Fast arithmetic computing with neural networks

The authors introduce a restricted model of a neuron which is more practical as a model of computation then the classical model of a neuron. The authors define a model of neural networks as a feedforward network of such neurons. Whereas any logic circuit of polynomial size (in n) that computes the product of two n-bit numbers requires unbounded delay, such computations can be done in a neural network with constant delay. The authors improve some known results by showing that the product of two n-bit numbers and sorting of n n-bit numbers can both be computed by a polynomial size neural network using only four unit delays, independent of n . Moreover, the weights of each threshold element in the neural networks require only O(log n)-bit (instead of n-bit) accuracy

On Low-End Obfuscation and Learning

Most recent works on cryptographic obfuscation focus on the high-end regime of obfuscating general circuits while guaranteeing computational indistinguishability between functionally equivalent circuits. Motivated by the goals of simplicity and efficiency, we initiate a systematic study of "low-end" obfuscation, focusing on simpler representation models and information-theoretic notions of security. We obtain the following results. - Positive results via "white-box" learning. We present a general technique for obtaining perfect indistinguishability obfuscation from exact learning algorithms that are given restricted access to the representation of the input function. We demonstrate the usefulness of this approach by obtaining simple obfuscation for decision trees and multilinear read-k arithmetic formulas. - Negative results via PAC learning. A proper obfuscation scheme obfuscates programs from a class C by programs from the same class. Assuming the existence of one-way functions, we show that there is no proper indistinguishability obfuscation scheme for k-CNF formulas for any constant k ? 3; in fact, even obfuscating 3-CNF by k-CNF is impossible. This result applies even to computationally secure obfuscation, and makes an unexpected use of PAC learning in the context of negative results for obfuscation. - Separations. We study the relations between different information-theoretic notions of indistinguishability obfuscation, giving cryptographic evidence for separations between them

Neural computation of arithmetic functions

A neuron is modeled as a linear threshold gate, and the network architecture considered is the layered feedforward network. It is shown how common arithmetic functions such as multiplication and sorting can be efficiently computed in a shallow neural network. Some known results are improved by showing that the product of two n-bit numbers and sorting of n n-bit numbers can be computed by a polynomial-size neural network using only four and five unit delays, respectively. Moreover, the weights of each threshold element in the neural networks require O(log n)-bit (instead of n -bit) accuracy. These results can be extended to more complicated functions such as multiple products, division, rational functions, and approximation of analytic functions