Friday, April 11, 2008

* One-way functions and trapdoor functions

ONE-WAY FUNCTIONS

A function f: Sigma^* -> Sigma^* is length-preserving if the lengths
of w and f(w) are equal for every w.  So f maps Sigma^n to Sigma^n.

A one-way permutation is a length-preserving permutation with the
following properties.

1. It is computable in polynomial time.

2. For every probabilistic poly-time TM M, every k, and sufficiently
large n, if we pick a random w of length n and run M on input w, then

Pr_{M,w}[M(f(w)) = w] <= 1/n^k.

A one-way function is a length-preserving function with the following
properties.

1. It is computable in polynomial time.

2. For every probabilistic poly-time TM M, every k, and sufficiently
large n, if we pick a random w of length n and run M on input w, then

Pr_{M,w}[f(M(f(w))) = w] <= 1/n^k.

Candidate one-way function is multiplication.  For example, mult(w) =
w_1*w_2, where w_1 and w_2 represent the first and second halves of w.
Pad the product with leading 0s to make it length-preserving.

No probabilistic poly-time algorithm is known to invert mult, even on
a polynomial fraction of the inputs.

http://qwiki.stanford.edu/wiki/Complexity_Zoo

TRAPDOOR FUNCTIONS

A trapdoor function is a variant of one-way functions which can be
efficiently inverted given some additional information.  

Call f: Sigma^* x Sigma^* -> Sigma^* an indexing function.  Here
f can be viewed as {f_i: i in Sigma^*}.

A trapdoor function is a triple -- indexing function f, auxiliary
probabilistic TM G and auxiliary function h: Sigma^* x Sigma^* ->
Sigma^* with the following properties.

1. f and h are computable in poly time.

2. For every poly-time probabilistic TM E and every k and sufficiently
large n, if we take a random output <i,t> of G on 1^n and a random w
in Sigma^n then

Pr[E(i,f_i(w)) = y, where f_i(y) = f_i(w)] <= n^{-k}.

3.  For every n, every w of length n, and every output <i,t> of G that
occurs with nonzero probability for some input to G

h(t,f_i(w)) = y, where f_i(y) = f_i(w).