How to accomplish the basic operations in cryptosystem becomes a hot research direction in the field of DNA-based cryptography. This paper proposes a DNA computing model that could fulfill parallel computation of point-addition and point-doubling, two fundamental point operations in conic curves cryptosystem, using tile self-assembly. The combination of the two operations is a key step to compute point-multiplication, a point operation to generate intractability of conic curve discrete logarithm problem. Point-addition is deduced by division in one sub-model and another sub-model calculates point-doubling directly and generates the parameters for division. The assembly time complexity of this model is \(2n^2+3n-3\) , and the space complexity is \(n^4+3n^3-3n+1\) .