On CC⁰ Lower Bounds for AND via Torus Polynomials

Abstract

We explore the torus polynomial approximation based approach towards a long-standing question: whether AND can be computed by CC⁰ circuits - the class of constant-depth polynomial size circuits containing MOD_m gates for some natural number m. Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach for proving lower bounds against ACC⁰ - a class containing CC⁰ where the circuits are also allowed AND, OR and NOT gates. 
We show how lower bounds for torus polynomials approximating AND can be used to make progress on this question. Using lower bounds on the degree of symmetric torus polynomials approximating AND, proved by Krishan and Vishwanathan (ITCS 2026), we prove size lower bounds for symmetric CC⁰-circuits computing AND. More precisely, we prove that any depth h symmetric CC⁰ circuit requires 2^Ω̃(n^{1/O(h)}) size to compute AND. 
A key ingredient in our proof is an argument that we can construct symmetric torus polynomials to approximate symmetric CC⁰ circuits. Our construction exhibits an explicit correspondence between the symmetry of the circuit and that of the polynomial. Using this, we also establish lower bounds for weaker notions of circuit symmetry. Lower bounds for symmetric CC⁰ circuits were also independently established by Pago (ICALP 2026) using different techniques.
In the asymmetric regime, we establish degree upper bounds for depth three circuits of the form MOD_p∘MOD_m∘AND_O(1) where m = pq is a semiprime. This circuit class is a special case of the constant degree hypothesis, introduced by Barrington, Straubing and Thérien (Information and Computation, 1990), where m could be an arbitrary composite number. We argue that improved lower bounds for asymmetric torus polynomials approximating AND imply size lower bounds for semiprime m and hence progress on the constant-degree hypothesis.