Some properties of the dot product over the free module Z_m~n
Zhao Ya-qun · Journal of Shandong University of Technology · 2003
Discusses some properties of the dot product on the free module Z n m with rank n over the ring Z m of residue modulo-m. With the dot product, a Z-module homomorphism η: E→Z m is defined, where E is a submodule of Z n m, kerη,E/kerη and some properties of η -1 are presented in the paper. And another Z-module homomorphism ρ:Z n m→Z k m(1kn) is also defined with the dot product, kerρ and Z n m/kerρ are given, corresponding properties of ρ -1 are discussed in the paper. In addition, the orthogonal submodule E ⊥ of E is defined by E ⊥={x∈Z n m|x·y=0,y∈E}, correlation properties of m-th root of unity u m over E are studied, and the formula (w∈Z n m)∑x∈Eu w·x m=|E|,w∈E ⊥, 0,w≠E ⊥. is obtained in the paper, which is very useful in the study of logical functions of cryptology.