QC-MDPC (quasi cyclic moderate density parity check) code-based McEliece cryptosystems are considered to be one of the candidates for post-quantum cryptography. Decreasing DFR (decoding failure rate) is one of important factor for their security, since recent attacks to these cryptosystems effectively use DFR information. In this paper, we pursue the possibility of optimization-base decoding, concretely we examine ADMM (alternating direction method of multipliers) based method, a recent developing method in optimization theory. Further, RSPA (reproducing sum-product algorithm), which efficiently reuse outputs of SPA (sum-product algorithm) is proposed for further reduction of DFR. By numerical simulations, we show that the proposing scheme shows considerable decrement in DFR compared to the conventional decoding methods such as BF (bit-flipping algorithm) or SPA.

Abstract Visual cryptography (VC) is a powerful technique with high security and requires no PC or device to reconstruct the secret information. Progressive visual cryptography (PVC) is a variation of the VC scheme in which the quality of the reconstructed image is improved by increasing the number of shared images. The previous study focused directly on maximizing the value of the quality in the completely reconstructed image; thus, there is a difference in the quality of the shared images. In this paper, we focus on the aforementioned issue and propose a new approach based on inductive reasoning. Our basic idea is to maximize the quality of the reconstructed images each time the number of shared images increases. We call this method the bottom-up approach. Moreover, hitherto, PVC has been evaluated based on the value of relative difference or by sight. Such evaluation methods are only subjective or difficult to execute without the knowledge of basis matrices. In addition, PVC users cannot easily confirm the effectiveness of their shared images. In this paper, we propose a new information-theoretic evaluation method for PVC, which only uses shared images, to solve the aforementioned problems. Our proposed method can objectively and quantitatively evaluate PVC based on the numerical value, and PVC users can easily confirm the effectiveness of their shared images.