Secure Autocorrelation Function (ACF)

• Secure ACF is a family of design criteria that ensures a controlled autocorrelation profile, whether by low sidelobes, random-like periodicity, or invariant second-order statistics.
• Techniques such as spectral shaping in NLFM radar, smoothing-spline optimization, and periodic power allocation in ISAC are employed to achieve robust target detection and adversarial deception.
• Designing secure ACF requires balancing trade-offs between resolution, sidelobe suppression, and SNR loss to meet diverse performance needs in sensing, cryptography, and communications.

Searching arXiv for recent and foundational papers on secure autocorrelation function across radar, cryptography, ISAC, and waveform design. A secure autocorrelation function (ACF) is not a single universally defined object but a family of design goals and analytical criteria that recur across radar, communications, cryptography, and integrated sensing and communication (ISAC). In the cited literature, security at the ACF level may mean a sharp and unambiguous main peak with very low sidelobes for robust ranging and low-probability-of-intercept operation, a random-like periodic autocorrelation profile resistant to correlation-based cryptanalytic exploitation, a data-independent second-order statistic that reveals no payload information, or a deliberately engineered ACF that misleads unauthorized sensing receivers while remaining invertible to a legitimate one (Ghavamirad et al., 2018, Caullery et al., 2024, Yin et al., 2022, Han et al., 2 Oct 2025, Zhao et al., 16 Jun 2026).

1. Definitions and domain-specific meanings

For a complex baseband waveform $s(t)$, the aperiodic autocorrelation function is

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$

In pulsed radar, sonar, and related waveform design problems, the key ACF attributes are the mainlobe width, the peak sidelobe level (PSL), the integrated sidelobe level (ISL), and the spectral shape implied by Wiener–Khinchin duality (Ghavamirad et al., 2018). In this setting, a secure or robust ACF is one with a sharp mainlobe, low sidelobes, controlled resolution, and spectral behavior compatible with low detectability or resistance to interference (Ghavamirad et al., 2018, Ghavamirad et al., 11 Jan 2025).

For binary sequences $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$, the periodic autocorrelation at shift $u\in F_m$ is

$C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$

with peak sidelobe statistic

$C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$

Here security is tied to resistance to cryptographic attacks: small nontrivial periodic autocorrelations indicate reduced exploitable structure, while the random-sequence benchmark specifies what “secure” random-like behavior should look like asymptotically (Caullery et al., 2024).

In communication systems using chaotic shape-forming filters, the ACF can be made invariant with respect to the transmitted information symbols. In that case, the transmitted second-order statistic is fixed by the base function $p(t)$, not by the data sequence, so the ACF becomes a data-independent system signature useful for blind channel estimation and, in a limited sense, for second-order statistical concealment (Yin et al., 2022).

In recent ISAC work, the term secure ACF is used in an explicitly adversarial sense. Instead of suppressing sidelobes, the transmit ACF is intentionally shaped to create periodic artificial peaks that degrade unauthorized passive radar estimation; the legitimate receiver removes those artifacts by mismatched filtering at the cost of signal-to-noise ratio (SNR) loss (Han et al., 2 Oct 2025). A related but non-adversarial line of work, Auto-correlation Function Keying (ACFK), embeds information directly into the periodic ACF domain while enforcing peak sidelobe constraints for sensing robustness (Zhao et al., 16 Jun 2026).

Domain	Security meaning	Representative paper
NLFM radar/sonar	Low-PSL, robust, low-ambiguity ACF	(Ghavamirad et al., 2018)
Random binary sequences	Random-like periodic ACF for cryptographic resistance	(Caullery et al., 2024)
Chaotic wireless communication	Data-independent ACF and blind CSI utility	(Yin et al., 2022)
Sensing-secure ISAC	Deliberately misleading ACF for unauthorized sensing	(Han et al., 2 Oct 2025)
Communication-centric ISAC	Exact nominal P-ACF control under PSL constraints	(Zhao et al., 16 Jun 2026)

This multiplicity of meanings is central. A secure ACF may be “clean,” “random-like,” “invariant,” or even intentionally “impure,” depending on whether the design objective is robust detection, cryptographic pseudorandomness, blind inference, or adversarial sensing impairment.

2. Low-sidelobe secure ACF in radar waveform design

In nonlinear frequency modulation (NLFM) waveform design, the ACF is controlled indirectly through spectral shaping. A constant-amplitude waveform is written as

$$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$

with $a(t)=A$ constant and instantaneous frequency

$f(t)=\frac{1}{2\pi}\varphi'(t).$

Under the stationary phase concept (SPC), the spectral magnitude is linked to the second derivative of phase, so a chosen power spectral density (PSD) window determines a group delay $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$0, then an instantaneous frequency law $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$1, and finally the phase $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$2 (Ghavamirad et al., 2018).

