A Completeness Theorem for Pseudo-Linear Functions with Applications to UC Security

We consider multivariate pseudo-linear functions over finite fields of characteristic two. A pseudo-linear polynomial is a sum of guarded linear-terms, where a guarded linear-term is a product of one or more linear-guards and a single linear term, and each linear-guard is again a linear term but raised to the power q-1, where q is the field size. Pseudo-linear functions over GF(2) are given by pseudo-linear polynomials defined over GF(2). Let f1, f2, ..., fk be k pseudo-linear functions in n variables, and let f be another pseudolinear function in the n variables. We show that if f is a function of the given k functions, then it must be a pseudo-linear function of the given k functions. This generalizes the straightforward claim for just linear functions. We also prove a more general theorem where the k functions can in addition take further arguments, and prove that if f can be represented as an iterated composition of these k functions, then it can be represented as a probabilistic pseudo-linear iterated composition of these functions. Proceeding further, we generalize the theorem to randomized pseudo-linear functions. Additionally, we allow f itself to be a randomized function, i.e. we give a procedure for deciding if f is a probabilistic sub-exponential in m time iterated function of the given k randomized functions, and the decision procedure runs in computational time independent of m. These theorems have implications for automatic proving of universally-composable security theorems for ideal and real functionalities composed of if-then-else programs with (uniform) random number generation and data objects from additive group of GF(2). The theorems imply that, within this language framework, there is a decision procedure to find out if a real functionality realizes an ideal functionality, and this procedure is in computational time independent of m (which is essentially the security parameter). ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 92 (2010)