Modular differential attack is one of the important methods in SHA-1 collision attacks. In allusion to width-reduced SHA-1 algorithm, the width-reduced local collisions theorems are deduced in this paper, and the probability of local collisions in width-reduced SHA-1 algorithm is proposed. Thus, modular differential attack is proved to be appropriate for width-reduced SHA-1 algorithm. By means of theoretical analysis and computer search, it is proved that for n-bit SHA-1 algorithm, the hamming weight of the optimum disturbance vector is 25 if and only if n>3; and when and only when n>8, the optimum disturbance vector has no other equivalence class than type- I and type-II.