The optimization-based refinement in “Sidelobe Level Reduction in ACF of NLFM Waveform” formulates waveform synthesis as a constrained least-squares problem that matches a desired spectral magnitude $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$3 while preserving constant envelope. The design error is

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$4

which is converted to a phase-matching problem using $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$5. In discrete form, with $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$6, the solution update is the phase of the inverse transform of $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$7, enforcing $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$8 at each iteration (Ghavamirad et al., 2018). The algorithm yields a nonincreasing minimum-error sequence,

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$9

and the paper states that the trend decrement of minimum error guarantees convergence (Ghavamirad et al., 2018).

For six windows—Raised-Cosine, Taylor, Chebyshev, Gaussian, Poisson, and Kaiser—the reported ACF peak sidelobe reductions relative to the stationary phase method average about $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$0 dB, with final PSL values near $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$1 dB for most cases (Ghavamirad et al., 2018). The detailed values are $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$2 dB for Raised-Cosine, $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$3 dB for Taylor, $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$4 dB for Chebyshev, $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$5 dB for Gaussian, $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$6 dB for Poisson, and $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$7 dB for Kaiser (Ghavamirad et al., 2018). In this literature, low sidelobes are called secure in the sense of lower ambiguity, reduced masking of weak nearby targets, and lower vulnerability to clutter, interference, and deceptive returns.

The 2025 smoothing-spline extension retains the stationary-phase framework but replaces polynomial fitting of the inverse group delay by cubic smoothing splines. The functional minimized is

$S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$8

so the instantaneous frequency law is obtained as a curvature-controlled fit to sampled $S_m=(s_0,\dots,s_{m-1})\in\{-1,1\}^m$9 design points (Ghavamirad et al., 11 Jan 2025). This paper reports substantially larger PSL improvements than polynomial fitting: for a Gaussian window, PSL improves from $u\in F_m$0 dB to $u\in F_m$1 dB for $u\in F_m$2, and from $u\in F_m$3 dB to $u\in F_m$4 dB for $u\in F_m$5; for a Taylor window, PSL improves from $u\in F_m$6 dB to $u\in F_m$7 dB and from $u\in F_m$8 dB to $u\in F_m$9 dB, respectively (Ghavamirad et al., 11 Jan 2025). The cost is increased normalized mainlobe width, ranging from $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$0 to $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$1 in the spline cases versus about $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$2 to $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$3 under polynomial fitting (Ghavamirad et al., 11 Jan 2025).

These results establish a classical secure-ACF interpretation: security is identified with sidelobe suppression under constant-envelope constraints. A plausible implication is that this notion aligns more with robust sensing and LPI-oriented waveform engineering than with secrecy in the cryptographic sense.

3. Random-like secure ACF in cryptography

For periodic binary sequences, the central security question is whether nontrivial periodic autocorrelations remain small enough to resist correlation-based distinguishers and related attacks. In “Periodic autocorrelation of sequences,” the sequence model is uniform on $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$4, with independent Rademacher coordinates, and the maximum periodic autocorrelation is analyzed for prime $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$5 (Caullery et al., 2024).

The principal theorem states that, as $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$6 through primes,

$C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$7

equivalently

$C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$8

Thus the peak sidelobe level of a random binary sequence is typically of order $C_u(S_m)=\sum_{i\in F_m} s_i s_{i+u},$9 (Caullery et al., 2024).

The upper tail is controlled by a Chernoff/Hoeffding-type argument: $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$0 for $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$1 (Caullery et al., 2024). The lower side is obtained through a refined large-deviation argument and pairwise control of large-autocorrelation events. In particular, for sufficiently large $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$2,

$C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$3

and

$C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$4

McDiarmid-type concentration then shows that $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$5 is concentrated around its expectation (Caullery et al., 2024).

For a fixed nonzero shift $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$6, the random variables $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$7 are mutually independent for $C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$8, which yields

$C(S_m)=\max_{u\in F_m^*}|C_u(S_m)|.$9

The paper states that $p(t)$0 is approximately Gaussian for large $p(t)$1, so individual periodic autocorrelation values are typically of order $p(t)$2, whereas the maximum over all nontrivial shifts is of order $p(t)$3 (Caullery et al., 2024).

In this setting, a secure ACF is one that behaves like that of a truly random sequence: most $p(t)$4 should be $p(t)$5, and the peak $p(t)$6 should not greatly exceed $p(t)$7 (Caullery et al., 2024). The paper explicitly states that “the autocorrelation of a sequence is a useful criterion, among all, of resistance to cryptographic attacks,” and positions its result as a benchmark for pseudorandom constructions. Extremely small periodic ACF may still be desirable, but the paper notes that such behavior is atypical for random sequences and may reflect special algebraic structure (Caullery et al., 2024). This suggests that cryptographic security cannot be inferred from ACF alone, even though abnormal periodic autocorrelation is an immediate warning sign.

