A Note on Negligible Functions

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

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 nc s.t. g(n) ≦n-c, for all n≧nc

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

4. Eventually less • g1: N->R is eventually less than g2:N->R, if there is an integer k s.t. g1(n) ≦ g2(n) for all n≧k written g1 ≦ev g2,

5. One-Way & Uniformly One-Way • f is one-way if for every inverter I the function InvI is negligible • f is uniformly one-way if there is a negligible function d s.t. InvI≦evd for every inverter I.

6. Another view of OW. & Uni. OW. • f is one-way:  inverters I negligible dI s.t. InvI≦evdI. • f is uniformly one-way:  negligible d s.t.  inverters I we have InvI≦evd. The order of quantification is different.

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

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 d s.t.  inverters I we have InvI≦evd.  inverters I negligible dI=d s.t. InvI≦evdI. Then f is one-way.

9. Negligibility of Function Collections • F={Fi: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 Fi is negligible for each i in N. • F is uniformly negligible: if there is a negligible function d s.t. Fi≦evd for all i in N.

10. Observation (only for countable case) • Let I={Ii: 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 Fi by Fi(n)= InvIi(n), F={Fi:i in N}={InvIi:i in N}. • f is one-way iff F is pointwise negligible. • f is uniformly one-way iff F is uniformly negligible

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. Definitions and Elementary Facts • Def 2.1. If f,g are functions we say that f is eventually less than g, written f≦evg, if there is an integer k s.t. f(n)<g(n) for all n>k. • Prop 2.2. The relation is ≦ev transitive: if f1≦evf2 and f2≦evf3, then f1≦evf3 .

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 ≦evg. Pf. (=>) setting g=f (<=) let c in N. f ≦evg and g ≦ev(.)-c, we have f ≦ev(.)-c.

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

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 d s.t. F≦evd for every F in F.

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

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 d that is limit point of F, Let F in F, we know that F≦evd. d is negligible, so F is negligible. F is pointwise negligible. it holds regardless of whether the collection is countable or uncountable.

18. The case of a Countable Collection • Thm 3.2. Let F={Fi: 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 d(n) = max{F1(n),F2(n),...,Fn(n)} = max{Fi(n): i in N}. Certainly Fi≦evd for each i in N but d is not negligible. e.g. Fi(j)=1 if j≦i and Fi(j)=u(j) if j>i, where u is negl. d(n)= max{F1(n),F2(n),...,Fn(n)}=1 is not negligible.

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 Fi(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. Proof of Thm 3.2. • For every i,c in N, we know that Fi≦ev (.)-c. i.e. Let Ni,c in N be s.t. Fi(n) ≦n-c for all n≧Ni,c. • Define h:{0}∪N->N recursively and let h(0)=0 and h(c)=max{N1,c,N2,c,...,Nc,c,1+h(c-1)} for c in N. • Claim 1. F1(n),...,Fc(n) ≦ n-c for all n≧h(c) and all c in N.

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. 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 d(n)=max{Fi(n):1≦i≦g(n)}. (4)

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

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

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. 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≦evM. • Thm 3.7. F is uniformly negligible iff it is majored by some pointwise negligible, countable collection of functions.

27. Proof of Thm 3.7. • (=>) F is uniformly negligible, it has a limit point d. We set M={md: 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 d. Since M majors F,  M in M s.t. F≦evM. M≦evd and F≦evM. We obtain F≦evd.

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