A DNA Computing System of Modular-Multiplication over Finite Field GF(2n)

Abstract

The enormous parallel computing ability and high memory density of DNA computing bring potential challenges and opportunities to traditional cryptography. Finite field GF(2n) is one of the most commonly used mathematic sets for cryptography. It is an open problem that how to implement the arithmetic operations over finite field GF(2n) based on DNA computing. Existing research has the problem that the lengths of parameters in the DNA tile assembly process could not match each other strictly. This paper proposes a parallel molecular computing system to compute the modular-multiplication, an operation combining multiplication and reduction over finite field GF(2n). The multiplication and the reduction are executed simultaneously in this system. One concrete example of \(1100 \cdot 1001 \ mod \ 10011\) is proposed to show the details of our tile assembly system. The time complexity of this system is Θ(n) and the space complexity is Θ(n 2). This system requires 210 types of computation tiles and 17 types of boundary tiles.KeywordsModular-multiplicationTile assembly modelTile assembly systemFinite field GF(2n)DNA tile