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

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

n is a prime

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

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]:

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

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

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

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

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)

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

(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)

(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

(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

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”

O(loglog n) space due to

- Modular representation (based on Chinese Remainder Theorem)

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

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)

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

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)

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

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

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

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

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

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

__ 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

__ 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

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

__ 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

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

__ 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

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

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}

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}      

__ 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

__ 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 = (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

__ 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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 = (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 = (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

__ 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

__ 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

__ 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

__ 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

__ 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 = (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

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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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

__ 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

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

__ 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

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

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

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

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 = (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

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

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

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

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

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

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 = (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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ’

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 ’

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

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

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

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

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

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

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

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ?

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Thank You for Your Attention

Thank You for Your Attention