2. 1 Slides by Iddo Tzameret and Gil Shklarski. Adapted from Oded Goldreich’s course lecture notes by Erez Waisbard and Gera Weiss.

3. 2 PRG - Stronger Notion Def: A deterministic polynomial-time algorithm G is called a non-uniformly strong pseudorandom generator if there exists a stretching function l: N  N, so that for any family {C k } of polynomial-size circuits, for any polynomial p, and for all sufficiently large k’s |Pr[C k (G(U k ))=1]-Pr[C k (U l(k) )=1]| < 1/p(k) This definition involves polynomial size circuits as distinguishers instead of probabilistic polynomial time TM. Recall that BPP  P/poly

4. 3 Implications of such PRG Theorem: If such non-uniformly strong pseudorandom generator exists then Proof: Suppose L  BPP. Let A(x,r) be the machine that decides L: x is the input and r is the sequence of coin tosses of the machine. r is of size l(|x|  ). Define a new algorithm A’ as follows: A’(x,r) := A(x,G(r)) Where We can construct such A that uses exactly l(|x|  ) coin tosses

5. 4 Proof Continued (1) Claim: For all but finitely many x’s |Pr[A(x,U l(k) )=1] - Pr[A’(x, U k )=1]| < 1/6 where k=|x| . Proof: Assume, by way of contradiction, that, for infinitely many x’s |Pr[A(x,U l(k) )=1] - Pr[A’(x, U k )=1]|  1/6 and construct a family of poly-size circuits x  C (x) (input) := A(x,input) then construct the family {C k } as follows: C k  {C (x) | A(x) uses l(k) coin tosses} Infinitely many x’s on which A and A’ differ imply infinitely many sizes of x’s on which they differ, and infinite number of such C k s.

6. 5 Proof Continued (2) For each such C k : C k (G(U k ))  A’(x,U k ) and C k (U l(k) )  A(x,U l(k) ) Hence we have a family of circuits s.t. |Pr[C k (G(U k ))=1]-Pr[C k (U l(k) )=1]|  1/6 In contradiction to the definition of our pseudorandom generator.  claim

7. 6 Proof Continued (3) Going back to proving the theorem: A is our BPP machine so for every x: x  L  Pr[A(x,U l(k) ) = 1]  2/3 x  L  Pr[A(x,U l(k) ) = 1] < 1/3 In particular, using the claim we get for all but finitely many x’s: x  L  Pr[A’(x,U k ) = 1] > Pr[A(x,U l(k) ) = 1]-1/6  1/2 x  L  Pr[A’(x,U k ) = 1] < Pr[A(x,U l(k) ) = 1]+1/6 < 1/2

8. 7 Proof Continued (4) Now, define a deterministic algorithm A’’ for deciding L: if x is one of those finitely x’s return a known pre-computed answer else { for all Run A’(x,r) return the majority of A’ answers. } A’’ deterministically decides L and run in time as required.  Theorem

9. 8 Goal: to design a new PRG construction, which would be used for derandomization New Method: generate random bits in parallel, instead of sequentially (compare with the “Pseudo Random Generators” lecture) Different Assumptions: weaker then before, since the new PRG can run in time exponential in its input size: Assume an unpredictable Boolean function. New Construct: called Design; consisting of nearly disjoint subsets of the random seed. New notion of PRG

10. 9 The new requirements for PRG: Indistinguishable by polynomial-size circuit. Can run in exponential time (2 O(k) on k-bit seed). One can construct such PRG under seemingly weaker assumption (than for the construction shown in the “Pseudo Random Generators” lecture): The existence of unpredictable Boolean function. For k=O(log(|x|)) it runs in polynomial-time. Instead of assuming the existence of one-way permutation.

