Parallel molecular computation of modular-multiplication based on tile assembly model

Abstract

DNA computing is a new method for computation using the technology in molecular biology. The enormous parallel computing ability of DNA computing brings new opportunities and challenges to the development of cryptography. DNA cryptography is a cutting-edge sciences which combines classical cryptogram and molecular computing. Finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) is one of the most commonly used mathematic sets for cryptography. This paper proposes a parallel molecular computing system to compute the modular-multiplication, an operation combining multiplication and reduction, over finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ). The operation of reduction is executed after the completion of the operation of multiplication. An instance of computing modular-multiplication is introduced to show the details of our system. The time complexity is Θ(n) and the space complexity is Θ(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ).