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A Note on Negligible Functions

Mihir Bellare

J. CRYPTOLOGY

2002

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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

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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.

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 n, and over the coin tosses of I.

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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,

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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 s.t.

InvI≦ev for every inverter I.

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Another view of OW. & Uni. OW.

f is one-way:

inverters I negligible I s.t. InvI≦evI.

f is uniformly one-way:

negligible s.t. inverters I we have

InvI≦ev.

The order of quantification is different.

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Observation Another way to see the difference f is not one-way:

inverters I and 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 inverter I s.t.

InvIn>n for infinitely many n. This does not directly say that there is one inverter achiev

ing non-negligible success.

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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

InvI≦ev.

inverters I negligible I=s.t. InvI≦evI.

Then f is one-way.

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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 s.t.

Fi≦ev for all i in N.

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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

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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.

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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 .

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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 ≦evg and g ≦ev(.)-c,

we have f ≦ev(.)-c.

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Definitions and Elementary Facts

A collection of functions is a set of functions whose cardinality could be countable or uncountable.

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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.

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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.

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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.

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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 (n) = max{F1(n),F2(n),...,Fn(n)}

= max{Fi(n): i in N}.

Certainly Fi≦ev for each i in N but is not negligible.

e.g. Fi(j)=1 if j≦i and Fi(j)=(j) if j>i, where is negl.

(n)= max{F1(n),F2(n),...,Fn(n)}=1 is not negligible.

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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 f

or n ≧ h(c). The sequence eventually covers the entire table.

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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.

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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.

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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{Fi(n):1≦i≦g(n)}. (4)

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Proof of Thm 3.2.

Claim 6.

The function is a limit point of F={Fi: i in N}.

pf:

i, we need to show ni s.t. Fi(n)≦(n) n≧ni.

set ni=h(i) and suppose n≧ni.

Applying Claim 3 and 5 we get

g(n)≧g(h(i))=i.

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Proof of Thm 3.2.

Claim 7.

The function is negligible.

c, we need to show nc s.t. (n)≦n-c n≧nc.

Set nc=h(c), assume n≧nc=h(c), we get

(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

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The Case of an Uncountable Collection of Functions Prop 3.5. There is an uncountable collection of fu

nctions 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.

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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.

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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≦evM. M≦ev and F≦evM. We obtain F≦ev.

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Non-Uniform Algorithms

The set of all negligible functions is uncountable??

The set of all polynomials is countable?