11. 10 Unpredictable Boolean function Def (Unpredictable Boolean function): An exp(l)-computable Boolean function b:{0,1} l  {0,1} is unpredictable by small circuits if for every polynomial p(.), for all sufficiently large l’s and for every circuits C of size p(l): Pr[C(U l )=b(U l )] < ½+1/p(l) Assume such Boolean functions exist

12. 11 Unpredictable Boolean function How strong is that assumption? We prove that it is not stronger than assuming the existence of a one-way permutation: Claim: if f 0 is a one-way permutation and b 0 is a hard-core of f 0, then b(x):=b 0 (f 0 -1 (x)) is an unpredictable Boolean function. ? one-way permutation unpredictable Boolean function

13. 12 One way permutation  unpredictable Boolean function Proof: Let f 0 be a one-way permutation and b 0 a hard-core of f 0. We’ll show the function b(x):=b 0 (f -1 0 (x)) is an unpredictable Boolean function. f 0 can be inverted in exponential time and b 0 can be computed in polynomial time so b is computable in exponential time. Unpredictability: Assume, by way of contradiction, that b is predictable. We’ll show the b 0 is not hard-core bit of f 0.

14. 13 Proof continued Assuming b is predictable we have a family of circuits {C k } of size p(k) s.t. for infinite number of k’s Pr[C k (U k )=b(U k )]  1/2 + 1/p(l). For y:=f 0 -1 (x) we get b(f 0 (y))=b 0 (y). f is a permutation so we get Pr[C k (f 0 (U k ))=b(f 0 (U k ))]  1/2 + 1/p(l) Pr[C k (f 0 (U k ))=b 0 (U k )]  1/2 + 1/p(l). Which is a contradiction to b 0 being a hard core. We defined hard-core bit with BPP machines and not P/poly so there is a problem here !

15. 14 The Design Generating a single random bit from a seed is easy assuming you have an unpredictable Boolean function. But how can we generate more than one bit? We will manage that, utlizing a collection of nearly disjoined subsets of the seed to get random bits that are almost mutually independent Almost means: indistinguishable by polynomial sized circuits

16. 15 The Design Def: A collection of m subsets {I 1,I 2,…,I m } of {1…k} is a (k,m,l)-design if the following hold: For every i  {1,…,m}:|I i | = l For every i  j  {1,…,m}: |I i  I j | = O(log k) The collection is constructible in exp(k)-time. Notation: For S= and I={i 1, …, i l }  {1,..,k}

17. 16 S (seed): The Design - Visualization INDEX I 1, I 2, …, I m : {1,4,7} {2,5,8} {3,9,10}...{1,8,9} {1,0,0} {0,0,1} {1,1,0}... {1,1,1} k l S[I 1 ], …, S[I m ]:

18. 17 Prop: let b: {0,1}  k  {0,1} be an unpredictable Boolean function, and {I 1,…,I m } be a (k,m,  k)-design then the following function is a strong non-uniform PRG: G(S)  Constructing the PRG 15.3

19. 18 Constructing the PRG: Visualization m 0 1 1 …………… 0 Pseudo random string l S (seed): INDEX I 1, I 2, …, I m : {1,4,7} {2,5,8} {3,9,10}...{1,8,9} {1,0,0} {0,0,1} {1,1,0}... {1,1,1} k S[I 1 ], …, S[I m ]: b( ) ………

20. 19 Proof (1) Proof: Computing G(s) takes time exponential in k, since: we have m=l(k) computations of b(S[I i ]); Computing each b(S[I i ]) takes exp( |S[I i ]| ) = O(exp(k)).

21. 20 Proof (2) we will show that no small circuit can distinguish the output of G from a random sequence. Assume by way of contradiction that there exists a family of poly-size circuits {C k } k  N and a polynomial p(.) such that for infinitely many k’s | Pr[C k (G(U k )) = 1] - Pr[C k (U l(k) )=1] | > 1/p(k) Without loss of generality we can remove the absolute sign. There are infinitely many k’s s.t. Pr[C k (G(U k )) = 1] - Pr[C k (U l(k) )=1] has the same sign for all k, however, we can fix the sign arbitrarily since we can take a sequence of circuits with reverse signs.

