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Physical Layer Security in Chirp Permuted AFBM

Pedro Schuves Marodin, Henrique L. Senger, Gustavo P. Gonçalves, Hyeon Seok Rou, Bruno S. Chang, Giuseppe Thadeu Freitas de Abreu, Didier Le Ruyet

arXiv:2610.33276v1eess.SP

TL;DR

AFBM provides spectral containment through a filter bank, but securing its waveform against eavesdropping motivates a physical-layer approach. The paper permutes chirp sequences at both DAFT stages with separate keys; with either key wrong, an eavesdropper’s BER remains near 0.5 and rate below 0.09 bit/symbol, at a PAPR cost compared with the single-permutation variant.

  • Problem

    The broadcast setting exposes AFBM communications to eavesdropping, while upper-layer cryptography adds computation and latency that conflict with IoT and ISAC requirements.

    Source: §I · p. 1

  • Method

    The scheme applies a separate secret chirp-sequence permutation to each of AFBM’s two DAFT stages.

    Source: §I · p. 1, A. Double Chirp-Permutation in AFBM · p. 3

  • Results

    With either key wrong, Eve’s BER stays between 0.47 and 0.51 and her rate below 0.09 bit/symbol, even with perfect CSI, co-location with Bob, and blind carrier-phase recovery.

    Source: B. Security · p. 4, C. Secrecy Rate · p. 4

  • Takeaways & Limitations

    A wrong key makes the precoder fail to restore complex orthogonality, leaving filter-bank interference at the detector; the double scheme preserves spectral containment but costs 1.9 dB in PAPR.

    Source: §I · p. 1, §V · p. 6

    The double scheme increases PAPR by 1.9 dB, while its search space is 10^93.3 against a key space of 10^96.5.

    Source: §V · pp. 5–6

Abstract

from arXiv · show

We propose a physical-layer security scheme for affine filter bank modulation (AFBM), a waveform proposed for sixth-generation (6G) integrated sensing and communications (ISAC). AFBM contains two discrete affine Fourier transforms, one for precoding and one for spreading, and the scheme permutes, with one secret key per transform, the chirp sequence that each transform places between two Fourier stages. A wrong key is then conjugated by a discrete Fourier transform into a dense matrix, the restoration of complex orthogonality fails, and the intrinsic filter-bank interference reaches the detector. Over a doubly-dispersive channel, an eavesdropper with perfect channel state information, co-located with the legitimate receiver and holding either key wrong, keeps a bit error rate near 0.5 and a rate below 0.09 bit/symbol, also under a blind carrier-phase attack. Compared with a single-permutation variant, the double scheme costs about 2 dB of peak-to-average power ratio and enlarges the search space from 10^61 to 10^93.

I. INTRODUCTION

AFBM combines DAFT spreading with a polyphase filter bank and a precoder that restores complex orthogonality, making it a candidate for secure communications with spectral containment. The paper proposes permuting chirps at both DAFT stages and evaluates the scheme against an eavesdropper under a worst-case channel assumption.

I. INTRODUCTION

  • AFBM adds a polyphase filter bank to AFDM spreading for improved spectral containment, while a precoder restores the complex orthogonality missing in FBMC.The cited work reports −100 dB OOBE when a long prototype filter is admissible.Source: §I · p. 1
  • Prior chirp-permuted AFDM increases the search space to N! but does not address spectral containment, motivating a permutation-based security scheme for AFBM.The introduction contrasts this with AFDM’s secret chirp parameters, whose search cost is O(N^2).Source: §I · p. 1
  • With either key wrong, Eve’s BER stays between 0.47 and 0.51 and her rate below 0.09 bit/symbol, even with perfect CSI and co-location with Bob.Blind carrier-phase recovery does not change this; the double scheme costs 1.9 dB in PAPR.Source: §I · p. 1

II. SYSTEM MODEL

  • The model assumes Alice lacks CSI, while Bob and Eve have perfect CSI and Eve is co-located with Bob, receives the same signal, and knows all parameters except the keys.Alice and Bob share the secret keys governing the DAFT chirp-sequence permutations.Source: §II · pp. 1–2

A. Transmit Signal Model

The AFBM transmitter maps QAM symbols onto active subcarriers, spreads them with a DAFT, and applies a filtered, precoded filter-bank signal. Zero-padding supports frequency containment without allowing the precoding DAFT to cancel the filter-bank IDAFT.

