This paper describes a one-way function for use in an image watermark to improve authentication ability. The oneway function is popular in cryptography and can prove encryption security. The existence of the one-way function has not been proved yet, while the public key encryption procedure is widely used in practical commerce. The authors proposed a quasi-one-way function, which is weakly defined for practical use. The differences between the strict one-way-function and the proposed quasi-one-way function are discussed. Applications of the quasi-one-way function to image watermarking are shown. Two watermarking systems with SVD method and Jordan canonical form are tried. The robustness of the proposed watermarking method is high.