1. 1 A Note on Negligible Functions Mihir Bellare J. CRYPTOLOGY 2002

2. 2 Negligible functions A function g: N->R is called negligible if it approaches zero faster than the reciprocal of any polynomial. That is A function g: N->R is called negligible if For every c in N, there is an integer n c s.t. g(n) ≦ n -c, for all n ≧ n c

3. 3 The Issue for One-Way Functions f:  * ->  * be a poly-time computable, length- preserving function. I:An inverter for f, a probabilistic, poly-time algorithm. Inv I : The success probability of I. Inv I (n): for any value n in N, Inv I (n)=Pr[f(I(f(x)))=f(x)], the prob. Being over a random choice of x from  n, and over the coin tosses of I.

4. 4 Eventually less g 1 : N->R is eventually less than g 2 :N->R, if there is an integer k s.t. g 1 (n) ≦ g 2 (n) for all n ≧ k written g 1 ≦ ev g 2,

5. 5 One-Way & Uniformly One-Way f is one-way if for every inverter I the function Inv I is negligible f is uniformly one-way if there is a negligible function  s.t. Inv I ≦ ev  for every inverter I.

6. 6 Another view of OW. & Uni. OW. f is one-way:  inverters I  negligible  I s.t. Inv I ≦ ev  I. f is uniformly one-way:  negligible  s.t.  inverters I we have Inv I ≦ ev . The order of quantification is different.

7. 7 Observation Another way to see the difference f is not one-way:  inverters I and a constant c s.t. Inv I (n)>n -c. for infinitely many n. i.e.  inverters whose success prob. is not negl. f is not uniformly one-way:  negligible   inverter I s.t. Inv I  n  >  n   for infinitely many n.  This does not directly say that there is one inverter achieving non-negligible success.

8. 8 Equivalence f is one-way iff f is uniformly one-way. (<=) It is not hard to see. Since f is uniformly one-way:  negligible  s.t.  inverters I we have Inv I ≦ ev .  inverters I  negligible  I =  s.t. Inv I ≦ ev  I. Then f is one-way.

9. 9 Negligibility of Function Collections F={F i :i in N} be a collection of functions, all mapping N to R. How to define negligibility of function Collection. F is pointwise negligible: if F i is negligible for each i in N. F is uniformly negligible: if there is a negligible function  s.t. F i ≦ ev  for all i in N.

10. 10 Observation (only for countable case) Let I={I i : i in N} be an enumeration of all inverters. (Since an inverter is a probabilistic, polynomial-time algorithm, the number of inverters is countable. For the non-uniform case, where there are uncountably many inverters.) For each i in N define F i by F i (n)= Inv I i (n), F={F i :i in N}={Inv I i :i in N}. f is one-way iff F is pointwise negligible. f is uniformly one-way iff F is uniformly negligible

11. 11 Equivalence F is pointwise negligible iff F is uniformly negligible ?? (<=) Clearly. (=>) It is true for countable collection (Thm 3.2). It is not true for uncountable collection.

12. 12 Definitions and Elementary Facts Def 2.1. If f,g are functions we say that f is eventually less than g, written f ≦ ev g, if there is an integer k s.t. f(n) k. Prop 2.2. The relation is ≦ ev transitive: if f 1 ≦ ev f 2 and f 2 ≦ ev f 3, then f 1 ≦ ev f 3.

13. 13 Definitions and Elementary Facts Def 2.3. A function f is negligible if f ≦ ev (.) -c for every integer c. Here (.) -c stands for the function n->n -c. Prop 2.4. A function f is negligible iff there is a negligible function g s.t. f ≦ ev g. Pf. (=>)setting g=f (<=)let c in N. f ≦ ev g and g ≦ ev (.) -c, we have f ≦ ev (.) -c.

14. 14 Definitions and Elementary Facts A collection of functions is a set of functions whose cardinality could be countable or uncountable.

15. 15 Definitions and Elementary Facts Def 2.5. A collection of functions F is pointwise negligible if for every F in F it is the case that F is negligible function. Def 2.6. A collection of functions F is uniformly negligible if there is a negligible function  s.t. F ≦ ev  for every F in F.

16. 16 Definitions and Elementary Facts Def 2.7. Let F be a collection of functions and let  be function. We say that  is a limit point of F if F ≦ ev  for each F in F. Prop 2.8. A collection of functions F is uniformly negligible iff it has a negligible limit point. Pf.(=>)  is the limit point. (<=)setting  =limit point.

