# Development Roadmap: Analogue Audio Encoding with Fingerprints and Cryptographic Verification

Introduction and Project Overview

This project aims to encode and recover information using sound-based analogue models combined with digital fingerprinting and cryptographic verification. We draw inspiration from analogue gravity (e.g. sonic or acoustic black holes in fluid flow) where sound waves can be trapped like light in a black hole , as well as from AI watermarking techniques that hide robust codes in model outputs . The goal is to simulate a “digital black hole channel” using audio: information (the fingerprint) is embedded into a sound waveform, sent through a lossy or absorbing audio environment analogous to a black hole event horizon , and then partially recovered. By combining error-correcting codes and cryptographic primitives with this analogue approach, we will ensure that even small remnants of the signal can reveal a high-confidence fingerprint and provide verifiable proof of origin or ownership.

Key components to integrate:
• Analogue Sound-Wave Simulation: Using phononic black hole concepts (trapping and absorbing sound) as a channel model for information loss  .
• Digital Fingerprinting & Error Correction: Embedding a robust fingerprint of the data (e.g. a BLAKE3 hash) with redundancy (e.g. RaptorQ fountain codes) so that partial recovery is sufficient  .
• Cryptographic Verification: Using Bitcoin/Ethereum-grade cryptography (e.g. secp256k1 ECDSA signatures) to sign the fingerprint, enabling ownership proof and authenticity checks .

Below is a structured technical plan broken into tasks. Each task includes technical goals, references to prior research or tools, and illustrative pseudocode or code snippets. The roadmap emphasizes how to combine partial fingerprint recovery with error correction for high-probability detection, how to use audio as a medium to simulate an information “black hole”, and how to integrate the analogue simulation with cryptographic verification.

Task 1: Research Foundations and Concept Design

Goal: Establish the theoretical and experimental foundation for the project, guiding the design of the analogue and digital components.
• Analog Gravity & Phononic Black Holes: Research how black hole analogues trap waves and encode information. In fluid acoustic analogues, a flow faster than sound creates a sonic event horizon beyond which phonons (sound quanta) cannot escape . Notably, infalling perturbations can imprint information on the field – e.g. quantum “hair” encoding a unique fingerprint of what fell in . These concepts inform our design of a sound-based channel that absorbs most information while leaving a subtle fingerprint. We will design an analogue model where a region in the medium acts like a black hole, trapping waves and only allowing minimal leakage (analogous to Hawking radiation) that carries an “echo” of the input signal.
• Sound/Diffusion-Based Information Recovery: Review prior experiments in recovering signals from heavily distorted or diffused media. In acoustic engineering, an Acoustic Black Hole (ABH) effect is achieved by gradually reducing wave velocity to near zero at a boundary, plus damping material, to absorb ~100% of incident sound . This means almost no direct signal comes through, challenging us to recover information from what little does leak or reflect. We note that robust watermarking methods often rely on combining multiple weak signals. For example, Westerfeld’s audiowmark project spreads a payload across multiple audio blocks and uses error-correcting codes so that merging several partial decodings yields the correct message  . This informs our strategy: embed the fingerprint in a redundant way across the audio signal so that even if the channel obliterates most of it, the fragments that do survive can be aggregated to reconstruct the fingerprint.
• AI Watermarking & Error-Correcting Codes: Study state-of-the-art watermarking for AI-generated content that use cryptographic approaches. A recent paradigm is to encode a message with an error-correcting code (ECC) and embed it in model output such that it can be recovered even after perturbations . Christ & Gunn (CRYPTO 2024) propose pseudorandom codes (error-correcting codewords that look random without a secret key) to achieve robustness, undetectability, and unforgeability in text or image watermarks  . This guides us to use strong ECC (for robustness) and secret keys (for security) in our fingerprint scheme. Prior audio watermarking research also shows the benefit of ECC: e.g. using convolutional or Hamming codes to correct errors in watermarked audio signals . We will choose a modern ECC, likely fountain codes (RaptorQ), due to their ability to reconstruct data from arbitrary subsets of packets  .
• Cryptographic Ownership Proofs: Decide how to incorporate cryptographic authentication. Bitcoin uses the elliptic curve secp256k1 for its public-key signatures (ECDSA) , and libraries like libsecp256k1 (with Python bindings such as coincurve) offer high-performance implementations . We plan to use a keypair to sign the fingerprint, so that possession of the corresponding private key is proven by a valid signature. This will ensure that even if the fingerprint is recovered from the “black hole” audio, it cannot be forged by an attacker without the key. Additionally, we consider using a watermarking key (symmetric secret) during embedding for security: as audiowmark suggests, a secret watermark key can control pseudo-random embedding patterns, preventing others from decoding or removing the watermark . In summary, cryptography will be used both to hide the fingerprint embedding (secret key) and to publicly verify the fingerprint’s origin (digital signature).

