Paper 2006/135 Original: eprint.iacr.org/2006/135 ↗

The Design Principle of Hash Function with Merkle-Damgård Construction

Abstract

The paper discusses the security of hash function with Merkle-Damgård construction and provides the complexity bound of finding a collision and primage of hash function based on the condition probability of compression function $y=F(x,k)$. we make a conclusion that in Merkle-Dammaård construction, the requirement of free start collision resistant and free start collision resistant on compression function is not necessary and it is enough if the compression function with properties of fix start collision resistant and fix start preimage resistant. However, the condition probability $P_{Y|X=x}(y)$ and $P_{Y|K=k}(y)$ of compression function $y=F(x,k)$ have much influence on the security of the hash function. The best design of compression function should have properties of that $P_{Y|X=x}(y)$ and $P_{Y|K=k}(y)$ are both uniformly distributed for all $x$ and $k$. At the end of the paper, we discussed the block cipher based hash function, point out among the the 20 schemes, selected by PGV\cite{Re:Preneel} and BPS\cite{Re:JBlack}, the best scheme is block cipher itself, if the block cipher with perfect security and perfect key distribution.