Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a1x1 + · · · + akxk ≡ b (mod n), with (xi, n) = ti (1 ≤ i ≤ k), where a1, t1, . . . , ak, tk, b, n (n ≥ 1) are arbitrary integers. Some special cases of this problem have been studied in many papers, and have found very interesting applications in number theory, combinatorics, and cryptography, among other areas. We also propose an authenticated encryption scheme, and using our explicit formula, analyze the integrity of this scheme.