A. Transmit Signal Model

  • The L-point DAFT combines chirp diagonals and a normalized DFT; its c1 chirp rate is constrained by the full-diversity condition.The c2 diagonal multiplies the data vector directly, while c1 lies between the DFT and the subsequent spreading stage.Source: A. Transmit Signal Model · p. 2
  • Frequency containment uses an IDAFT with frequency-domain zero-padding, where L < P < N prevents the precoding DAFT from canceling the filter-bank IDAFT.T_NP is the N × P zero-padding selection matrix.Source: A. Transmit Signal Model · p. 2
  • AFBM uses a polyphase network and prototype filter with a precoder to restore the complex orthogonality that standard FBMC lacks.The prototype filter’s polyphase blocks form a single-symbol matrix and a block-Toeplitz frame matrix with consecutive symbols delayed by N/2 samples.Source: A. Transmit Signal Model · p. 2
  • The precoder confines intrinsic interference to a single coefficient, requiring an overlap factor O ≤1.5 and blocks transmitted every N/2 samples.The all-block symbol mapping is Ξ = IK⊗¯Ξ.Source: A. Transmit Signal Model · p. 2
  • The transmitter needs no cyclic prefix because the prototype filter is well localized in time.The filter-bank frame uses consecutive symbols delayed by N/2 samples.Source: A. Transmit Signal Model · p. 2

B. Channel and Receiver Model

The receiver model sends the filtered signal through a doubly-dispersive channel and applies matched receive operators before LMMSE detection. The simulations account for the colored noise at the detector input.

B. Channel and Receiver Model

  • The signal propagates through a doubly-dispersive channel with Pch paths and a prefix of at least ℓmax, yielding a circular M × M channel matrix.The model specifies random path gains, uniformly distributed delays, and Doppler shifts.Source: B. Channel and Receiver Model · p. 2
  • For a chirp-periodic prefix, the channel model applies a prefix phase compensation matrix; a plain cyclic prefix uses the identity matrix.The channel includes a fractional-Doppler matrix.Source: B. Channel and Receiver Model · p. 2
  • The received signal is processed by matched receive operators to form an effective channel, then detected with an LMMSE receiver.The detector model treats its input noise as white, but simulations use the exact colored-noise covariance σ2 nRRH.Source: B. Channel and Receiver Model · p. 2

A. Double Chirp-Permutation in AFBM

The double scheme applies one secret chirp-sequence permutation in the precoding DAFT and another in the spreading transform. Its compensation is derived from both keyed transforms, while a power-complementary prototype makes that compensation independent of the keys.

A. Double Chirp-Permutation in AFBM

  • Two secret lexicographic permutation keys, k1 and k2, permute the chirp sequences in the precoding and spreading DAFT stages, respectively.The keys range over L! and P! permutations, and each permutation is placed between a DFT and the following operator.Source: A. Double Chirp-Permutation in AFBM · p. 3
  • The compensation is generally recomputed from both keys and defines the precoder from the full keyed structure.This compensation supports the precoding operation after both transforms are permuted.Source: A. Double Chirp-Permutation in AFBM · p. 3
  • For a power-complementary prototype, the compensation vector is constant and independent of both keys, so the security resides in the keyed transforms themselves.This follows from the constant polyphase energy, unitarity of the precoding DAFT, and QP,k2^H QP,k2 = IL.Source: A. Double Chirp-Permutation in AFBM · p. 3

B. Single Chirp-Permutation in AFBM

The single-permutation variant uses one key to permute the shared c2 chirp in both DAFT stages. Its protection comes from the spreading stage, while leaving the precoding-stage PAPR unchanged.

