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A straightforward way to prove this when you can prove AND as well as OR statements about discrete logarithms is to take all the $K=\binom{M}{N}$ subsets $A_i=\{A_{i_1},\ldots,A_{i_N}\}$ with $N$ elements of points from the set of your $M$ points and prove the statement $$PK\{(\alpha_1,\ldots,\alpha_N): \bigvee_{j\in K} \big( \bigwedge_{A_{j_i}\in A_j} ...



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