Fragile watermarking is a technique of authenticating the originality of the media (e. g., image). Although the watermark is destroyed with any small modi cation (tamper), it may be used to recover the original image. There is no method yet, based on our knowledge, to guarantee the Perfect recovery of small tampers. Although data-bits are embedded in Least Signi cant Bits of some other pixel(s), a tamper may destroy both data and authentication sets which makes recovery impossible. In this paper, a novel fragile watermarking scheme is proposed for both tamper detection and tampered image recovery. Here, all bits are reorganized in virtual pixels distributed in the image called as Distributed Pixels (DP). Distance of each pair of bits in a DP is su ciently large. This is why; tampers smaller than a threshold, cannot destroy more than one bit of a DP. Hamming code guarantees that changing at most one bit can be Perfectly detected and recovered. Then, Hamming(7, 4) is extended to (8, 5) to support embedding in eight-bits pixels. According to the experimental results, the proposed method could Perfectly detect and recover the tampered parts not greater than a quarter of image in diameter. It also achieved acceptable performance in other conditions, compared to state-of-the-art methods.

In this paper, we study Perfect 2-coloring of the quartic graphs with at most 8 vertices. The problem of the existence of Perfect coloring is a generalization of the concept of completely regular codes, given by Delsarte.