What Is Homomorphic Encryption?

Homomorphic encryption is a form of public-key cryptography that permits arithmetic operations to be performed directly on ciphertext, such that decrypting the result yields the same value as performing the equivalent operations on the original plaintext. This property allows a data owner to send encrypted data to an untrusted computing environment, obtain a result, and decrypt it locally without ever exposing the underlying information to the party performing the computation. The field draws on abstract algebra, computational complexity theory, lattice-based cryptography, and applied mathematics.

The concept of homomorphic encryption was identified as a theoretical goal in 1978 by Rivest, Adleman, and Dertouzos, shortly after the publication of the RSA public-key cryptosystem, but a construction supporting arbitrary computations was not demonstrated until Craig Gentry's doctoral thesis in 2009. Gentry's work introduced the first fully homomorphic encryption scheme, relying on ideal lattice mathematics and a technique called bootstrapping to refresh the noise that accumulates in ciphertext under repeated operations. That breakthrough catalyzed a substantial research program aimed at reducing the computational cost of fully homomorphic encryption to levels suitable for practical deployment.

Partially and Fully Homomorphic Encryption

Homomorphic encryption schemes are classified by the class of computations they support. Partially homomorphic encryption (PHE) supports only a single type of operation, either addition or multiplication, an unlimited number of times. RSA, for example, is multiplicatively homomorphic over the integers. Somewhat homomorphic encryption (SHE) supports a limited number of both addition and multiplication operations before noise in the ciphertext exceeds a correctable threshold. Fully homomorphic encryption (FHE) extends SHE by incorporating a bootstrapping step that refreshes the ciphertext and allows an unbounded number of operations of both types, which is equivalent to arbitrary computable functions. The practical distinction matters because FHE is the only form that enables general-purpose computation, including the evaluation of arbitrary machine learning inference or database query, on encrypted data.

Computational Architecture and Performance

Fully homomorphic encryption carries a significant performance overhead relative to conventional cryptographic operations. Bootstrapping, which involves evaluating the decryption circuit homomorphically, is computationally intensive: early FHE implementations were approximately one million times slower than plaintext computation, rendering them impractical for most real-time applications. Research in the years following Gentry's construction has produced several more efficient schemes, including BGV (Brakerski-Gentry-Vaikuntanathan), BFV (Brakerski-Fan-Vercauteren), and CKKS (Cheon-Kim-Kim-Song), which is optimized for approximate arithmetic on real-valued data and is well-suited to machine learning inference. Hardware acceleration using graphics processing units, field-programmable gate arrays, and purpose-built accelerator chips has reduced latency by several orders of magnitude, and the NIST post-quantum cryptography standardization effort has highlighted lattice-based cryptographic schemes as candidates that share structural properties with FHE constructions.

Cloud and Data Privacy Applications

The primary motivation for homomorphic encryption is enabling computation in environments where the data owner cannot fully trust the computing infrastructure. Cloud service providers can perform analytics on customer data that remains encrypted throughout, giving customers cryptographic guarantees of confidentiality rather than relying solely on contractual or policy controls. In healthcare, homomorphic encryption allows a hospital to outsource genomic analysis or diagnostic inference to a cloud platform without exposing patient records in decrypted form. Financial institutions can compute risk models or fraud detection scores on encrypted transaction histories. The Cloud Security Alliance's analysis of FHE versus confidential computing outlines the complementary roles these two privacy-preserving technologies play in cloud deployments.

Applications

Homomorphic encryption has applications in a range of privacy-sensitive computing contexts, including:

• Secure machine learning inference, allowing models to operate on encrypted input data from users
• Privacy-preserving genomics and clinical data analysis in distributed healthcare research
• Encrypted financial computation, supporting fraud detection and risk modeling without exposing raw transaction data
• Secure multiparty computation, enabling collaborative analytics across organizations that cannot share underlying data
• Election systems, where encrypted ballots can be tallied without decryption of individual votes