A form ula fora specialsolution to nonhom ogeneous linear recurrence equations w ith constant coefficients is derived from a generalorder-reducing form ula presented here.

We give a theoretical basis for special solution of the linear non-homogeneous recursion equation with constant coefficient.

Invariable coefficient the number of times is different linear recursion sequence will transform using the sequence difference as often the coefficient inhomogeneous linear difference equation,Thus obtains one kind of computation Invariable coefficient the number of times is different linear recursion sequence special another interpretation simple method.

This paper principally reviews the identity authentication protocol and message authentication protocol based on nonhomogeneous linear equations in [1],analyzes the security defects in these two protocols in combination with [2],and modifies them by introducing in trap-door one-way function.

The problem two fewer and one more in information hiding was transformed to find the solution vector,which had the least weight,of the nonhomogeneous linear equations whose elements amount wasn t fixed,on binary domain,built a mathematical model,and gave the solution models.

By introducing first the concept of coset in the sub-space of linear space,we have testified an im- portant property of the coset and finally give the necessary and sufficient condition for the same solution of the solva- ble nonhomogeneous linear equations.