B. Single Chirp-Permutation in AFBM

  • One key permutes the shared c2 chirp in both DAFT stages, with the full permutation restricted to the first L entries for the spreading stage.The key permutes λc2,P in WP; its restriction permutes λc2,L in WL.Source: B. Single Chirp-Permutation in AFBM · p. 3
  • In the spreading stage, the mismatched chirp acts as a per-subcarrier phase rotation that carrier recovery can remove.In the precoding stage, the permuted chirp lies between Fourier stages, so a mismatch cannot be commuted out in the same way.Source: B. Single Chirp-Permutation in AFBM · p. 3
  • Protection therefore comes from the spreading stage alone; the variant uses one key and leaves PAPR unchanged because it does not permute λc1,L.The c1 chirp is the one used in DAFT spreading to flatten the envelope.Source: B. Single Chirp-Permutation in AFBM · p. 3

C. Mismatched Effective Channel

A key mismatch alters the effective channel used for eavesdropper detection, while the intrinsic filter-bank interference remains unconditioned. Depending on which key is wrong, the mismatch becomes dense interference rather than a removable diagonal phase.

C. Mismatched Effective Channel

  • The intrinsic interference between overlapping blocks is not conditioned by the precoder, and it limits AFBM at high SNR for both Bob and Eve.The model compares Bob’s matched effective channel with Eve’s channel formed using guessed keys.Source: C. Mismatched Effective Channel · p. 3
  • A k1 mismatch inserts a non-constant diagonal between Fourier stages; Fourier conjugation makes it circulant, spreading ICI across subcarriers with no per-subcarrier correction.For the stated parameters, the mismatch error has 1.2 times the Frobenius norm of U, with about a third of its energy off-diagonal.Source: C. Mismatched Effective Channel · p. 3
  • A k2 mismatch likewise places a diagonal between Fourier stages, making the eavesdropper’s product dense and allowing intrinsic filter-bank interference into detection.The text describes this failure for mismatches in either key.Source: C. Mismatched Effective Channel · p. 3

D. Key Space versus Search Space

The section distinguishes the number of key configurations from the number of distinct hypotheses an eavesdropper must test, then defines security measures for mismatched-key detection. The achievable-rate difference is not a Wyner secrecy capacity.

D. Key Space versus Search Space

  • For the double scheme, the search space excludes permutations of repeated chirp entries, while the key space counts all L! × P! configurations; the single scheme has P! keys.The rational c1 chirp has only 23 distinct entries among 32, whereas the irrational c2 chirp entries are pairwise distinct.Source: D. Key Space versus Search Space · p. 3

E. Eavesdropper Model and Security Metrics

  • The blind fourth-power carrier estimator tests whether an apparent degradation persists after carrier recovery, using QPSK’s E{x^4} = −1.A short known preamble resolves the residual π/2 ambiguity; the estimator costs O(LT) and searches no keys.Source: E. Eavesdropper Model and Security… · pp. 3–4
  • The achievable-rate measure uses post-equalization SINR from the true channel response and the eavesdropper’s guessed-key equalizer, but it is not Wyner secrecy capacity.Each rate is a lower bound on its corresponding mutual information, and their difference bounds neither side of the difference.Source: E. Eavesdropper Model and Security… · p. 4

IV. NUMERICAL RESULTS

The simulations compare paired waveforms under the same doubly-dispersive channel realization. Bob’s BER falls with SNR, while mismatched-key eavesdroppers remain near chance and their achievable rate stays low.

