Fault-tolerant mosaic encoding in knot-based cryptography

TL;DR

This paper enhances a knot-based cryptographic protocol by introducing a mosaic encoding scheme that improves efficiency and error robustness, enabling fault-tolerant data transmission and unambiguous knot diagram representation.

Contribution

The authors develop a mosaic encoding method using prototiles and Reed-Muller coding, improving efficiency and fault tolerance in knot-based cryptography.

Findings

Mosaic encoding unambiguously represents knot diagrams.

Encoding is fault-tolerant under random 1-bit flips.

Complexity relates polynomially to knot crossings.

Abstract

The cryptographic protocol based on topological knot theory,recently proposed by the authors, is improved for what concerns the efficiency of the encoding of knot diagrams and its error robustness. The standard Dowker-Thistlethwaite code, based on the ordered assignment of two numbers to each crossing of a knot diagram and not unique for some classes of knots, is replaced by a system of eight prototiles (knot mosaics) which, once assembled according to a set of combinatorial rules, reproduces unambiguously any unoriented knot diagram. A Reed-Muller scheme is used to encode with redundancy the eight prototiles into blocks and, once the blank tile is added and suitably encoded, the knot diagram is turned into an N X N mosaic, uniquely associated with a string of length 4 N^2 bits. The complexity of the knot, measured topologically by the number of crossings, is in turn polynomially related to the number of tiles of the associated mosaic, and for knot diagrams of higher complexity the mosaic encoding provides a design of the knot-based protocol which is fault-tolerant under random 1-bit flips. It is also argued that the knot mosaic alphabet might be used in other applications which require high-capacity data transmission.