---
title: 'Secure AFDM: High-Mobility Physical-Layer Security'
url: https://www.emergentmind.com/topics/secure-affine-frequency-division-multiplexing-se-afdm
type: topic
---

# Secure AFDM: High-Mobility Physical-Layer Security

Secure Affine Frequency Division Multiplexing (SE-AFDM) denotes a class of physical-layer secure extensions of affine frequency division multiplexing in which the affine or chirp-domain degrees of freedom of the waveform—most commonly the second AFDM parameter \(c_2\)—are exploited as secret, time-varying, or difficult-to-estimate controls. In the published formulations, the legitimate receiver preserves AFDM’s robustness in doubly selective and high-mobility channels by compensating the secure parameterization with synchronized side information, while an eavesdropper that lacks that information faces either severe bit-error-rate degradation or sharply increased brute-force demodulation complexity [2509.18555, 2510.02023, 2605.14837].

## 1. AFDM foundations and the meaning of “secure” in SE-AFDM

AFDM is a chirp-based multicarrier waveform built on the discrete affine Fourier transform (DAFT). In its standard form, an affine-domain symbol vector \(\mathbf{x}\) is mapped to the time domain through the inverse DAFT,
\[
\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\mathbf{F}^{H}\mathbf{\Lambda}_{c_2}^{H}\mathbf{x},
\]
or, equivalently in scalar form,
\[
s[n]=\frac{1}{\sqrt N}\sum_{m=0}^{N-1}x[m]e^{j2\pi\left(c_1n^2+\frac{mn}{N}+c_2m^2\right)}.
\]
A chirp-periodic prefix (CPP) is appended so that transmission over multipath doubly selective channels can be represented by a structured effective channel in the affine domain [2104.11331, 2404.10232].

The technical attraction of AFDM is that its parameterized chirp basis is matched to channels with both delay spread and Doppler spread. Foundational AFDM work shows that \(c_1\) can be selected so that delay-Doppler paths become separable in the DAFT domain, yielding a sparse or structured input-output relation and full diversity under stated conditions. A 6G-oriented survey further identifies AFDM’s chirp parametrization as a native design lever for physical-layer security, in contrast to OFDM’s fixed DFT basis and OTFS’s more constrained configuration space [2104.11331, 2507.21704].

Within this literature, “SE-AFDM” refers to *secure* AFDM. That terminology requires care because in AFDM channel-estimation work the abbreviation “SE” is also used for *spectral efficiency* rather than security. In particular, “Channel Estimation for AFDM With Superimposed Pilots” treats SE as spectral efficiency and contains no adversarial or secrecy model [2404.10232].

## 2. Core waveform constructions

The central secure design idea is to preserve the conventional AFDM role of \(c_1\) for reliable communication in doubly selective channels while converting \(c_2\) into the security-bearing parameter. In the first explicit SE-AFDM construction, \(c_2\) is no longer a scalar but an index-varying vector
\[
\mathbf{c}_2^A=[c_2^A[0],c_2^A[1],\ldots,c_2^A[N-1]]^T,
\]
with entries selected from a public codebook
\[
\mathbb{C}_2=\{c_{2,1},c_{2,2},\ldots,c_{2,M}\}
\]
that uniformly discretizes \([ -c_{2,\max},\,c_{2,\max}]\). The selection is driven by a secret synchronized long-period pseudo-noise (LPPN) sequence, yielding a transmit-side diagonal matrix
\[
\mathbf{\Lambda}_{c_2,A}^{H}=\mathrm{diag}\!\left(e^{-j2\pi c_2^A[m]m^2},\,m=0,\ldots,N-1\right).
\]
The resulting SE-AFDM signal is
\[
s_A[n]=\frac{1}{\sqrt N}\sum_{m=0}^{N-1}x[m]e^{j2\pi\left(c_1n^2+c_2^A[m]m^2+\frac{mn}{N}\right)}.
\]
This is described as *parameter-domain spreading*: the data symbols themselves are not spread as in DSSS; rather, the waveform parameter is pseudo-randomly varied, so the payload spectral efficiency remains the same as in conventional AFDM [2509.18555, 2510.02023].