References & Tools: Academic literature on analogue gravity (e.g. sonic black holes by Unruh, Steinhauer, etc.), documentation of audio watermarking tools (audiowmark  ), Cloudflare’s blog on cryptographic watermarks , the BLAKE3 hash and RFC6330 RaptorQ specs, and Bitcoin’s secp256k1 ECDSA guidelines. These will guide the technical choices in subsequent tasks.

Task 2: Simulate a Sound-Based “Black Hole” Channel

Goal: Develop a wave-based simulation that mimics a black hole-like channel, i.e. a medium where most of the input sound energy is absorbed or lost, leaving only a faint, recoverable signal. This provides a testbed for analogue information encoding and recovery.
• Design the Acoustic Simulation: We will simulate the propagation of sound waves in a 1D or 2D medium with spatially-varying properties to create an analog event horizon. One approach is to implement a finite-difference time-domain (FDTD) solver for the 2D acoustic wave equation. For example, we set up a grid with a speed of sound c(x,y) that decreases towards a center region and drops to near zero at the core. According to acoustic black hole theory, a gradual power-law tapering of wave velocity to zero, combined with damping, yields nearly total absorption of waves . In practice, this means in our simulation region we define a circular “black hole” area where waves slow down and a high damping coefficient is applied, simulating an absorption horizon.
• Open-Source Tools: We can use Python with libraries like NumPy for the core simulation math. If performance is an issue, consider using optimized libraries or frameworks: e.g. k-Wave (an open-source acoustic FDTD toolbox) for MATLAB/Python, or a finite-element solver (like FEniCS) if a higher-level approach is desired. However, a custom NumPy FDTD is feasible for 2D grids on modern hardware. We’ll also incorporate absorbing boundary conditions (or a PML – perfectly matched layer) at the edges of the simulation to avoid reflections. This ensures any wave escaping the black hole region can be cleanly recorded.
• Simulated Wave Input: We will inject an input waveform that carries the encoded fingerprint (from Task 4) at one side of the grid (or as an initial condition). For example, the source could be a small speaker-emulating boundary emitting the signal towards the center. The simulation will run for a set duration, during which most of the wave energy will enter the absorbing region (the black hole) and be lost. We will place one or more “detectors” in the simulation – e.g. probe points far from the hole – to record any escaping waves (analogous to Hawking radiation or scattered signals). The output at these probes is expected to be extremely weak compared to the input.
• Verify the Analogue Behavior: As a test, we can simulate a simple pulse or wave packet and observe how it behaves. We expect to see that the wave greatly diminishes upon reaching the black hole region, with only a very small ripple making it to the far side. We will tune parameters (gradient of sound speed, damping factor) to maximize absorption. Literature suggests that the closer the core sound speed is to 0 and the longer the taper region, the better the absorption . Ensuring a bit of controlled leakage is helpful (if absorption is 100%, recovery is impossible), so we might reduce damping slightly to allow a tiny fraction of the signal through or around the horizon.
• Illustrative Pseudocode: Below is a conceptual snippet for a 2D FDTD wave simulation with an absorbing “black hole” region:

import numpy as np

# Simulation grid and parameters
Nx, Ny = 200, 200
dx = dy = 0.01 # spatial step
dt = 0.0001 # time step
c0 = 340.0 # base speed of sound (m/s)
# Define spatially varying speed: slower in center (black hole)
c = c0 * np.ones((Nx, Ny))
center = (Nx//2, Ny//2)
r_max = 50 # radius of influence for the black hole region
for i in range(Nx):
for j in range(Ny):
r = np.hypot(i-center[0], j-center[1])
if r < r_max:
# reduce sound speed following a taper (e.g., quadratic taper)
c[i,j] = c0 * (r / r_max)**2 # approaches 0 at center
# Damping region: high damping at center to absorb waves
damping = np.zeros((Nx, Ny))
damping[c < 0.2 * c0] = 0.02 # small damping everywhere sound speed is very low

