Linear congruence from a discrete-logarithm Fourier sample ($c_{1}y\equiv -c_2(\mathrm{mod}M)$)

Let $d=\gcd (c_1,M)$. A supported sample guarantees $d\mid c_2$. Dividing its linear congruence by $d$ gives an invertible coefficient modulo $M/d$ and a unique residue $y_0$ there. The $d$ compatible discrete logarithms modulo $M$ are $y_0+jM/d$ for $0\le j<d$. Only $d=1$ determines the discrete logarithm from the sample alone; $(c_1,c_2)=(0,0)$ conveys no information for $M>1$.