This paper introduces the first formal study of circuit privacy for approximate homomorphic encryption (HE). The authors propose formal models to reason about circuit privacy and show that approximate HE can be made circuit private using differential privacy (DP) techniques with appropriately chosen parameters. They demonstrate that applying exponential Gaussian noise to ciphertexts removes useful information about the circuit, and that the noise parameter must be exponential in the security parameter to ensure circuit privacy. They also extend their analysis to multikey and threshold HE schemes, showing that applying a DP mechanism to partial decryptions provides IND-CPA-style security. The paper also shows that the noise required for circuit privacy is tight, and that lower noise parameters can be exploited by efficient adversaries. The authors conclude that the noise in CKKS-type schemes significantly impacts their precision. The paper addresses the challenge of achieving circuit privacy in approximate HE, which is critical for applications such as private set intersection, neural network inference, and genomic data analysis. The results provide a foundation for secure and efficient approximate HE schemes.This paper introduces the first formal study of circuit privacy for approximate homomorphic encryption (HE). The authors propose formal models to reason about circuit privacy and show that approximate HE can be made circuit private using differential privacy (DP) techniques with appropriately chosen parameters. They demonstrate that applying exponential Gaussian noise to ciphertexts removes useful information about the circuit, and that the noise parameter must be exponential in the security parameter to ensure circuit privacy. They also extend their analysis to multikey and threshold HE schemes, showing that applying a DP mechanism to partial decryptions provides IND-CPA-style security. The paper also shows that the noise required for circuit privacy is tight, and that lower noise parameters can be exploited by efficient adversaries. The authors conclude that the noise in CKKS-type schemes significantly impacts their precision. The paper addresses the challenge of achieving circuit privacy in approximate HE, which is critical for applications such as private set intersection, neural network inference, and genomic data analysis. The results provide a foundation for secure and efficient approximate HE schemes.