A. Setup

  • The simulation uses 64 subcarriers, 32 active, eight blocks, and paired channel realizations, with 2,000 BER trials and 300 rate trials.Detection is LMMSE, and the effective channel matrices are exactly invertible at the selected N.Source: A. Setup · p. 4

B. Security

  • 0.47–0.51 BER: every double-scheme key-mismatched Eve configuration stays near chance across SNR, including under the blind attack, while Bob falls from 0.20 to 2.0×10−3.The setup gives Eve perfect CSI and co-location with Bob; the stated explanation is filter-bank interference rather than removable phase rotation.Source: Figure 2 · p. 4, B. Security · p. 4
  • Below 0.09 bit/symbol: Eve’s rate remains nearly flat in SNR across the three key-mismatch cases.The figure caption defines Eve’s rate as the gap between each secrecy-rate curve and Bob’s achievable rate.Source: Figure 3 · p. 4

C. Secrecy Rate

Eve’s rate remains below 0.09 bit/symbol while the secrecy rate follows Bob’s across the tested SNR range. The double scheme’s search space is smaller than its key space, but remains beyond exhaustive search.

C. Secrecy Rate

  • Eve’s rate stays below 0.09 bit/symbol, while the secrecy rate rises from 0.74 bit/symbol at 0 dB to 3.68 bit/symbol at 21 dB in the least favourable mismatch case.Eve’s rate is nearly flat in SNR because residual filter-bank interference scales with signal power, not noise.Source: C. Secrecy Rate · p. 4
  • The power-complementary prototype gives a relative spread of about 10^-16 and identical ˜b across key pairs, whereas PHYDYAS gives a spread of 1.1 × 10^-1 and key-dependent ˜b.The comparison checks the stated property for prototypes with flat versus non-flat polyphase energy.Source: C. Secrecy Rate · p. 4

D. Exhaustive Search Complexity

  • At L = 32, the double scheme’s key space is 10^96.5 and its search space is 10^93.3, while the single permutation’s key and search space is 10^61.1.The 3.2-order gap in the double scheme comes from repeated entries of λc1,L.Source: D. Exhaustive Search Complexity · p. 5
  • Grover adaptive search requires 10^46.7 queries over the double search space and 10^48.3 over its key space, versus 10^30.5 for the single permutation.At an assumed rate of 10^9 queries per second, the estimated runtimes are about 10^30.2 years for the double scheme and 10^14.0 years for the single permutation.Source: D. Exhaustive Search Complexity · p. 5
  • Both schemes remain far beyond exhaustive search, and the reduced double-scheme search space is not operationally decisive because full-diversity conditions make c1 rational.That rational chirp rate couples the permutation domain and chirp-rate design.Source: D. Exhaustive Search Complexity · p. 5

E. PAPR and Out-of-Band Emission

The double permutation raises PAPR but retains AFBM’s spectral containment. The conclusion also reports that its wrong-key protection persists under blind carrier-phase recovery, with a PAPR cost relative to the single-permutation variant.

E. PAPR and Out-of-Band Emission

  • At a CCDF of 10^-2, double CP-AFBM reaches 10.20 dB PAPR, 1.9 dB above unpermuted AFBM, while single CP-AFBM reaches 8.18 dB.The comparison uses a common ×4 oversampling grid; permuting λc1,L destroys the quadratic phase progression that flattens the envelope.Source: Figure 5 · p. 5, E. PAPR and Out-of-Band Emission · p. 5
  • AFBM’s OOBE is −49.5 dB versus −20.3 dB for band-limited OFDM, and CP-AFBM reaches −51.5 dB.Permutation leaves spectral containment intact because the occupied band is fixed by zero-padding and the prototype.Source: Figure 6 · p. 5, E. PAPR and Out-of-Band Emission · p. 5

V. CONCLUSION

  • With either double-scheme key wrong, Eve’s BER is 0.47–0.51 and her rate stays below 0.09 bit/symbol, even under blind carrier-phase recovery.The double scheme’s PAPR cost is 1.9 dB; the single permutation avoids that increase, but blind phase attack lowers Eve’s BER to 0.27–0.33.Source: §V · pp. 5–6
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