Computing All MOD-Functions Simultaneously

The fundamental symmetric functions are EXk (equal to 1 if the sum of n input bits is exactly k), THk (the sum is at least k), and MODm,r (the sum is congruent to r modulo m). It is well known that all these functions and in fact any symmetric Boolean function have linear circuit size. Simple counting shows that the circuit complexity of computing n symmetric functions is Ω(n2−o(1)) for almost all tuples of n symmetric functions. It is well-known that all EX and TH functions (i.e., for all 0 ≤ k ≤ n) of n input bits can be computed by linear size circuits. In this short note, we investigate the circuit complexity of computing all MOD functions (for all 1 ≤ m ≤ n). We prove an O(n) upper bound for computing the set of functions {MOD1,r,MOD2,r, . . . ,MODn,r} and an O(n log logn) upper bound for {MODn1,r1 ,MOD n 2,r2 , . . . ,MOD n n,rn}, where r1, r2, . . . , rn are arbitrary integers.