A Recursive Definition of the Holographic Standard Signature

TL;DR

This paper introduces a recursive framework for understanding holographic standard signatures, enabling bounds, testing, and counting of signatures over finite fields, advancing the theoretical understanding of holographic algorithms.

Contribution

It provides a recursive characterization of standard signatures, facilitating bounds, testing algorithms, and counting methods over finite fields, which was not previously available.

Findings

Proves tight bounds on planar matchgate sizes

Develops efficient algorithms for testing standard signatures

Counts and analyzes the expected sparsity of signatures over finite fields

Abstract

We provide a recursive description of the signatures realizable on the standard basis by a holographic algorithm. The description allows us to prove tight bounds on the size of planar matchgates and efficiently test for standard signatures. Over finite fields, it allows us to count the number of n-bit standard signatures and calculate their expected sparsity.