How FHE Works: A Breakdown Of The Mathematical Foundations
It is probably fair to say that Fully Homomorphic Encryption (FHE) is a relatively new and groundbreaking development because of its complicated nature. A practical FHE implementation stands upon the shoulders of several mathematical foundations and scientific principles that have only been brought together in the last 20 years.
This blog post will seek to unravel FHE in a way that shines a clear light on its building blocks. On our way, we will explain each technology’s supporting pillars. Thus, we shall talk through key concepts such as homomorphism, advanced number theory, lattice-based cryptography, computational complexity theory, and more.
FHE Foundations, Turn-By-Turn
Homomorphism
Homomorphism could be called the backbone of FHE, and aptly it is represented by the ‘H’ in the center of the acronym. It is a concept within abstract algebra which refers to a structure-preserving map between two algebraic structures. In the context of FHE, this means that operations (like addition and multiplication) performed on encrypted data yield the same result as if they had been carried out on unencrypted plaintexts, then encrypted.
This fundamental idea enables FHE schemes to process data without ever exposing the raw input. It’s not just a mathematical curiosity—it’s a crucial requirement for privacy-preserving computation.
Number Theory
FHE is deeply rooted in number theory, and particularly the use of modular arithmetic, polynomial rings, and prime fields. These constructs are essential for encoding and manipulating data in ciphertext form.
Many FHE schemes operate over rings of polynomials with coefficients modulo a prime or integer, such as
The algebraic properties of these rings allow FHE systems to implement encrypted operations efficiently while controlling noise growth.
Number theory also underpins ciphertext packing, bootstrapping transformations, and the secure construction of key-switching (such as relinearization). Without this layer of math, FHE wouldn’t be functional or efficient.
Lattice-Based Cryptography
Lattice-based cryptography provides the security foundations for most modern FHE schemes. A lattice is a grid-like structure of points in high-dimensional space. Cryptographic hardness is derived from the difficulty of solving lattice-based problems like:
- The Shortest Vector Problem (SVP) – finding the shortest non-zero vector in a high-dimensional lattice.
- Learning With Errors (LWE) – a problem that involves solving systems of linear equations with small amounts of noise added.
- Ring Learning with Errors (RLWE) – a variation of LWE where polynomials are used instead of vectors.
The above problems are considered hard, even for quantum computers, making lattice-based encryption schemes quantum-resistant. They also enable compact ciphertexts, embrace support for parallelizable operations, and deliver provable security guarantees, all of which are critical for scaling FHE in real-world applications.
Computational Complexity Theory
This area of computer science evaluates how difficult it is to solve certain problems, particularly with regard to the time and resources required. FHE’s security is based on asymmetry in complexity: operations like encryption and decryption are easy (polynomial-time), but attempting to reverse encryption without the secret key (i.e., solving the underlying lattice problems) is infeasible, believed to require super-polynomial or exponential computation time.
Moreover, FHE schemes must carefully balance computational overhead and noise management. Homomorphic operations accumulate noise in ciphertexts, and bootstrapping (a refresh step) is expensive. Understanding this complex landscape allows system designers to optimize performance and ensure security margins are preserved.
FHE: Bringing It All Together
FHE is not a monolithic concept, but rather an intricate synthesis of multiple domains of math and computer science.
- Homomorphism enables operations on ciphertexts;
- Number theory structures the encrypted data space;
- Lattice-based cryptography ensures security and efficiency; and computational complexity theory undergirds both the hardness assumptions and practical performance strategies.
These foundations work in tandem to create what was once considered impossible: performing arbitrary computation on encrypted data while it remains confidential.
In Summary
Being able to process encrypted data without it ever being decrypted yet attaining the same results as if the processing had been carried out on plain texts sounds rather magical. However, we know that it works in the real-world and this experience is backed up by mathematical proof and scientific theory, as outlined above.
Next up, we will be publishing a blog post about the computational cost and other real-world challenges of implementing FHE, and how these will be overcome in the not-too-distant future.