The paper also places itself in the line of work of Caullery–Férard–Rodier and Schmidt on autocorrelation and nonlinearity of random Boolean functions, presenting the sequence case as an analogue in which normalized autocorrelation concentrates around a constant (Caullery et al., 2024).

4. Data-independent ACF in chaotic communication systems

A different secure-ACF concept appears in chaos-based communication. In “Autocorrelation Invariance Property of Chaos for Wireless Communication,” the transmitted baseband signal is

$p(t)$8

where $p(t)$9 are independent information symbols and $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$0 is the base function of a chaotic shape-forming filter (CSF) (Yin et al., 2022). The paper’s central claim is that the ACF of the transmitted signal is identical to the ACF of the base function, regardless of the encoded information: $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$1

The derivation separates diagonal and off-diagonal terms in

$$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$2

Because $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$3 and $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$4 for $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$5, the off-diagonal terms vanish in expectation or under long averaging, so the second-order statistic depends only on $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$6 (Yin et al., 2022). The resulting ACF is therefore fixed by CSF parameters $$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$7, not by the transmitted bit pattern.

This invariance is then used to derive the received ACF over a multipath wireless channel,

$$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$8

with received signal

$$x(t)=a(t)e^{j\varphi(t)}, \qquad -\frac{T}{2}\le t \le \frac{T}{2},$$9

The received autocorrelation is

$a(t)=A$0

or, after rearrangement, a sum of shifted copies of the known function $a(t)=A$1 plus noise (Yin et al., 2022). Since $a(t)=A$2 is known a priori and AWGN contributes only at zero lag in expectation, the channel state information can be identified blindly, without explicit probe symbols.

The paper highlights two benefits: the CSI can be identified without the probe information known to the receiver, improving bandwidth efficiency, and the correlation operation is insensitive to channel noise, improving identification accuracy compared with commonly used methods (Yin et al., 2022). The channel-estimation error metric is

$a(t)=A$3

Qualitatively, the proposed ACF-based blind estimator is reported to outperform blind MNPE, MPSV, and ML-PSV, while approaching the non-blind least-squares benchmark with Gaussian input at low to moderate SNR (Yin et al., 2022).

The security interpretation is narrower than in cryptography. The paper states that second-order statistics of the transmitted signal are independent of the information symbols, so an observer using only ACF or PSD cannot infer the data (Yin et al., 2022). At the same time, the paper does not claim information-theoretic secrecy; it explicitly notes that an adversary with full CSF knowledge and sufficient SNR could still demodulate the system. Accordingly, the secure ACF here is best understood as a data-independent second-order signature, useful for blind inference and limited second-order concealment, rather than as a complete secrecy mechanism.

5. Adversarial secure ACF engineering in ISAC

In sensing-secure ISAC, the secure ACF is deliberately designed to be harmful to an unauthorized passive sensing receiver. For an OFDM waveform with subcarrier power allocation $a(t)=A$4 and symbols $a(t)=A$5, the discrete ACF, which is the zero-Doppler cut of the ambiguity function, is

$a(t)=A$6

with

$a(t)=A$7

Conventional radar would seek an impulse-like $a(t)=A$8; by contrast, “Sensing-Secure ISAC: Ambiguity Function Engineering for Impairing Unauthorized Sensing” designs $a(t)=A$9 to contain strong periodic artificial peaks that appear as ghost targets to an eavesdropper using matched filtering (Han et al., 2 Oct 2025).

The target ACF is a Dirac comb: $f(t)=\frac{1}{2\pi}\varphi'(t).$0 where $f(t)=\frac{1}{2\pi}\varphi'(t).$1. This compresses the effective unambiguous range to $f(t)=\frac{1}{2\pi}\varphi'(t).$2, with artificial ranges

$f(t)=\frac{1}{2\pi}\varphi'(t).$3

For unit-amplitude constellations, Theorem 1 gives a subcarrier-power pattern with high power $f(t)=\frac{1}{2\pi}\varphi'(t).$4 on a periodic index set $f(t)=\frac{1}{2\pi}\varphi'(t).$5 and low power $f(t)=\frac{1}{2\pi}\varphi'(t).$6 on its complement: $f(t)=\frac{1}{2\pi}\varphi'(t).$7 with

$f(t)=\frac{1}{2\pi}\varphi'(t).$8

This periodic power allocation shapes the ACF into the desired comb (Han et al., 2 Oct 2025).

