Many modern applications that deal with sensitive data, such as healthcare and government services, outsource computation to cloud platforms. In such untrusted environments, privacy is of vital importance. One solution to this problem is homomorphic encryption (HE), a family of cryptographic schemes that support certain algebraic operations on encrypted data without the need for decryption. However, despite major advancements, encryption in modern HE schemes still comes with a non-trivial computational overhead that can hamper data-intensive workloads. To resolve this, recent research has shown that leveraging caching techniques, such as Rache, can significantly enhance the performance of HE schemes while maintaining security. Rache unfortunately displays a key limitation in the time complexity of its caching procedure, which scales with the size of the plaintext space. Smuche is another caching scheme that simultaneously improves the scalability of the caching procedure and turns the encryption process into a constant-time operation, utilizing only a single scalar multiplication. Even still, more can be done. In this paper, we present an encryption method we call ``Zinc" which entirely forgoes the multiple caching process, replacing it with a single scalar addition, and then injecting randomness that takes constant time with respect to the plaintext space. This injection of randomness is similar to Smuche, and a great improvement from Rache, allowing Zinc to achieve efficiency without compromising security. We implement the scheme using Microsoft SEAL and compare its performance to vanilla CKKS.