# Initialize wavefield arrays
u_prev = np.zeros((Nx, Ny)) # wave at t-dt
u_curr = np.zeros((Nx, Ny)) # wave at t
u_next = np.zeros((Nx, Ny)) # wave at t+dt

# Source: a pulse at left boundary (to inject the encoded signal)
source_x, source_y = 5, Ny//2
signal = generate_audio_signal(length=10000) # from Task 4 encoding
signal_ptr = 0

# Time-stepping loop
for t in range(10000): # number of time steps
# Update interior using 2D wave eq: u_next = 2*u_curr - u_prev + c^2 * laplacian(u_curr) - damping*u_curr
laplacian = (
np.roll(u_curr, 1, axis=0) + np.roll(u_curr, -1, axis=0) +
np.roll(u_curr, 1, axis=1) + np.roll(u_curr, -1, axis=1) -
4 * u_curr
) / (dx**2)
u_next = (2 * u_curr - u_prev) + (c**2 * dt**2) * laplacian - damping * u_curr

# Source injection (e.g., adding the next sample of our audio signal at the source location)
if signal_ptr < len(signal):
u_next[source_x, source_y] += signal[signal_ptr]
signal_ptr += 1

# Swap time steps for next iteration
u_prev, u_curr = u_curr, u_next

# Record at detector (e.g., on the far right side of grid)
detector_output = u_curr[Nx-5, Ny//2] # sample near opposite boundary
record(detector_output)

In this pseudocode, generate_audio_signal would produce the waveform encoding the fingerprint (to be implemented in Task 4). The simulation loop updates the wavefield using the finite difference approximation of the wave equation. A damping term is included where c is very low (center) to absorb energy. A detector at Nx-5 (far right) records any transmitted wave. After the simulation, recorded_output will contain the audio that made it through the analogue channel.
• Outcome: By the end of this task, we will have a working analogue simulation representing a harsh channel that swallows most of the signal. This environment will be the test channel for our fingerprint. We expect that without error correction, almost no intelligible information would be recoverable, setting the stage for the next tasks.

References: Sonic black hole concept (Unruh 1981) , acoustic black hole absorption in engineering , and open-source acoustic simulation tools (e.g., k-Wave, FEniCS). These inform the configuration of our simulation.

Task 3: Fingerprint Generation and Error-Correction Encoding

Goal: Create a robust digital fingerprint of the content, and encode it with redundancy so that it can be recovered from partial data. We will use a cryptographic hash for the fingerprint and a fountain code (like RaptorQ) for forward error correction (FEC).
• Generate a Content Fingerprint (BLAKE3): We choose BLAKE3 as the hashing algorithm for its speed and security . The fingerprint could be, for example, a hash of the original data we want to prove ownership of (if the content is large, we hash it) or a unique identifier for the AI model or user. BLAKE3 by default produces a 256-bit output (32 bytes) . This hash will serve as the core message we want to hide in the audio. If needed, we can include additional info in the message (like a timestamp or short identifier) before hashing, but the final size should be small (tens of bytes) for embedding feasibility.
• Error Correction with Fountain Codes: To ensure the 32-byte fingerprint can be recovered from partial pieces, we apply a fountain code. We specifically plan to use RaptorQ (RFC 6330) because it allows generating a potentially limitless stream of encoded packets such that any subset of roughly k packets (equal to the original number of symbols) is enough to reconstruct the data  . RaptorQ has excellent efficiency: e.g., a 1 MB file can be recovered from just 1.002 MB of arbitrary fragments , indicating only slightly more than the original count of symbols is needed. We will tune the redundancy for our scenario: since our channel is extremely lossy, we might generate, say, 3–5× redundancy (e.g., encoding the 32-byte data into 100+ encoded symbols, each maybe a few bytes). Even if only 20–30% of those symbols survive, the decoder can still succeed.
• Open-Source Libraries: We can leverage existing implementations. libRaptorQ (C++11) with Python CFFI bindings is available on PyPI . Also, a Rust implementation with Python bindings exists (e.g., via pyo3) . We will use one of these to avoid reinventing the code. Similarly, for BLAKE3, the official library provides a Rust and C implementation with Python bindings; on PyPI the blake3 module can be used (it’s a wrapper around the official implementation). These libraries ensure efficient and reliable hashing and FEC encoding/decoding.
• Encoding Process: Take the fingerprint (hash output bytes) and feed it into the RaptorQ encoder to produce a large set of encoding symbols. We will treat the fingerprint bytes as the “source data” for FEC. We can configure the symbol size and number of symbols. For example, using symbol size of 1 byte (or a few bytes), RaptorQ can create many symbols equal to that size. We request a certain number of output symbols (some multiple of the original number of bytes). Each symbol will have an ID and data. We may shuffle or interleave these symbols over time/frequency when embedding (in Task 4). The decoder on the other end will require any subset of symbols that meets the reconstruction threshold. According to RaptorQ, in most cases receiving just k symbols (where k is number of source symbols) is enough; only rarely a few extra symbols are needed . We will generate, say, n = 2*k symbols to be safe, knowing the channel might drop well over half.
• Illustrative Code Snippet: Using pseudocode for clarity:

