On the Learnability of Monotone Functions

A longstanding lacuna in the field of computational learning theory is the learnability of succinctly representable monotone Boolean functions, i.e., functions that preserve the given order of the input. This thesis makes significant progress towards understanding both the possibilities and the limitations of learning various classes of monotone functions by carefully considering the complexity measures used to evaluate them. We show that Boolean functions computed by polynomial-size monotone circuits are hard to learn assuming the existence of one-way functions. Having shown the hardness of learning general polynomial-size monotone circuits, we show that the class of Boolean functions computed by polynomial-size depth-3 monotone circuits are hard to learn using statistical queries. As a counterpoint, we give a statistical query learning algorithm that can learn random polynomial-size depth-2 monotone circuits (i.e., monotone DNF formulas). As a preliminary step towards a fully polynomial-time, proper learning algorithm for learning polynomial-size monotone decision trees, we also show the relationship between the average depth of a monotone decision tree, its average sensitivity, and its variance. Finally, we return to monotone DNF formulas, and we show that they are teachable (a different model of learning) in the average case. We also show that non-monotone DNF formulas, juntas, and sparse GF2 formulas are teachable in the average case

On pseudorandomization of information-theoretically secure schemes without hardness assumptions

A recent work by Nuida and Hanaoka (in ICITS 2009) provided a proof technique for security of information-theoretically secure cryptographic schemes in which the random input tape is implemented by a pseudorandom generator (PRG). In this paper, we revisit their proof technique and generalize it by introducing some trade-off factor, which involves the original proof technique as a special case and provides a room of improvement of the preceding result. Secondly, we consider two issues of the preceding result; one is the requirement of some hardness assumption in their proof; another is the gap between non-uniform and uniform computational models appearing when transferring from the exact security formulation adopted in the preceding result to the usual asymptotic security. We point out that these two issues can be resolved by using a PRG proposed by Impagliazzo, Nisan and Wigderson (in STOC 1994) against memory-bounded distinguishers, instead of usual PRGs against time-bounded distinguishers. We also give a precise formulation of a computational model explained by Impagliazzo et al., and by using this, perform a numerical comparison showing that, despite the significant advantage of removing hardness assumptions, our result is still better than, or at least competitive to, the preceding result from quantitative viewpoints. The results of this paper would suggest a new motivation to use PRGs against distinguishers with computational constraints other than time complexity in practical situations rather than just theoretical works. Keywords: Information-theoretic security, pseudorandomization, unconditional security, Impagliazzo– Nisan–Wigderson pseudorandom generato