Fraud Detection on Ethereum Using Homomorphic SVM with RBF Kernel

The article details the solution provided by the winner of the Ethereum Fraud Detection via SVM challenge.

Author: Victor Correa

Model Training

For fraud detection on Ethereum transactions, a Support Vector Machine (SVM) with a Radial Basis Function (RBF) kernel was selected due to its superior performance in capturing complex, non-linear patterns in transaction data. The RBF kernel is defined as:

$$K(x, x_i) = \exp\left(-\gamma \|x - x_i\|^2\right)$$

Optimal Hyperparameters

• Kernel Coefficient (γ): 0.5
• Regularization (C): 1200
• Number of Support Vectors: 512

This configuration resulted in the highest observed F1-score. Despite the numerous features and support vectors, the SIMD (Single Instruction, Multiple Data) capabilities of CKKS homomorphic encryption ensured minimal computational overhead.

Homomorphic Inference

The SVM decision function is computed as:

$$f(x) = \text{sign}\left(\sum_{i=1}^n y_i K(x, x_i) + b\right)$$

Where:

• The Ethereum transaction feature vector x is encrypted
• Support vectors ( x_i ), dual coeffs ( y_i ) and bias ( b ) are in clear

Key Challenges & Solutions

1. Non-Polynomial Kernel Approximation

   • The RBF kernel is non-polynomial and cannot be directly evaluated in CKKS.
   • Solution: A polynomial approximation over the interval [-30000, 0] with a degree-1024 polynomial was employed to ensure high precision.

2. Sign Function Approximation

   • The sign function is discontinous and must be approximated.
   • Practical approximation: Polynomial approximation of the hyperbolic tangent (tanh) approximation with:
     • Polynomial Degree: 256
     • Approximation Interval: [-70, 70]
     • Sharpness Parameter (α): 6 (higher α improves approximation to sign but must be chosen with respect to the degree)

3. Efficient SIMD Packing

   • SIMD batching allows to parallelize kernel evaluations, significantly reducing latency in encrypted inference. Specifically, the 64 features of the 512 support vectors can fit int 64×512=32684 slots, taking full advantage of the available plaintext capacity. Each block of 64 slots computed the kernel between the encrypted transaction and a support vector.

SIMD Inference Steps

The following outlines the key SIMD inference steps involved in efficiently computing the final prediction:

• SIMD squaring of feature differences
• Local summation to compute squared distances
• Kernel evaluation for each computed distance
• SIMD multiplication of kernels with dual coefficients
• Global summation across relevant slots, followed by bias addition
• Homomorphic sign approximation for the final prediction

Implementation with OpenFHE

All polynomial approximations and evaluations were performed using the OpenFHE library’s EvalChebyshevFunction method. The approximation intervals were selected taking into account the outputs of the decision function on the training dataset.