Statistical investigation of the double random\ud phase encoding technique

摘要

The amplitude-encoding case of the double random phase encoding technique is examined by defining a cost\udfunction as a metric to compare an attempted decryption against the corresponding original input image. For\udthe case when a cipher–text pair has been obtained and the correct decryption key is unknown, an iterative\udattack technique can be employed to ascertain the key. During such an attack the noise in the output field for\udan attempted decryption can be used as a measure of a possible decryption key’s correctness. For relatively\udsmall systems, i.e., systems involving fewer than 55 pixels, the output decryption of every possible key can\udbe examined to evaluate the distribution of the keys in key space in relation to their relative performance when\udcarrying out decryption. However, in order to do this for large systems, checking every single key is currently\udimpractical. One metric used to quantify the correctness of a decryption key is the normalized root mean\udsquared (NRMS) error. The NRMS is a measure of the cumulative intensity difference between the input and\uddecrypted images. We identify a core term in the NRMS, which we refer to as the difference parameter, d.\udExpressions for the expected value (or mean) and variance of d are derived in terms of the mean and variance\udof the output field noise, which is shown to be circular Gaussian. These expressions assume a large sample set\ud(number of pixels and keys). We show that as we increase the number of samples used, the decryption error\udobeys the statistically predicted characteristic values. Finally, we corroborate previously reported simulations\udin the literature by using the statistically derived expressions