Security is quantified by the eavesdropper’s peak sidelobe level and integrated sidelobe level,

$f(t)=\frac{1}{2\pi}\varphi'(t).$9

For large $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$00,

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$01

and

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$02

with

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$03

in Corollary 4 (Han et al., 2 Oct 2025). For PSK, $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$04, so the random-signal sidelobe floor vanishes; for QAM, $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$05, adding an elevated sidelobe floor that further harms the unauthorized receiver (Han et al., 2 Oct 2025).

The legitimate receiver avoids these ACF artifacts with a reciprocal filter

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$06

yielding

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$07

so the effective channel term no longer carries $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$08 and the artificial targets disappear (Han et al., 2 Oct 2025). The price is SNR loss. If $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$09 and $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$10 are the matched-filter and reciprocal-filter SNRs, then

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$11

and under the comb power pattern,

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$12

Hence stronger or more numerous artificial peaks imply greater legitimate SNR loss (Han et al., 2 Oct 2025).

The paper formulates a convex optimization problem to maximize a weighted combination of communication rate and legitimate sensing performance while guaranteeing minimum sensing-security levels $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$13 and $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$14 (Han et al., 2 Oct 2025). Numerical results show that with secure ACF design, Eve’s detection probability collapses and range estimation RMSE increases by more than $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$15 m in one reported two-target scenario, while Alice experiences only a modest detection penalty if the design avoids very large $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$16 or very small $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$17 (Han et al., 2 Oct 2025). This is the strongest adversarial interpretation of secure ACF in the cited literature: the ACF is intentionally made deceptive for an unauthorized receiver.

6. Direct ACF-domain signaling and unifying trade-offs

Auto-correlation Function Keying extends ACF design from waveform shaping to modulation architecture. For a frequency-domain transmit vector $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$18, the periodic ACF of the time-domain OFDM signal is

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$19

The expected sidelobe level, peak sidelobe level, and peak sidelobe level ratio are

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$20

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$21

The paper’s motivation is that ESL alone does not control large spurious sidelobe peaks in individual payload realizations, which may deteriorate weak-target detection performance (Zhao et al., 16 Jun 2026).

The modulation defines a nominal ACF-domain vector $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$22 with conjugate-symmetric sidelobe entries carrying data. The actual transmitted vector is

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$23

where $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$24 is a PSK phase vector and $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$25 is the ACF-domain constellation stream (Zhao et al., 16 Jun 2026). If the spectral non-negativity constraint

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$26

holds, then the nominal P-ACF equals the actual P-ACF exactly: $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$27 This gives exact control of the nominal periodic ACF (Zhao et al., 16 Jun 2026).

For ACFK, the nominal sidelobe control is explicit: $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$28 and for any nonzero sidelobe bin,

$$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$29

The spectral non-negativity violation probability decays exponentially in $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$30: $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$31 so for moderate $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$32 the actual P-ACF coincides with the nominal design with high probability (Zhao et al., 16 Jun 2026). When violations occur, the actual P-ACF equals the nominal one plus a perturbation term $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$33, and the resulting PSLR degradation is bounded probabilistically (Zhao et al., 16 Jun 2026).

The information-theoretic formulation asks for mutual-information maximization under per-realization PSL constraints and a total power budget. In quasi-static frequency-flat channels, the paper proves that a continuous ACF-domain uniform construction is asymptotically optimal at high SNR, and ACFK is introduced as a finite-constellation realization of that principle (Zhao et al., 16 Jun 2026). Relative to a generalized PAS baseline, the reported results show much tighter PSLR control and improved weak-target detection performance under comparable sensing and communication settings (Zhao et al., 16 Jun 2026).

Across the cited works, three recurring technical trade-offs define the secure-ACF problem. First, sidelobe suppression versus resolution: in NLFM radar, lower PSL generally broadens the mainlobe (Ghavamirad et al., 2018, Ghavamirad et al., 11 Jan 2025). Second, sensing security versus legitimate performance: in sensing-secure ISAC, higher PSL and ISL for Eve imply higher SNR loss for Alice (Han et al., 2 Oct 2025). Third, ACF control versus communication efficiency: in ACFK, increasing $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$34 tightens PSLR but reduces the effective SNR of ACF-domain symbols (Zhao et al., 16 Jun 2026).

A secure ACF is therefore best treated as a domain-dependent design object rather than a fixed formal definition. In radar and sonar it is typically a low-sidelobe, low-ambiguity correlation profile; in cryptography it is a random-like periodic structure whose extremes follow the $$R_s(\tau) = \int_{-\infty}^{\infty} s(t)\, s^*(t-\tau)\, dt.$$35 law; in chaos-based communications it is a data-independent second-order invariant; and in secure ISAC it may be intentionally deceptive, with engineered sidelobes that are removable by a privileged receiver but damaging to an unauthorized one (Ghavamirad et al., 2018, Caullery et al., 2024, Yin et al., 2022, Han et al., 2 Oct 2025, Zhao et al., 16 Jun 2026).