Learning Noisy Characters, Multiplication Codes, and Cryptographic Hardcore Predicates

We present results in cryptography, coding theory and sublinear algorithms. In cryptography, we introduce a unifying framework for proving that a Boolean predicate is hardcore for a one-way function and apply it to a broad family of functions and predicates, showing new hardcore predicates for well known one-way function candidates such as RSA and discrete-log as well as reproving old results in an entirely different way. Our proof framework extends the list-decoding method of Goldreich and Levin [38] for showing hardcore predicates, by introducing a new class of error correcting codes and new list-decoding algorithm we develop for these codes. In coding theory, we introduce a novel class of error correcting codes that we name: Multiplication codes (MPC). We develop decoding algorithms for MPC codes, showing they achieve desirable combinatorial and algorithmic properties, including: (1) binary MPC of constant distance and exponential encoding length for which we provide efficient local list decoding and local self correcting algorithms; (2) binary MPC of constant distance and polynomial encoding length for which we provide efficient decoding algorithm in random noise model; (3) binary MPC of constant rate and distance. MPC codes are unique in particular in achieving properties as above while having a large group as their underlying algebraic structure. In sublinear algorithms, we present the SFT algorithm for finding the sparse Fourier approximation of complex multi-dimensional signals in time logarithmic in the signal length. We also present additional algorithms for related settings, differing in the model by which the input signal is given, in the considered approximation measure, and in the class of addressed signals. The sublinear algorithms we present are central components in achieving our results in cryptography and coding theory. Reaching beyond theoretical computer science, we suggest employing our algorithms as tools for performance enhancement in data intensive applications, in particular, we suggest replacing the O(N logN)-time FFT algorithm with our Θ̃(logN)-time SFT algorithm for settings where a sparse approximation suffices. Thesis Supervisor: Shafi Goldwasser Title: RSA Professor of Electrical Engineering and Computer Science