22. 21 Using a Hybrid Distribution - proof (3) For any 0  i  m we define a “hybrid” distribution as follows: the first i bits are chosen to be the first i bits of G(U k ) and the other m-i bits are chosen uniformly at random. H i k  G(U k ) [1,…,i]  U m-i also f k (i)  Pr[C k (H k i )=1] Using these definitions we can write: f k (m) - f k (0) > 1/p(k) there must be some 0  i k  m s.t: f k (i k +1) - f k (i k ) > 1/m * 1/p(k)

23. 22 Approximating the Next bit from the previous bits Defining p’(k):=m  p(k) and i:=i k we get: Pr[C k (H k i+1 )=1]- Pr[C k (H k i )=1] > 1/p’(k) Now, we can construct from C k a circuit C’ k which can approximate the next bit with large enough probability: When R i are independent uniformly distributed bits. It can be shown that Pr[C’ k (G(U k ) [1,…i] ) = G(U k ) i+1 ] > 1/2 + 1/p’(k) Probability over random bits R i and U k

24. 23 Approximating the Next bit from the previous bits ½-  ½+  b(S[I 1 ]) …… b(S[I ik ]) Circuit C‘ k Next bit b(S[I ik+1 ])  :=1/p’(k)

25. 24 Approximating b(S[I i+1 ]) from S and b(S[I i ])’s We can construct a circuit C’’ which inputs S in addition to b(S[I 1 ]),…, b(S[I i ]) and can approximate the unpredictable boolean function b(S[I i+1 ]). This can be done by ‘ignoring’ those new inputs and using b(S[I 1 ]),…, b(S[I i ]) and C’. The formal definition is: C’’ k (S°G(S) [1..i] ) := C’ k (G(S) [1..i] ) We get: Pr s [C’’ k (S°G(S) [1..i] ) = G(S) i+1 ] > 1/2 + 1/p’(k) Pr s [C’’ k (S°G(S) [1..i] ) = b(S[I i+1 ])] > 1/2 + 1/p’(k) Probabilities over random bits R i and S

26. 25 Approximating b(S[I i+1 ]) from S[I i+1 ] and b(S[I j ])’s There exist   {0,1} k-|Ii| s.t. Pr s [C’’ k (S°G(S) [1..i] ) = b(S[I i+1 ]) | S[I i+1 ]=  ] > 1/2 + 1/p’(k) We’ll hard-code this  into our circuit and get a circuit that takes b(S[I 1 ]),…, b(S[I i ]) and S[I i+1 ] as inputs and approximate b(S[I i+1 ]) with some bias. Applying the Law of Averages: Pr[C’’k(S°G(S)[1..i] ) = b(S[Ii+1])] =  Pr [C’’k(S°G(S)[1..i] ) = b(S[Ii+1]) | S[Ii+1]=  ]Pr[S[Ii+1]=  ] If for all  : Pr [C’’k(S°G(S)[1..i] ) = b(S[Ii+1]) | S[Ii+1]=  ]  1/2+1/p’(k) We’d get Pr[C’’k(S°G(S)[1..i] ) = b(S[Ii+1])]  1/2+1/p’(k).

27. 26 Visualization of C’’ b(S[I 1 ])… …… b(S[I i ]) ½-  ½+  Circuit C‘ k Next bit b(S[I i+1 ])  S[I i+1 ]) S: Circuit C‘’ k S[I i+1 ])

28. 27 Approximating b(S[I i+1 ]) from S[I i+1 ] We know how to approximate b(S[I i+1 ]) from its input S[I i+1 ] and from b(S[I 1 ]),…, b(S[I i ]). Can we approximate it using only S[I i+1 ] ?

29. 28 Computing S[I j ]’s from S[I i+1 ] S: S[I i+1 ]  = S[I i+1 ]) S[I 1 ] ? S[I 2 ] ? O(log(k)) ……… S[I i ] ? ? After hard-coding , there is only a small number of free bits in S[I 1 ]…S[I i ]. The design gives us iO(log(k)) as a bound.

