FHE Schemes Deep Dive: BFV, CKKS, and the Trade-offs in Approximate vs. Exact Computation
Introduction
Homomorphic encryption (FHE) schemes have revolutionized the way we think about secure computation. By enabling computations to be performed directly on encrypted data, FHE has opened up new possibilities for secure data analysis, machine learning, and more. However, not all FHE schemes are created equal, and the choice of scheme can have significant implications for performance, security, and usability. In this post, we'll take a deep dive into two of the most widely used FHE schemes, BFV and CKKS, and explore the trade-offs between approximate and exact computation.
The BFV Scheme
The Brakerski-Fan-Vercauteren (BFV) scheme is one of the most widely used FHE schemes, known for its high performance and exact integer arithmetic capabilities. BFV is based on the hardness of the ring-learning-with-errors (RLWE) problem, and it uses a combination of polynomial and ring operations to enable homomorphic computation.
The RLWE Problem
The RLWE problem is defined as follows: given a ring element $\mathbf{a}$ and an error term $\mathbf{e}$, find the closest ring element $\mathbf{x}$ such that $\mathbf{a} \cdot \mathbf{x} \equiv \mathbf{e} \pmod{q}$, where $q$ is a large prime number. The RLWE problem is a variant of the learning with errors (LWE) problem, which is known to be hard to solve on average.
The BFV Scheme
The BFV scheme uses the RLWE problem to construct a public-key cryptosystem. The public key consists of a ring element $\mathbf{a}$, and the private key consists of a ring element $\mathbf{x}$ that satisfies the RLWE problem. To encrypt a message, the message is represented as a polynomial $m(x) = \sum_{i=0}^{n-1} m_i x^i$, where $n$ is the degree of the polynomial. The ciphertext is then computed as $c = \mathbf{a} \cdot m(x) + \mathbf{e}$, where $\mathbf{e}$ is an error term.
Exact Integer Arithmetic
One of the key features of the BFV scheme is its ability to perform exact integer arithmetic. This is achieved through the use of a technique called "modular arithmetic", which enables the computation of polynomial products and divisions modulo a large prime number. This allows the BFV scheme to support exact integer arithmetic, which is essential for many applications such as financial calculations and cryptographic protocols.
Limitations of the BFV Scheme
While the BFV scheme is highly performant and provides exact integer arithmetic, it has some limitations. For example, the BFV scheme is not suitable for applications that require floating-point operations or approximate arithmetic. Additionally, the BFV scheme is not as secure as some other FHE schemes, as it is based on the RLWE problem, which is considered to be less secure than other problems such as the learning with errors (LWE) problem.
The CKKS Scheme
The Cheon-Kim-Kim-Song (CKKS) scheme is another widely used FHE scheme, known for its ability to support approximate arithmetic and floating-point operations. CKKS is based on the hardness of the ring-learning-with-errors (RLWE) problem, similar to the BFV scheme.
Approximate Arithmetic
The CKKS scheme uses a technique called "approximate arithmetic" to enable floating-point operations. This involves representing floating-point numbers as polynomials with rational coefficients, and then performing arithmetic operations on these polynomials. The resulting ciphertexts are then approximated to the nearest integer, allowing for approximate arithmetic operations to be performed.
Floating-Point Operations
The CKKS scheme supports floating-point operations through the use of a technique called "scaling and rounding". This involves scaling the ciphertexts by a large factor, and then rounding the resulting values to the nearest integer. This allows for the computation of floating-point operations such as addition and multiplication.
Applications of the CKKS Scheme
The CKKS scheme is well-suited for applications that require floating-point operations or approximate arithmetic, such as machine learning and statistical analysis. The CKKS scheme has been used in a variety of applications, including secure machine learning, secure data analysis, and secure scientific computing.
Limitations of the CKKS Scheme
While the CKKS scheme is highly performant and provides approximate arithmetic capabilities, it has some limitations. For example, the CKKS scheme is not suitable for applications that require exact integer arithmetic. Additionally, the CKKS scheme is less secure than some other FHE schemes, as it is based on the RLWE problem, which is considered to be less secure than other problems such as the learning with errors (LWE) problem.
Conclusion
In this post, we've explored the trade-offs between approximate and exact computation in FHE schemes. We've seen how the BFV scheme provides exact integer arithmetic capabilities, but is not suitable for applications that require floating-point operations or approximate arithmetic. We've also seen how the CKKS scheme provides approximate arithmetic capabilities, but is not suitable for applications that require exact integer arithmetic. By understanding the trade-offs between approximate and exact computation, developers can choose the most suitable FHE scheme for their specific application.
Code Examples
Here are some code examples in Python using the HElib library:
from helib import Ciphertext, Plaintext, CKKS, BFV
# Create a CKKS ciphertext
plaintext = Plaintext([1.0, 2.0, 3.0])
ciphertext = CKKS.encrypt(plaintext, 128)
# Perform approximate arithmetic operations
ciphertext.add(ciphertext)
ciphertext.multiply(ciphertext)
# Create a BFV ciphertext
plaintext = Plaintext([1, 2, 3])
ciphertext = BFV.encrypt(plaintext, 128)
# Perform exact integer arithmetic operations
ciphertext.add(ciphertext)
ciphertext.multiply(ciphertext)
Best Practices
When choosing an FHE scheme, it's essential to consider the trade-offs between approximate and exact computation. Here are some best practices to keep in mind:
- Choose the BFV scheme for applications that require exact integer arithmetic.
- Choose the CKKS scheme for applications that require floating-point operations or approximate arithmetic.
- Consider the security implications of each scheme and choose the most secure option.
- Consider the performance implications of each scheme and choose the most performant option.
- Consider the usability implications of each scheme and choose the most user-friendly option.
By following these best practices, developers can choose the most suitable FHE scheme for their specific application and ensure the security and performance of their system.