On the pseudorandomness of the Lai-Massey scheme

The pseudoranness of the Lai-Massey schme is studied in this paper. First, itis proved that if σ is an affine almost orthomorphism, 3-round Lai-Massey scheme can not reach the pseudorandomness, which is a counterexample of the Lai-Massey scheme designer's result. Then, it is proved that at least 3-round Lai-Massey scheme is need for the property of pseudorandomness when the bijective σ is an arbitrary orthomorphism, and at least 4-round Lai-Massey scheme is need for the property of super pseudorandomness when the bijective σ is an orthomorphism. From the above results, it is preferable to design a bijective σ which is nonlinear orthomorphism or almost orthomorphism to construct the Lai-Massey scheme with better pseudorandomness.