Let q=pr, where p is a prime number and ß=(β1,…,βr) be a basis of Fq over Fp. Any ξ∈Fq has a unique representation ξ=∑ri=1xiβi with x1,…,xr∈Fp. The coefficients x1,…,xr are called the digits of ξ with respect to the basis ß. The analog of the Rudin-Shapiro function is R(ξ)=x1x2+⋯+xr−1xr. For f∈Fq[X], non constant and c∈Fp, we obtain some formulas for the number of solutions in Fq of R(f(ξ))=c. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in Fp with several variables.