Calculating the number of solutions of a difference equation

The hash algorithms of the MDx family involve cyclic shifts, computation of primitive Boolean functions, and addition of constants. So far, very few works have been published in which the authors attempt to explain the impact that the choice of constants, shifts, and Boolean functions has on the cryptographic properties of the algorithms. G. A. Karpunin and H. T. Nguyen suggested a model in which the resistance against differential cryptanalysis may be quantitatively estimated in terms of the number of solutions of a special equation. In this work, in the framework of the aforementioned model, an equation for the MD5 hash function is derived. Examination of one Boolean function and one value of the cyclic shift through exhaustive search requires 2(128) operations of computation of the step of the hash function. The formulas suggested in the present work allow to reduce the complexity of the examination to 2(44) arithmetic operations.