Advances in network communications have necessitated secure local-storage and transmission of multimedia content. In particular, military networks need to securely store sensitive imagery which at a later stage may be transmitted over bandwidth-constrained wireless networks. This work investigates compression efficiency of JPEG and JPEG 2000 standards for encrypted images. An encryption technique proposed by Kuo <i>et al.</i> in  is employed. The technique scrambles the phase spectrum of an image by addition of the phase of an all-pass pre-filter. The post-filter inverts the encryption process, provided the correct pseudo-random filter coefficients are available at the receiver. Additional benefits of pre/post-filter encryption include the prevention of blocking effects and better robustness to channel noise . Since both JPEG and JPEG 2000 exploit spatial and perceptual redundancies for compression, pre/post-filtered (encrypted) images are susceptible to compression inefficiencies. The PSNR difference between the unencrypted and pre/post-filtered images after decompression is determined for various compression rates. Compression efficiency decreases with an increase in compression rate. For JPEG and JPEG 2000 compression rates between 0.5 to 2.5 bpp, the difference in PSNR is negligible. Partial encryption is proposed wherein a subset of image phase coefficients are scrambled. Due to the phase sensitivity of images, even partial scrambling of the phase information results in unintelligible data. The effect of compression on partially encrypted images is observed for various bit-rates. When 25% of image phase coefficients are scrambled, the JPEG and JPEG 2000 compression performance of encrypted images is almost similar to that of unencrypted images for compression rates in the 0.5 to 3.5 bpp range.