17. 17 Relations between the Two Notions of Negligible Collections F is uniformly negligible iff F is pointwise negligible?? Prop 3.1. if F is uniformly negligible, then it is pointwise negligible. (=>) Pf. By Prop 2.8,  a negligible function  that is limit point of F, Let F in F, we know that F ≦ ev .  is negligible, so F is negligible. F is pointwise negligible. it holds regardless of whether the collection is countable or uncountable.

18. 18 The case of a Countable Collection Thm 3.2. Let F={F i : i in N} be a countable collection of functions. Then F is pointwise negligible iff it is uniformly negligible. Remark 3.3. First thought: Set  (n)= max{F 1 (n),F 2 (n),...,F n (n)} = max{F i (n): i in N}. Certainly F i ≦ ev  for each i in N but  is not negligible. e.g. F i (j)=1 if j ≦ i and F i (j)=  (j) if j>i, where  is negl.  (n)= max{F 1 (n),F 2 (n),...,F n (n)}=1 is not negligible.

19. 19 Proof of Thm 3.2. Imagine a table with rows indexed by the values i = 1, 2, …; columns indexed by the values of n = 1, 2, …; and entry (i, n) of the table containing F i (n). For any c, the entries in each row eventually drop below n –c. However, it happens differs from row to row. In stage c we will consider only the first c functions. We will find h(c) s.t. all these functions are less than (.) -c for n ≧ h(c). The sequence eventually covers the entire table.

20. 20 Proof of Thm 3.2. For every i,c in N, we know that F i ≦ ev (.) -c. i.e. Let N i,c in N be s.t. F i (n) ≦ n -c for all n ≧ N i,c. Define h:{0} ∪ N->N recursively and let h(0)=0 and h(c)=max{N 1,c,N 2,c,...,N c,c,1+h(c-1)} for c in N. Claim 1. F 1 (n),...,F c (n) ≦ n -c for all n ≧ h(c) and all c in N.

21. 21 Proof of Thm 3.2. Claim 2. h is an increasing (strict increasing) function, meaning h(c)<h(c+1) for all c in N ∪ {0}. For any n in N, we let g(n)=max{ j in N: h(j) ≦ n}.(3) Claim 3. g is a non-decreasing (increasing) function, meaning g(n) ≦ g(n+1) for all n in N.

22. 22 Proof of Thm 3.2. Claim 4. h(g(n)) ≦ n for all n in N. It is clear from (3). Letting n =h(c) in (3) and using Claim 2, we get: Claim 5. g(h(c))=c for all c in N. For any n in N we let  (n)=max{F i (n):1 ≦ i ≦ g(n)}.(4)

23. 23 Proof of Thm 3.2. Claim 6. The function  is a limit point of F={F i : i in N}. pf:  i, we need to show  n i s.t. F i (n) ≦  (n)  n ≧ n i. set n i =h(i) and suppose n ≧ n i. Applying Claim 3 and 5 we get g(n) ≧ g(h(i))=i.

24. 24 Proof of Thm 3.2. Claim 7. The function  is negligible.  c, we need to show  n c s.t.  (n) ≦ n -c  n ≧ n c. Set n c =h(c), assume n ≧ n c =h(c), we get  (n)=max{F i (n):1 ≦ i ≦ g(n)} ≦ n -g(n) ≦ n -c. DEF(4) Claim 4 and 1 Since n ≧ n c =h(c), applying claim 3 and 5 we get g(n) ≧ g(h(c))=c

25. 25 The Case of an Uncountable Collection of Functions Prop 3.5. There is an uncountable collection of functions F that is pointwise negligible but not uniformly negligible. pf: Let F be the set of all negligible functions. F is pointwise negligible. Assume g is limit point of F, f=2g is negligible, f is not eventually less than g. Hence, F has no limit point.

26. 26 Uncountable Collection Def 3.6. Let F,M be collections of functions. We say that F is majored by M, or M majors F, if for every F in F there is an M in M s.t. F ≦ ev M. Thm 3.7. F is uniformly negligible iff it is majored by some pointwise negligible, countable collection of functions.

27. 27 Proof of Thm 3.7. (=>) F is uniformly negligible, it has a limit point . We set M={m  : m in N}. This countable, pointwise negligible collection of function, and it majors F. (<=) M is countable, it is uniformly negligible by Thm 3.2. M has a negligible limit point . Since M majors F,  M in M s.t. F ≦ ev M. M ≦ ev  and F ≦ ev M. We obtain F ≦ ev .

28. 28 Non-Uniform Algorithms The set of all negligible functions is uncountable?? The set of all polynomials is countable?