Is SHA-1 conceptually sound?

We argue that if the message expansion code of SHA-1 is replaced by a linear code with a better minimum distance, then the resulting hash function is collision resistant. To support this argument, we characterize the disturbance vectors which are used to build local collision attacks as a linear code. This linear code is the xor-sum of two codes, the message expansion code and a linear code representing the underlying block cipher in SHA-1. We also show that the following constraint satisfaction problem is NP-hard. The constraints are restricted to being XOR constraints, or Majority constraints on at most three variables each. The instances are further restricted by requiring that the constraints can be listed in a sequence C 1 ; C 2 ; ; C m , such that for every constraint C i , two of the variables in it occur only in constraints C j , with jj ? ij < 48. This problem is similar to the problem modeling the one-way function property of SHA-1.