Optical image encryption using different twiddle factors in the butterfly algorithm of fast Fourier transform

Abstract

We studied the flow of data in the paths of a fast Fourier transform (FFT) and inverse FFT in the butterfly algorithm in order to establish a relationship between the forward and backward data. We show that an FFT image can be encrypted by modifying the data of a path of FFT and decrypted by restoring the data during the inverse FFT. For encryption, the data in a path of FFT are shuffled by enforcing right bit shifts on the bit representation of the data index. Moreover, the phases of the data are changed by multiples of a fixed fraction of 2π; the data index is used as the multiplication constant, and the divisor for the fraction is used as an encryption key. The path number and number of right bit shifts are two other encryption keys. The three integers constitute a set of encryption keys for two-dimensional images. Four different images were encrypted with two sets of encryption keys. The decryptions were performed with partially correct keys in order to test the robustness of our method against brute attacks. The correlation between the original and decrypted images was evaluated using the correlation coefficient.