A second secure construction generalizes the conventional quadratic \(c_2m^2\) term into a generic phase function \(f(c_2,m)\), producing
\[
s[n]=\frac{1}{\sqrt N}\sum_{m=0}^{N-1}x_m e^{j2\pi\left(c_1n^2+\frac{mn}{N}+f(c_2,m)\right)}.
\]
In that framework, the design objective is not synchronization asymmetry via an LPPN sequence but sensitivity amplification: the eavesdropper’s admissible parameter mismatch interval is controlled by the first derivative \(\partial f(c_2,m)/\partial c_2\). The proposed family
\[
f(c_2,m)=\kappa m^a\cos\!\left(\pi c_2 m^b\right)
\]
preserves AFDM’s chirp structure while enlarging brute-force demodulation complexity beyond the conventional quadratic-phase case [2605.14837].

These constructions suggest a broader taxonomy of SE-AFDM. One branch secures AFDM by *dynamic secret parameter trajectories* generated from LPPN-controlled codebooks; another secures AFDM by *phase-function redesign* that makes parameter mismatch dramatically more destructive.

## 3. Legitimate reception, compensation, and synchronization

For the legitimate receiver, the crucial property is that the nonlinear effect introduced by secure \(c_2\) parameterization is known and can therefore be absorbed into the effective channel rather than appearing as unknown distortion. In the LPPN-driven design, Bob uses his own synchronized de-chirp matrix
\[
\mathbf{\Lambda}_{c_2,B}=\mathrm{diag}\!\left(e^{-j2\pi c_2^B[m]m^2},\,m=0,\ldots,N-1\right),
\]
and after CPP removal, multiplication by \(\mathbf{\Lambda}_{c_1}\), DFT processing, and secure de-chirping, obtains
\[
\mathbf{y}_B=\mathbf{H}_{\rm eff,B}\mathbf{x}+\bar{\mathbf w}_B.
\]
Detection is then performed by a standard linear receiver such as MMSE,
\[
\hat{\mathbf{x}}_B=\mathbf{H}_{\rm eff,B}^{H}\left(\mathbf{H}_{\rm eff,B}\mathbf{H}_{\rm eff,B}^{H}+\sigma_{n,B}^2\mathbf I_N\right)^{-1}\mathbf y_B.
\]
When LPPN synchronization holds, \(c_2^A[q]=c_2^B[q]\), so the secure phase law is part of a known effective channel and the legitimate BER remains essentially unchanged relative to AFDM [2509.18555].

The later formulation extends this idea with an explicit synchronization architecture. A frame is divided into a frame-synchronization block, an LPPN-sequence synchronization block, and a secure communication block,
\[
\mathbf{S}=\left[\mathbf{S}_{\rm head},\mathbf{S}_{\rm LPPN},\mathbf{S}_{\rm com}\right].
\]
During the first two blocks, the transmitter fixes \(\mathbf c_{2,\mu}^A=\mathbf u_{N\times 1}\), allowing Bob to perform header acquisition and to recover the LPPN generator state. The state vector
\[
\mathbf w=\left[\mathbf n^T,(\mathbf s_{\rm X1A}^k)^T,(\mathbf s_{\rm X1B}^k)^T,(\mathbf s_{\rm X2A}^k)^T,(\mathbf s_{\rm X2B}^k)^T\right]^T
\]
is spread with DSSS only for synchronization reliability, not for payload transmission. After desynchronization and local generator initialization, Bob reproduces the same dynamic \(c_2\) trajectory as Alice and removes it before payload detection [2510.02023].

This architecture is significant because early SE-AFDM analysis assumed LPPN synchronization and left its realization for future work, whereas the later design turns synchronization into an explicit system component. A plausible implication is that SE-AFDM’s practical feasibility depends as much on secure parameter alignment as on the secure waveform itself.

## 4. Eavesdropper model and security mechanism

The adversarial model in SE-AFDM is strong in waveform terms. Eve is typically assumed to know the public parameters \(c_1\), \(N\), the CPP length, and the \(c_2\) codebook, and may also estimate her own channel. What she does not know is the secret LPPN realization or generator configuration that determines Alice’s dynamic \(c_2\) sequence [2509.18555, 2510.02023].

Under this asymmetry, Eve’s affine-domain observation can be rewritten as
\[
\mathbf y_E=\mathbf H'_{\rm eff,E}\mathbf x' + \bar{\mathbf w}_E,
\]
where the data-like quantity presented to Eve is not \(\mathbf x\) but
\[
\mathbf x'=\mathbf x\odot \mathbf x_c
\qquad\text{or equivalently}\qquad
\mathbf x'=\mathbf{\Lambda}_{c_2,A}^{H}\mathbf x,
\]
with
\[
x_c[q]=e^{j2\pi c_2^A[q]q^2}.
\]
After equalization, Eve therefore recovers \(\hat{\mathbf x}'_E\), not \(\mathbf x\). Componentwise,
\[
\hat x'_E[q]\approx e^{j2\pi c_2^A[q]q^2}x[q].
\]
The published argument is an identifiability argument rather than a secrecy-capacity theorem: for each \(q\ge 1\), the observation contains two unknowns, namely the information symbol \(x[q]\) and the secure chirp factor determined by \(c_2^A[q]\), so separation is impossible without the synchronized side information [2509.18555].

