Arithmetic Computation Using Self-Assembly of DNA Tiles: Modular-Square over Finite Field GF(2n)

Abstract

Tile assembly model is a highly distributed parallel model of DNA computing. This paper proposes how the tile assembly process could be used for computing the modular square, an operation combining square 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> ). In this molecular computing system, reduction is executed after the completion of square. The time complexity of this molecular computing system is ⊙(n) and the space complexity is ⊙(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). This system requires 75 types of computation tiles and 9 types of boundary tiles.