The invention relates to efficient zero knowledge verification of composite statements that involve both arithmetic circuit satisfiability and dependent statements about the validity of public keys (key-statement proofs) simultaneously. The method enables a prover to prove this particular statement in zero-knowledge. More specifically, the invention relates to a computer-implemented method for enabling zero-knowledge proof or verification of a statement (S) in which a prover proves to a verifier that a statement is true while keeping a witness (W) to the statement a secret. The invention also relates to the reciprocal method employed by a verifier who verifies the proof. The method includes the prover sending to the verifier a statement (S) having an arithmetic circuit with m gates and n wires configured to implement a function circuit and determine whether for a given function circuit output (h) and an elliptic curve point (P), the function circuit input (s) to a wire of the function circuit is equal to the corresponding elliptic curve point multiplier (s). The prover also sends individual wire commitments and/or a batched commitment for wires of the circuit, an input for a wire in the arithmetic circuit; and a function circuit output (h). The prover receives from the verifier a challenge value (x) and responding with an opening or additionally sends a proving key (PrK) to the verifier. The statement and the data enables the verifier to determine that the circuit is satisfied and calculate the elliptic curve point (P) and validate the statement, thus determining that the prover holds the witness (W) to the statement.