An Alternative to Arithmetic Coding with Local Decodability

We describe a simple, but powerful local encoding technique, implying two surprising results: 1. We show how to represent a vector of n values from Σ using dn log2 Σe bits, such that reading or writing any entry takes O(1) time. This demonstrates, for instance, an “equivalence” between decimal and binary computers, and has been a central toy problem in the field of succinct data structures. Previous solutions required space of n log2 Σ + n/ lg O(1) n bits for constant access. 2. Given a stream of n bits arriving online (for any n, not known in advance), we can output a prefix-free encoding that uses n+log2 n+O(lg lgn) bits. The encoding and decoding algorithms only require O(lg n) bits of memory, and run in constant time per word. This result is interesting in cryptographic applications, as prefix-free codes are the simplest counter-measure to extensions attacks on hash functions, message authentication codes and pseudorandom functions. Our result refutes a conjecture of [Maurer, Sjödin 2005] on the hardness of online prefix-free encodings.