Abstract: Cryptographic algorithm is generally based on mathematical difficult problems in order to insure security. At present, the existing mathematical problem can be reduced to solving one or a series of unknown number. Different from the existing mathematical difficult problem, this paper presents a new class of mathematical difficult problem which is to solve unknown function, but not unknown number. There is no feasible method to solve the problem. In this paper, the obstacles are overcome and the security under exhaustive attack is considered, and a block cipher scheme is proposed based on the combination of solving unknown number and unknown function. The function of encryption and decryption are unknown. In the scheme, the unknown function is decided by the key and the codebreaker does not know the key, and hence cannot confirm the cipher function. As the encryption and decryption function is unknown, so precondition of most cryptanalysis method is unknown, and hence the analysis is infeasible. Under the scheme a new one-wayness about the cipher function is formed. The men to encrypt and decrypt can easily determine the cipher function, but the cryptanalyst cannot determine. The scheme has perfect security for the existing cryptanalysis. The conceivable attack methods are imagined, several attack thoughts are given, and the optimization of the cryptosystem for these attacks is given. Aiming at the conceivable attacks, some principled suggestions are given to avoid the concrete form is determined by cryptanalysis using various clues or the unknown function is unified into a known function.