) --optimal, cannot be improved below loglog n, only regular languages accepted (even with the help of alternation)

153 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n)

154 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n) Primes

155 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n) Primes un-Primes

156 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n) Primes un-Primes +factoring

157 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n) Primes  ? un-Primes +factoring

158 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Positive answer: Primes  ? un-Primes +factoring

159 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Positive answer: -- deterministic factoring in polynomial time Primes  ? un-Primes +factoring

160 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Positive answer: -- deterministic factoring in polynomial time (breaking cryptographic security) Primes  ? un-Primes +factoring

161 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Negative answer: Primes  ? un-Primes +factoring

162 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Negative answer: un-L  accept-ASPACE(loglog n) Primes  ? un-Primes +factoring

163 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Negative answer: un-L  accept-ASPACE(loglog n) L  ASPACE(log n) = P Primes  ? un-Primes +factoring

164 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Negative answer: un-L  accept-ASPACE(loglog n) L  ASPACE(log n) = P  NSPACE(log n) Primes  ? un-Primes +factoring

165 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) Negative answer: un-L  accept-ASPACE(loglog n) L  ASPACE(log n) = P  NSPACE(log n) Primes  ? NL versus NP un-Primes +factoring

166 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n) Primes  ? un-Primes +factoring

167 LDSPACE(log n)  un-DSPACE(loglog n) NLNSPACE(log n)  un-NSPACE(loglog n) PASPACE(log n)  un-ASPACE(loglog n) PspaceDSPACE(n O(1) )  un-DSPACE(log O(1) n)|| NSPACE(n O(1) )un-NSPACE(log O(1) n) ExptimeASPACE(n O(1) )  un-ASPACE(log O(1) n) Primes  ? un-Primes +factoring Primes ?

168 Thank You for Your Attention

169 Thank You for Your Attention

170 Thank You for Your Attention

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