On the entropy of keys derived from laser speckle; statistical properties of Gabor-transformed speckle

Laser speckle has been proposed in a number of papers as a high entropy source of unpredictable bits for use in security applications. Bit strings derived from speckle can be used for a variety of security purposes such as in identification, authentication, anti-counterfeiting, secure key storage, random number generation and tamper protection. The choice of laser speckle as a source of random keys is quite natural, given the chaotic properties of speckle. However, this same chaotic behaviour also causes reproducibility problems. Cryptographic protocols require either zero noise or very low noise in their inputs; hence the issue of error rates is critical to applications of laser speckle in cryptography. Most of the literature uses an error reduction method based on Gabor filtering. Though the method is successful, it has not been thoroughly analysed. In this paper we present a statistical analysis of Gabor-filtered speckle patterns. We introduce a model in which perturbations are described as random phase changes in the source plane. Using this model we compute the second- and fourth-order statistics of Gabor coefficients. We determine the mutual information of perturbed and unperturbed Gabor coefficients and the bit error rate in the derived bit string. The mutual information provides an absolute upper bound on the number of secure bits that can be reproducibly extracted from noisy measurements.