Deterministic oblivious distribution (and tight compaction) in linear time

TL;DR

This paper presents a deterministic, oblivious algorithm for array permutation and tight compaction that operates in linear time, resolving open questions and improving bounds for cryptographic and data organization tasks.

Contribution

It introduces a deterministic, oblivious linear-time algorithm for array permutation and tight compaction, answering an open problem and enhancing existing bounds.

Findings

Deterministic oblivious permutation in O(N) time

Deterministic oblivious tight compaction in O(N) time

Improved bounds for randomized and deterministic compaction algorithms

Abstract

In an array of N elements, M positions and M elements are "marked". We show how to permute the elements in the array so that all marked elements end in marked positions, in time O(N) (in the standard word-RAM model), deterministically, and obliviously - i.e. with a sequence of memory accesses that depends only on N and not on which elements or positions are marked. As a corollary, we answer affirmatively to an open question about the existence of a deterministic oblivious algorithm with O(N) running time for tight compaction (move the M marked elements to the first M positions of the array), a building block for several cryptographic constructions. Our O(N) result improves the running-time upper bounds for deterministic tight compaction, for randomized tight compaction, and for the simpler problem of randomized loose compaction (move the M marked elements to the first O(M) positions) - until now respectively O(N lg N), O(N lg lg N), and O(N lg*N).