Making AFDM Secure Against Eavesdroppers: A Phase Function Design Approach

Technical Summary: Making AFDM Secure Against Eavesdroppers: A Phase Function Design Approach

Problem Statement
Affine Frequency Division Multiplexing (AFDM) has emerged as a robust waveform for high-mobility communications and Integrated Sensing and Communication (ISAC) due to its resilience to Doppler effects. However, the integration of sensing capabilities introduces Physical Layer Security (PLS) threats, where eavesdroppers may exploit sensing information to estimate Channel State Information (CSI). While AFDM utilizes two adjustable chirp parameters ($c_1$ and $c_2$) for modulation, $c_1$ is constrained by the channel's maximum Doppler shift, leaving $c_2$ as the primary candidate for security enhancement.

Existing PLS strategies for AFDM, such as parameter hopping, permutations, or codebook selection, have limitations. Hopping increases uncertainty over time but leaves the brute-force complexity of individual symbols unchanged. Codebook approaches rely on pseudo-random selection rather than systematic waveform design, failing to explicitly control complexity. Furthermore, schemes dependent on channel reciprocity are vulnerable in ISAC scenarios where eavesdroppers may access channel-related information. Consequently, there is a need for a waveform design that enhances robustness against brute-force demodulation in a channel-independent manner, even if the eavesdropper possesses perfect CSI.

Methodology
The paper proposes a novel phase function design for the $c_2$ parameter to increase the computational difficulty of brute-force demodulation for an eavesdropper. The core methodology involves the following steps:

1. Derivation of a Design Criterion: The authors derive a theoretical relationship (Proposition 1) demonstrating that the admissible mismatch interval ($\Delta c_2$) for an eavesdropper to successfully demodulate a signal is inversely proportional to the magnitude of the first derivative of the phase function with respect to $c_2$. Specifically, $|\Delta c_2| \propto 1 / |\partial f(c_2, m) / \partial c_2|$.
2. Phase Function Redesign: In conventional AFDM, the phase function is $f(c_2, m) = c_2 m^2$, resulting in a brute-force complexity scaling of $O(N^2)$. To improve this, the authors introduce a family of phase functions:
   $$f(c_2, m) = \kappa m^a \cos(\pi c_2 m^b)$$
   where $(\kappa, a, b) \in \mathbb{R}^3$ and $b \geq 0$.
3. Complexity Analysis: By analyzing the derivative of the proposed function, the authors show that the mismatch interval scales as $O(1/N^{a+b})$. Consequently, the brute-force search complexity scales as $O(N^{a+b})$. To preserve the quadratic chirp structure of conventional AFDM, the parameter $a$ is fixed at 2, resulting in a complexity of $O(N^{2+b})$.
4. Parameter Optimization: The parameter $\kappa$ is suggested to be an irrational number to further expand the search space. The authors identify that operating points where $\sin(\pi c_2 m^b) = 0$ must be avoided to maintain high sensitivity to parameter mismatches.

Key Contributions

• Theoretical Criterion: The paper establishes a fundamental design criterion linking the brute-force demodulation complexity directly to the first derivative of the phase function.
• Unbounded Complexity Control: The proposed phase function family allows the brute-force search complexity to be increased in a controllable and theoretically unbounded manner by adjusting the parameter $b$, without altering the underlying AFDM modulation framework.
• Channel Independence: Unlike previous channel-dependent security schemes, this approach enhances security regardless of the eavesdropper's knowledge of the channel state, provided they lack the specific phase function parameters.
• Compatibility: The method is designed to be compatible with existing security mechanisms, such as chirp-parameter hopping and chirp permutation, allowing for layered security enhancements.

Simulation Results
The authors evaluated the proposed design using Monte Carlo simulations over a four-tap linear time-varying (LTV) channel with $N=64$ subcarriers and QPSK modulation.

• Mismatch Sensitivity: Compared to conventional AFDM, which exhibits a mismatch interval of approximately $4 \times 10^{-5}$ (at a BER threshold of $10^{-3}$), the proposed design with $b=1$ reduced this interval to $1.7 \times 10^{-7}$, representing a two-order-of-magnitude increase in complexity.
• Scalability: Increasing $b$ to 10 resulted in an "impulse-like" BER curve, indicating an extremely small mismatch interval where even a slight deviation ($\Delta c_2 = 10^{-5}$) caused the eavesdropper's BER to remain at a high error floor, whereas conventional AFDM remained robust to this deviation.
• Legitimate Performance: The simulations confirmed that the proposed design does not degrade the communication performance of the legitimate user when parameters are perfectly matched ($\Delta c_2 = 0$).
• Parameter Constraints: The results verified that the parameter $c_2$ should not be set to 1, as this violates the condition required for high sensitivity, leading to deteriorated PLS performance.

Significance
The paper claims that this phase function design significantly improves the PLS performance of AFDM by several orders of magnitude in terms of brute-force demodulation complexity. The significance lies in providing a systematic, waveform-based approach to security that is independent of channel conditions. This allows legitimate users to flexibly tune the difficulty of demodulation faced by eavesdroppers while maintaining the robustness and sensing capabilities inherent to AFDM. The authors emphasize that this approach addresses a critical gap in current AFDM security research by moving beyond simple parameter hopping to a fundamental redesign of the phase structure.