In computational complexity theory and cryptography, averaging argument is a standard argument for proving theorems. It usually allows us to convert probabilistic polynomial-time algorithms into non-uniform polynomial-size circuits.

Example

edit

Example: If every person likes at least 1/3 of the books in a library, then the library has a book, which at least 1/3 of people like.

Proof: Suppose there are   people and   books. Each person likes at least   of the books. Let people leave a mark on the book they like. Then, there will be at least   marks. The averaging argument claims that there exists a book with at least   marks on it. Assume, to the contradiction, that no such book exists. Then, every book has fewer than   marks. However, since there are   books, the total number of marks will be fewer than  , contradicting the fact that there are at least   marks.  

Formalized definition of averaging argument

edit

Let X and Y be sets, let p be a predicate on X × Y and let f be a real number in the interval [0, 1]. If for each x in X and at least f |Y| of the elements y in Y satisfy p(x, y), then there exists a y in Y such that there exist at least f |X| elements x in X that satisfy p(x, y).

There is another definition, defined using the terminology of probability theory.[1]

Let   be some function. The averaging argument is the following claim: if we have a circuit   such that   with probability at least  , where   is chosen at random and   is chosen independently from some distribution   over   (which might not even be efficiently sampleable) then there exists a single string   such that  .

Indeed, for every   define   to be   then

 

and then this reduces to the claim that for every random variable  , if   then   (this holds since   is the weighted average of   and clearly if the average of some values is at least   then one of the values must be at least  ).

Application

edit

This argument has wide use in complexity theory (e.g. proving  ) and cryptography (e.g. proving that indistinguishable encryption results in semantic security). A plethora of such applications can be found in Goldreich's books.[2][3][4]

References

edit
  1. ^ Barak, Boaz (March 2006). "Note on the averaging and hybrid arguments and prediction vs. distinguishing" (PDF). COS522. Princeton University.
  2. ^ Oded Goldreich, Foundations of Cryptography, Volume 1: Basic Tools, Cambridge University Press, 2001, ISBN 0-521-79172-3
  3. ^ Oded Goldreich, Foundations of Cryptography, Volume 2: Basic Applications, Cambridge University Press, 2004, ISBN 0-521-83084-2
  4. ^ Oded Goldreich, Computational Complexity: A Conceptual Perspective, Cambridge University Press, 2008, ISBN 0-521-88473-X