Fast Parallel Molecular Algorithms for DNA-Based Computation: Solving the Elliptic Curve Discrete Logarithm Problem over GF(2n)

The analogs based on elliptic curve over finite fields of public-key crypto systems are algorithms that converts input data to an unrecognizable encryption and converts the unrecognizable data back into its original decryption form. The security of the elliptic curve public-key cryptosystem is based on the difficulty of the discrete logarithm problem on elliptic curve, especially over GF(2ⁿ), n isin Z⁺ . This paper demonstrates to find the discrete logarithm on elliptic curve, and is a breakthrough in basic biological operations using a molecular computer. In order to achieve this, we propose three DNA-based algorithms for parallel adder, parallel multiplier, and parallel getting inverse over GF(2ⁿ). The biological operation time of these algorithms are all polynomial with respect to n. This work indicates that the cryptosystems using public-key are perhaps insecure and also presents clear evidence of the ability of molecular computing to perform complicated mathematical operations.