Homomorphic encryption (HE) allows secure computation on encrypted data without revealing the original data, providing significant benefits for privacy-sensitive applications. Many cloud computing applications (e.g., DNA read mapping, biometric matching, web search) use exact string matching as a key operation. However, prior string matching algorithms that use homomorphic encryption are limited by high computational latency caused by the use of complex operations and data movement bottlenecks due to the large encrypted data size. In this work, we provide an efficient algorithm-hardware codesign to accelerate HE-based secure exact string matching. We propose CIPHERMATCH, which (i) reduces the increase in memory footprint after encryption using an optimized software-based data packing scheme, (ii) eliminates the use of costly homomorphic operations (e.g., multiplication and rotation), and (iii) reduces data movement by designing a new in-flash processing (IFP) architecture. We demonstrate the benefits of CIPHERMATCH using two case studies: (1) Exact DNA string matching and (2) encrypted database search. Our pure software-based CIPHERMATCH implementation that uses our memory-efficient data packing scheme improves performance and reduces energy consumption by 42.9X and 17.6X, respectively, compared to the state-of-the-art software baseline. Integrating CIPHERMATCH with IFP improves performance and reduces energy consumption by 136.9X and 256.4X, respectively, compared to the software-based CIPHERMATCH implementation.