The same effect is expressed through an effective SINR analysis. Writing
\[
\hat x'_E[p]=x[p]+\left(e^{j2\pi c_2^A[p]p^2}-1\right)x[p]+w_E[p],
\]
the unknown secure rotation becomes self-interference. For the AWGN-based analysis, the eavesdropper SINR for symbol \(p\) is
\[
\mathrm{SINR}_{E,p}=
\frac{\mathbb E\{|x[p]|^2\}}
{\mathbb E\{|x[p]|^2\}\,\mathbb E\!\left\{\left|e^{j2\pi c_2^A[p]p^2}-1\right|^2\right\}+\sigma_{n,E}^2 }.
\]
When \(c_2^A[p]\) is uniformly distributed over \([-c_{2,\max},c_{2,\max}]\), the analysis shows that Eve’s SINR decreases as the \(c_2\) range expands, while Bob’s output SINR remains \(\gamma_B\) because Bob does not incur self-interference from the known parameter sequence [2509.18555].

In the phase-function design approach, the security mechanism is instead formulated as brute-force demodulation resistance. If Eve uses \(\hat c_2=c_2+\Delta c_2\), then after a first-order Taylor expansion,
\[
\hat x_k=x_k e^{-j2\pi \Delta c_2 \frac{\partial f(c_2,k)}{\partial c_2}},
\]
so correct demodulation requires
\[
|\Delta c_2|\le
\frac{|\varepsilon|}{2\pi\left|\frac{\partial f(c_2,k)}{\partial c_2}\right|}.
\]
The smaller this admissible mismatch interval, the denser the brute-force search grid must be [2605.14837].

## 5. Parameter design and secure design variants

A distinct line of secure AFDM work studies the design space of AFDM parameters themselves. In that analysis, four parameters are emphasized:
\[
c_1,\quad c_2,\quad \alpha_{\max},\quad L_{\max}.
\]
The main conclusions are that \(c_1\) is bounded by the actual Doppler support and the preset Doppler guard, \(c_2\) has minimum periodicity \(1\) and therefore effective range \([0,1]\), and excessive \(L_{\max}\) enlarges a “security-risk interval” for eavesdropping [2503.19364].

More specifically, if
\[
\alpha_{c_1}\triangleq \frac{2Nc_1-1}{2}\in\mathbb N^+,
\]
then secure-yet-decodable operation requires
\[
\alpha_{\max}^C\le \alpha_{c_1}\le \alpha_{\max},
\]
or equivalently
\[
\frac{2\alpha_{\max}^C+1}{2N}
\le
c_1=\frac{2\alpha_{c_1}+1}{2N}
\le
\frac{2\alpha_{\max}+1}{2N}.
\]
If \(c_1\) is too small, some Doppler components are unresolved; if it is too large, the estimation region exceeds the preset guard interval and false paths are introduced [2503.19364].

For \(c_2\), the same work shows
\[
c_2\equiv c_2+z,\qquad z\in\mathbb Z,
\]
so \(c_2\) need only be searched or configured on \([0,1]\). This periodicity is later echoed in the phase-function security design, where the effective \(c_2\) search interval is also treated as periodic with period \(1\), and where \(c_2=1\) is specifically identified as a poor operating point because the sine term in the derivative criterion can vanish [2503.19364, 2605.14837].

Delay-guard design is also security-relevant. If \(L_{\max}^C\) is the actual maximum delay and \(L_{\max}\) is the preset delay guard, then the interval
\[
[L_{\max}^C,L_{\max}]
\]
is characterized as a “security-risk interval”: any receiver, including an eavesdropper, that chooses an estimation limit inside that interval can fully estimate all delays. The recommended design is therefore
\[
L_{\max}=L_{\max}^C.
\]
This shifts SE-AFDM parameterization away from purely communication-driven overprovisioning and toward security-aware guard sizing [2503.19364].

