Compression of the sign information of discrete cosine transform coefficients is an intractable problem in image compression schemes due to the equiprobable occurrence of the sign bits. To overcome this difficulty, we propose an efficient compression method for such sign information based on phase retrieval, which is a classical signal restoration problem attempting to find the phase information of discrete Fourier transform coefficients from their magnitudes. In our compression strategy, the sign bits of all the AC components in the cosine domain are excluded from a bitstream at the encoder and are complemented at the decoder by solving a sign recovery problem, which we call sign retrieval. The experimental results demonstrate that the proposed method outperforms previous techniques for sign compression in terms of a rate-distortion criterion. Our method implemented in Python language is available from https://github.com/ctsutake/sr.