The Relationship between the Cryptographic Function over Finite Field and Vector Function over Its Prime Field

The paper firstly reveals the relationship between the cryptographic function over finite field and vector function over its prime field. Then a class of functions taking all vectors as their linear structures is suggested, which are the weakest functions over finite field even if the algebraic degree is high. At last, the cryptographic properties of logical function over finite field and the corresponding vector function over its prime field are explored, including the Chrestenson spectra characteristics and correlation immunity, which is a significant criterion for cryptographic function to resist correlation attack.