How does homomorphic encryption work?

22 Tem 23:36

I tried out the Lattigo library in a university project to add encrypted income without decrypting it; it uses ring operations (e.g., the Paillier scheme) that allow ciphertexts to be summed directly, which is great for aggregated statistics but the computational cost and ciphertext size increase rapidly, limiting its use in real-time applications.

23 Tem 00:02

Homomorphic Encryption (HE) relies on mathematical structures called "rings" or "fields," where addition and multiplication operations on encrypted texts are defined the same way they are executed on the original plaintext. The core idea is to choose a public key that generates encrypted elements such that performing addition or multiplication on two encrypted elements yields the same result as performing the operation on the unencrypted values after decryption. In practice, the most popular algorithms use "lattice-based" structures like the BFV or CKKS schemes; the former supports integer calculations, while the latter allows for approximate floating-point computations and is more commonly used in machine learning. From my experience with an IoT energy consumption monitoring project, we used CKKS to train a predictive model on encrypted household consumption data in the cloud. The result was a 2-3% drop in accuracy compared to using open data, but privacy was guaranteed since the cloud never saw the actual values. Practical limitations include: 1. **Performance overhead** – Encrypted multiplication operations take several times longer than regular operations, so batching is preferred to reduce the number of encryption/decryption cycles. 2. **Key size** – Public keys can reach hundreds of kilobytes, making them cumbersome to transfer over low-speed networks. 3. **Operation restrictions** – Most HE schemes don’t support comparisons (e.g., > or <) unless complex "bootstrapping" techniques are used. If your application requires conditional logic, HE might not be suitable. My practical advice: If your goal is simple aggregate statistics (averages, sums) or training a basic ML model on encrypted data, start with CKKS using a library like Microsoft SEAL or PALISADE, and leverage "ciphertext packing" to bundle multiple values into a single ciphertext. If you need logical operations or ultra-low latency, consider combining HE with Trusted Execution Environments (TEE) to offload some of the computational burden from HE.

23 Tem 02:00

In my experience with libraries like BFV, the scheme relies on polynomial rings and the Learning With Errors (LWE) problem to perform arithmetic operations (addition and multiplication) on encrypted data. However, the computational cost and the increase in ciphertext size are significant, so their practical use is often limited to confidential data analysis in the cloud or algorithms that can tolerate some overhead.

23 Tem 02:32

The core idea behind fully homomorphic encryption (FHE) is that the cryptographic algorithm preserves algebraic operations; that is, if we encrypt two values x and y, performing an addition or multiplication on the ciphertexts yields a ciphertext that, when decrypted, corresponds to x + y or x × y. Mathematically, it relies on rings and groups with "homomorphic" properties, such as residues modulo a large prime (in Paillier schemes) or polynomial rings over finite fields (in BGV and CKKS schemes). In practice, compared to traditional techniques like symmetric encryption + decryption on the server, homomorphic encryption allows data analysis (e.g., summing metrics, training ML models) without ever exposing the plaintext, making it ideal for cloud computing environments or compliance with privacy regulations. However, the computational overhead remains significant: ciphertexts are hundreds or thousands of times larger than the original data, and each operation involves expensive modular exponentiation or Fourier transforms. That’s why, in latency-sensitive applications (e.g., real-time transactions), approaches like secure multiparty computation or trusted hardware enclaves (e.g., Intel SGX) are often preferred—they’re much faster but rely on hardware trust. In short, FHE shines when privacy is the absolute priority and higher computation times are tolerable, but for intensive and fast processing, combining it with other protection techniques is more practical.

23 Tem 04:59

Homomorphic encryption relies on algebraic rings where the encryption operation is a homomorphism: if \(E\) is the encryption function, then \(E(m_1) \circ E(m_2) = E(m_1 \star m_2)\), where \(\circ\) is an operation on ciphertexts (e.g., modular multiplication) and \(\star\) is the corresponding operation on plaintexts (addition or multiplication). In common schemes like BFV or CKKS, lattice structures and the Learning With Errors (LWE) problem are used as the security foundation; the noise added in each operation is controlled so that decryption remains possible as long as the noise level does not exceed a critical threshold. In practice, homomorphism is useful when data must remain encrypted while computations are performed, such as in cloud-based medical data analysis or federated learning without revealing trained models. Compared to Secure Multi-Party Computation (SMPC), homomorphism allows a single entity (the server) to perform calculations without coordinating multiple participants, simplifying the architecture but at a performance cost: computational overhead is typically 10–100× greater than operations on plaintext data, and noise management restricts circuit depth (few multiplication layers). That’s why it’s mainly used today in cases where privacy is critical and latency can be tolerated, while for intensive, real-time processes, SMPC or other secure enclave techniques are usually preferred.