The phase-function branch provides a different but complementary guideline. For the family
\[
f(c_2,m)=\kappa m^a\cos(\pi c_2 m^b),
\]
the derivative
\[
\frac{\partial f(c_2,m)}{\partial c_2}
=
\kappa\pi m^{a+b}\sin(\pi c_2 m^b)
\]
implies eavesdropper search complexity \(\mathcal O(N^{a+b})\), and with \(a=2\),
\[
\mathcal O(N^{2+b}).
\]
If \(\kappa\) is also unknown, the joint exhaustive-search complexity becomes
\[
\mathcal O(N^{b+4}).
\]
This places SE-AFDM on a spectrum ranging from codebook-driven secret parameter hopping to continuous secure phase-law design [2605.14837].

## 6. Performance, implementation status, and limitations

Across the reported simulations, the dominant empirical result is consistent: Bob’s BER remains almost the same as that of conventional AFDM, whereas Eve’s BER worsens sharply as the secure parameter range expands and often tends toward \(0.5\), corresponding to random-guess performance for QPSK bits. In the original SE-AFDM simulations, Eve’s effective SINR under a single-path AWGN-oriented study with \(\gamma_E=25\) dB decreases monotonically with \(c_{2,\max}\) and eventually approaches about \(-0.93\) dB. Under estimated CSI, using pilot SNR \(=30\) dB and \(c_{2,\max}=4.88\times 10^{-6}\), Bob’s BER still coincides with AFDM while Eve’s BER remains near \(0.5\) [2509.18555].

The later synchronized SE-AFDM design reports a full high-mobility simulation setting with QPSK, \(f_c=24\) GHz, bandwidth \(15.36\) MHz, \(\Delta f=15\) kHz, \(N=1024\), \(N_{\rm cp}=17\), \(M=1024\), \(P=3\), delays \([0,1,2]\), and maximum integer normalized Doppler \(\alpha_{\max}=2\), corresponding to \(1350\) km/h. In that system, Bob again tracks conventional AFDM performance while Eve’s BER approaches \(0.5\) as \(c_{2,\max}\) increases. The same work studies approximate brute-force search and reports that for \(c_{2,\max}=4.88\times 10^{-5}\), \(M=10^6\), and codebook interval \(9.76\times 10^{-11}\), Eve requires a search interval below \(9.77\times 10^{-8}\) to obtain BER below \(1.77\times 10^{-5}\); BER exceeds \(0.1\) once the search interval is above \(7.8\times 10^{-7}\) [2510.02023].

The phase-function design branch reports the same matched-user asymmetry in a different metric. For \(N=64\), QPSK, MMSE equalization, and the phase law
\[
f(c_2,m)=(\sqrt{2}-1)m^2\cos(0.2\pi m^b),
\]
conventional AFDM has a mismatch interval around \(4\times 10^{-5}\), whereas the proposed design with \(b=1\) reduces it to about \(1.7\times 10^{-7}\), and with \(b=10\) the BER-versus-mismatch curve becomes impulse-like. In matched reception, conventional AFDM and the secure phase design exhibit essentially the same BER; under mismatch \(\Delta c_2=10^{-5}\), conventional AFDM still decodes well while the secure design remains at a very high error floor [2605.14837].

Implementation evidence exists but is still limited. The synchronized SE-AFDM system has been validated on an SDR prototype with \(f_c=5\) GHz, \(B=49.152\) MHz, \(\Delta f=48\) kHz, \(N=1024\), \(N_{\rm cp}=13\), \(M=1024\), and \(N_{\rm sym}=256\), with an introduced frequency offset of \(22.222\) kHz to emulate approximately \(4800\) km/h. In that prototype, Bob’s BER decreases with SNR while Eve exhibits an error floor around \(0.1\), indicating that synchronization failure, not merely thermal noise, remains the dominant impairment for the eavesdropper [2510.02023].

The limitations are equally clear in the literature. Published SE-AFDM analyses are not secrecy-capacity theorems; they are ambiguity-based, BER-based, SINR-based, or brute-force-complexity-based physical-layer security arguments. Security depends critically on protecting the LPPN generator configuration or the secure phase-function parameters. AWGN-based eavesdropper SINR derivations are available, but full multipath-fading security analysis remains incomplete. Active attacks, pilot contamination, authentication, covert communication, and multiuser secure AFDM are largely outside the present scope. The broader AFDM survey therefore treats physical-layer security as an inherent opportunity of chirp-parametrized AFDM rather than a completed security stack [2509.18555, 2510.02023, 2605.14837, 2507.21704].

Source: https://www.emergentmind.com/topics/secure-affine-frequency-division-multiplexing-se-afdm