In this application, example methods for performing quantum Montgomery arithmetic are disclosed. Additionally, circuit implementations are disclosed for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. This application also shows that elliptic curve discrete logarithms on an elliptic curve defined over an n-bit prime field can be computed on a quantum computer with at most 9n+2┌log2(n)┐+10 qubits using a quantum circuit of at most 512n3 log2(n)+3572n3 Toffoli gates.