Demystifying Goldschmidt Division with CKKS: A Step-by-Step Guide
The world of cryptography is constantly evolving, with new techniques emerging to enhance security and efficiency. One such technique is homomorphic encryption, which allows computations on encrypted data without decryption. This powerful tool opens doors for secure data processing in sensitive environments.
Among the various homomorphic encryption schemes, CKKS (Cheon-Kim-Kim-Song) stands out for its ability to handle real numbers with high precision. This makes it ideal for applications requiring numerical computations, including division, which is a fundamental operation in many algorithms.
This article explores the implementation of Goldschmidt division using CKKS. We will delve into the core concepts, code examples, and practical implications of this method.
The Problem: Dividing Encrypted Numbers with CKKS
Imagine you need to perform a calculation involving two encrypted numbers, say 'a' and 'b'. Directly applying division on these encrypted values is not possible with current homomorphic encryption schemes. However, Goldschmidt division offers a clever workaround to perform division while preserving data privacy.
Understanding Goldschmidt Division
Goldschmidt division is an iterative algorithm that approximates the quotient of two numbers by transforming the division into a series of multiplications. This method avoids the need for direct division, which is often computationally expensive and can be challenging to implement within the constraints of homomorphic encryption.
The Algorithm Explained
The key to Goldschmidt division lies in iteratively finding a series of factors that converge to the quotient. Let's break it down:
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Initialization:
- Find an initial approximation 'r' for the reciprocal of the divisor 'b', i.e., 1/b.
- Calculate the first two factors: f1 = 1 - b * r and f2 = 1 + b * r.
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Iteration:
- Multiply the dividend 'a' and the approximation 'r' by the factors f1 and f2 respectively.
- Update the approximation 'r' by multiplying it with f1 * f2.
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Convergence:
- Repeat step 2 for a predefined number of iterations, each iteration bringing the approximation closer to the actual quotient.
Implementing Goldschmidt Division with CKKS
Here's a simplified code example using the SEAL library, which offers a robust implementation of CKKS:
from seal import *
# Define the CKKS scheme
context = EncryptionParameters(scheme_type.CKKS)
context.set_poly_modulus_degree(8192)
context.set_coeff_modulus(CoeffModulus.Create(8192, [60, 40, 40, 60]))
context.set_plain_modulus(PlainModulus.Batch(20))
encryptor = Encryptor(context)
evaluator = Evaluator(context)
decryptor = Decryptor(context)
# Input encrypted values
a_encrypted = ... # Encrypted dividend
b_encrypted = ... # Encrypted divisor
# Initial approximation for reciprocal
r = 1.0 # Initial value can be chosen based on the expected range of b
r_encrypted = encryptor.encrypt(Plaintext(r))
# Iterations for convergence
num_iterations = 5
# Goldschmidt division algorithm
for i in range(num_iterations):
# Calculate factors
f1 = evaluator.multiply_plain(r_encrypted, Plaintext(1 - b_encrypted * r_encrypted))
f2 = evaluator.multiply_plain(r_encrypted, Plaintext(1 + b_encrypted * r_encrypted))
# Update approximation and dividend
r_encrypted = evaluator.multiply(r_encrypted, f1)
r_encrypted = evaluator.multiply(r_encrypted, f2)
a_encrypted = evaluator.multiply(a_encrypted, f1)
# Result: a_encrypted now holds the approximate quotient of a / b
Advantages of Goldschmidt Division with CKKS
- Privacy-preserving: The entire division process is performed on encrypted data, protecting sensitive information.
- Flexibility: This method works for both integer and fractional values, offering broader applicability.
- Scalability: The number of iterations can be adjusted for desired accuracy, balancing performance and precision.
Limitations and Considerations
- Computational cost: The iterative nature of Goldschmidt division requires multiple multiplications, increasing the computational overhead.
- Precision: The accuracy of the result is dependent on the number of iterations and the underlying CKKS parameters.
Further Research and Applications
The integration of Goldschmidt division with CKKS opens up exciting possibilities for secure data analysis and machine learning. Future research can explore:
- Optimization: Developing techniques to minimize the number of iterations and improve the efficiency of the algorithm.
- Error analysis: Characterizing the error propagation and establishing bounds for the accuracy of the results.
- Real-world applications: Integrating this method into secure data processing pipelines for various domains like healthcare, finance, and cloud computing.
Conclusion
Goldschmidt division provides a valuable approach for performing division on encrypted data using CKKS. This technique paves the way for secure and efficient computations in privacy-sensitive scenarios. By understanding the algorithm and its implementation, researchers and developers can leverage this powerful tool to enhance security and enable confidential data analysis.
Resources:
- SEAL Library: https://github.com/microsoft/SEAL
- CKKS Homomorphic Encryption: https://eprint.iacr.org/2017/621.pdf
- Goldschmidt Division Algorithm: https://en.wikipedia.org/wiki/Goldschmidt%27s_algorithm
