Two modified Gerchberg-Saxton (GS) iterative phase retrieval algorithms are proposed. The first we refer to as the spatial phase perturbation GS algorithm (SPP GSA). The second is a combined GS hybrid input-output algorithm (GS/HIOA). In this paper (Part I), it is demonstrated that the SPP GS and GS/HIO algorithms are both much better at avoiding stagnation during phase retrieval, allowing them to successfully locate superior solutions compared with either the GS or the HIO algorithms. The performances of the SPP GS and GS/HIO algorithms are also compared. Then, the error reduction (ER) algorithm is combined with the HIO algorithm (ER/HIOA) to retrieve the input object image and the phase, given only some knowledge of its extent and the amplitude in the Fourier domain. In Part II, the algorithms developed here are applied to carry out known plaintext and ciphertext attacks on amplitude encoding and phase encoding double random phase encryption systems. Significantly, ER/HIOA is then used to carry out a ciphertext-only attack on AE DRPE systems.