A STUDY ON HASH FUNCTIONS

In this paper, we study hash function, which will take a message of arbitrary length and produce a massage digest of a specified size. The message digest will then be signed. We have to be careful that the use of a hash function h does not weaken the security of the signature scheme, for it is the message digest that is signed, not the message. It will be necessary for h to satisfy certain properties in order to prevent various forgeries. In order to prevent various type of attack, we require that hash function satisfy collision-free property. In section 1, we introduce some definitions and collision-free properties of hash function. In section 2, we study a discrete log hash function and introduce the main theorem as follows : Theorem Suppose $h:X{\rightarrow}Z$ is a hash function. For any $z{\in}Z$, let $$h^{-1}(z)={\lbrace}x:h(x)=z{\rbrace}$$ and denote $s_z={\mid}h^{-1}(z){\mid}$. Define $$N={\mid}{\lbrace}{\lbrace}x_1,x_2{\rbrace}:h(x_1)=h(x_2){\rbrace}{\mid}$$. Then (1) $\sum\limits_{z{\in}Z}s_z={\mid}x{\mid}$ and the mean of the $s_z$'s is $\bar{s}=\frac{{\mid}X{\mid}}{{\mid}Z{\mid}}$ (2) $N=\sum\limits_{z{\in}Z}{\small{s_z}}C_2=\frac{1}{2}\sum\limits_{z{\in}Z}S_z{^2}-\frac{{\mid}X{\mid}}{2}$. (2) $\sum\limits_{z{\in}Z}(S_z-\bar{s})^2=2N+{\mid}X{\mid}-\frac{{\mid}X{\mid}^2}{{\mid}Z{\mid}}$.