Constant-depth circuits for arithmetic in finite fields of characteristic two

We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,...,αt ∈ F2 n, compute α1 · α2 · · ·αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a t-bit integer k, compute αk ∈ F2 n

Delegating computation reliably : paradigms and constructions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 285-297).In an emerging computing paradigm, computational capabilities, from processing power to storage capacities, are offered to users over communication networks as a service. This new paradigm holds enormous promise for increasing the utility of computationally weak devices. A natural approach is for weak devices to delegate expensive tasks, such as storing a large file or running a complex computation, to more powerful entities (say servers) connected to the same network. While the delegation approach seems promising, it raises an immediate concern: when and how can a weak device verify that a computational task was completed correctly? This practically motivated question touches on foundational questions in cryptography and complexity theory. The focus of this thesis is verifying the correctness of delegated computations. We construct efficient protocols (interactive proofs) for delegating computational tasks. In particular, we present: e A protocol for delegating any computation, where the work needed to verify the correctness of the output is linear in the input length, polynomial in the computation's depth, and only poly-logarithmic in the computation's size. The space needed for verification is only logarithmic in the computation size. Thus, for any computation of polynomial size and poly-logarithmic depth (the rich complexity class N/C), the work required to verify the correctness of the output is only quasi-linear in the input length. The work required to prove the output's correctness is only polynomial in the original computation's size. This protocol also has applications to constructing one-round arguments for delegating computation, and efficient zero-knowledge proofs. * A general transformation, reducing the parallel running time (or computation depth) of the verifier in protocols for delegating computation (interactive proofs) to be constant. Next, we explore the power of the delegation paradigm in settings where mutually distrustful parties interact. In particular, we consider the settings of checking the correctness of computer programs and of designing error-correcting codes. We show: * A new methodology for checking the correctness of programs (program checking), in which work is delegated from the program checker to the untrusted program being checked. Using this methodology we obtain program checkers for an entire complexity class (the class of N/C¹-computations that are WNC-hard), and for a slew of specific functions such as matrix multiplication, inversion, determinant and rank, as well as graph functions such as connectivity, perfect matching and bounded-degree graph isomorphism. * A methodology for designing error-correcting codes with efficient decoding procedures, in which work is delegated from the decoder to the encoder. We use this methodology to obtain constant-depth (AC⁰) locally decodable and locally-list decodable codes. We also show that the parameters of these codes are optimal (up to polynomial factors) for constant-depth decoding.by Guy N. Rothblum.Ph.D

Electronic Colloquium on Computational Complexity, Report No. 87 (2005) Constant-Depth Circuits for Arithmetic in Finite Fields of Characteristic Two

We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,..., αt ∈ F2 n, compute α1 · α2 · · · αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a t-bit integer k, compute αk ∈ F2 n