30. 29 Computing S[I j ]’s from S[I i+1 ] Example S: S[I i+1 ]  = S[I i+1 ]) S[I 1 ] ? S[I 2 ] ? ……… S[I i ] ? ? S: S[I 1 ]S[I 2 ] … S[I i ] O(log(k)) 0 0 1 ……… 0 1 ???? 0 1 1 precomputed b( ) 1 S[I i+1 ] 1 b( )

31. 30 Computing b(S[I i+1 ])’s from S[I i+1 ] S: S[I 1 ] S[I 2 ] … S[I i ]  1  2  3 ………  j ????  j+1 …  k-l S[I i+1 ] Exp(log(k))= poly(k) circuit S[I 1 ]………S[I i ] b(S[I 1 ])……… Lookup table: for every possible S[I i ] return precomputed value of b(S[I i ]) b(S[I i ]) There are only poly(k) possible such S[I i ]’s, given S[I i+1 ]= .

32. 31 ½-  Circuit C‘ Next bit b(S[I i+1 ]) Final Circuit: Approximating b(S[I i+1 ]) from S[I i+1 ] S[I i+1 ] poly(k) circuit S[I 1 ]………S[I i ] Lookup table b(S[I 1 ]) … b(S[I i ]) ½+ 

33. 32 Design construction: greedy algorithm For the following parameters: k = l 2 m = poly(k) We want that for all i to have |I i |=l and for i  j, |I i  I j |=O(log k). For i = 1 to m For all I  [k], |I|=l do flag := FALSE for j = 1 to i-1 if |I i  I j | > log k then flag:=TRUE if flag = TRUE then I i = I The algorithm:

34. 33 Greedy algorithm: proof Assuming that for i  m we have I 1, I 2,…, I i-1 such that –for every j<i: |I j | = l –for every j 1,j 2 < i: |I j1  I j2 | < 2+log m We’ll show that there exists another set |I i |=l s.t. for every j < i: |I j  I i | < 2+log m Proof by the probabilistic method: Let S be a fixed set of size l. Let R be a set which is selected at random so that for every i  [k]: Pr[i  R] = 2/l. R length ~ binomial(k,2/l).

35. 34 Proof continued (1) Let S i be the i’th element in S sorted in some order. We’ll define the sequence {X i } i=1..l of random variables: X i are independent Bernoulli variables with Pr[X i =1]= 2/l for each i. Using Chernoff’s bound :

36. 35 Proof continued (2) For R selected as above the probability that there exists I j s.t. |I j  R| > 2+log m us bounded above by (i-1)/2m < 1/2. R is not necessarily of size l. We can show that with high probability |R|  l so it contains a subset of size l that we can choose as our I i. Considering the sequence {X i } i=1..l : Using Chernoff’s bound: For R selected as above the probability of too many collisions or being too small is strictly smaller than one. Therefore, there exists such R to be selected as I i.  Note: The algorithm itself is deterministic. We use the randomness as a tool in showing the algorithm will always find what it is looking for.

37. 36 Second Design Construction: using GF(l) arithmetic For the following parameters: k = l 2 m = poly(k) Let F:=GF(l) then |F  F| = k There is a 1-1 correspondence between {1,…,k} and F  F For every polynomial p(.) of degree d over F, I p is the graph of p(.) over F: I p := { | e  F } |I p | = |F| = l

38. 37 Second Design Construction: using GF(l) arithmetic For every two polynomials p(.)  q(.) of degree d intersects in at most d points, hence: |I p  I q |  d by the Fundamental Theorem of Algebra, hence we can choose d=O(log(k)). Note that for every polynomial m(k) we can construct m(k)= m(l 2 ) such sets, since there are |F| d+1 = l d+1 polynomials over GF(l), so by choosing an appropriate d the number of sets is greater then m(l 2 ). The sets are constructible in exponential in k, since we use simple arithmetic over GF(l).