Siyao Guo

Candidate weak pseudorandom functions in AC 0 ○ MOD 2

Pseudorandom functions (PRFs) play a fundamental role in symmetric-key cryptography. However, they are inherently complex and cannot be implemented in the class AC0 (MOD2). Weak pseudorandom functions (weak PRFs) do not suffer from this complexity limitation, yet they suffice for many cryptographic applications. We study the minimal complexity requirements for constructing weak PRFs. To this end We conjecture that the function family FA(x) = g(Ax), where A is a random square GF(2) matrix and g is a carefully chosen function of constant depth, is a weak PRF. In support of our conjecture, we show that functions in this family are inapproximable by GF(2) polynomials of low degree and do not correlate with any fixed Boolean function family of subexponential size. We study the class AC0 ○ MOD2 that captures the complexity of our construction. We conjecture that all functions in this class have a Fourier coefficient of magnitude exp(- poly log n) and prove this conjecture in the case when the MOD2 function is typical. We investigate the relation between the hardness of learning noisy parities and the existence of weak PRFs in AC0 ○ MOD2. We argue that such a complexity-driven approach can play a role in bridging the gap between the theory and practice of cryptography.

The Power of Negations in Cryptography

The study of monotonicity and negation complexity for Bool-ean functions has been prevalent in complexity theory as well as in computational learning theory, but little attention has been given to it in the cryptographic context. Recently, Goldreich and Izsak (2012) have initiated a study of whether cryptographic primitives can be monotone, and showed that one-way functions can be monotone (assuming they exist), but a pseudorandom generator cannot.

Sparse extractor families for all the entropy

We consider the problem of extracting entropy by sparse transformations, namely functions with a small number of overall input-output dependencies. In contrast to previous works, we seek extractors for essentially all the entropy without any assumption on the underlying distribution beyond a min-entropy requirement. We give two simple constructions of sparse extractor families. These are collections of sparse functions such that for any distribution X on inputs of sufficiently high min-entropy, the output of most functions from the collection on input X is statistically close to uniform. For strong extractor families (i.e., functions in the family do not take additional randomness) we give upper and lower bounds on the sparsity that are tight up to a constant factor for a wide range of min-entropies. We then prove that for some min-entropies weak extractor families can achieve better sparsity. We show how this construction can be used towards more efficient parallel transformation of (non-uniform) one-way functions into pseudorandom generators. More generally, sparse extractor families can be used instead of pairwise independence in various randomized or nonuniform settings where sparsity or preserving locality (i.e., parallelism) is of interest.