from blake3 import blake3
from raptorq import Encoder

data = original_content_bytes # content we want to fingerprint
# 1. Compute BLAKE3 hash (32 bytes output)
digest = blake3(data).digest() # 256-bit fingerprint
print("Content fingerprint (hex):", digest.hex())

# 2. Set up RaptorQ fountain encoder
encoder = Encoder(digest, symbol_size=2) # e.g., split data into 2-byte symbols
# Generate encoded symbols (with redundancy)
num_source_symbols = encoder.symbols # number of original symbols (digest length/2)
num_encoded_symbols = int(num_source_symbols * 3.0) # 3x redundancy
encoded_blocks = []
for sym_id in range(num_encoded_symbols):
packet = encoder.encode(sym_id)
encoded_blocks.append(packet)
print(f"Generated {len(encoded_blocks)} encoded symbols from fingerprint.")

In this snippet, Encoder.encode(sym_id) would produce an encoding packet for a given symbol ID (the exact API depends on the library). We chose a symbol size of 2 bytes for demonstration; the library will pad if needed. The output packet likely contains an ID and symbol payload bytes. These encoded_blocks now represent the fingerprint in a redundant way.
• Result: By the end of this task, we have a method to obtain a large set of small fingerprint-encoding units. We have also verified the tools: for example, hashing some test data with BLAKE3 and successfully decoding it with RaptorQ in a simulated erasure scenario (dropping random symbols to ensure that decoding still works as expected). This step sets the stage for embedding those symbols into the audio channel.

References: BLAKE3 official documentation (notes high speed and 256-bit output)  , RaptorQ RFC and libRaptorQ usage (fountain codes allow recovery from any k out of N pieces)  . These justify our choice of fingerprinting and FEC approach.

Task 4: Embed the Fingerprint in an Audio Signal and Transmit through the Analogue Channel

Goal: Translate the error-corrected fingerprint symbols into a sound waveform, send it through the simulated black-hole channel (from Task 2), and effectively embed the information within the audio. This involves designing an acoustic encoding scheme (how the bits/symbols modulate the audio) and ensuring it survives the channel’s distortions as much as possible.
• Audio Encoding Strategy: We need to map the encoded_blocks (from Task 3) onto an audio signal. Each encoded symbol consists of a small payload (a few bytes). We will design a modulation scheme robust to the channel effects:
• One approach is Spread Spectrum embedding: spread each symbol’s bits across a wide time or frequency band with a pseudo-random sequence (using a secret key for sequence generation). This technique, akin to CDMA, can allow the symbol to be recovered even if parts of the frequency band are obliterated by the channel. Given our channel might absorb certain frequency ranges (depending on how the horizon is set up), spreading ensures no one frequency carries all the information.
• Another approach is to use frequency or phase modulation over short time slots: e.g., assign each possible symbol value to a distinct chirp or tone sequence. For instance, use a set of orthogonal frequencies or orthogonal codes to represent different symbol bits, so that even if amplitude is damped, a correlation detector might pick them up.
• For simplicity and clarity, we might start with a Frequency-Shift Keying (FSK) style: e.g., to send bit 1 play a 3 kHz tone for 1 ms, to send bit 0 play a 4 kHz tone, etc. However, we have many symbols to send, so more sophisticated multiplexing is needed.
• Time-Frequency Allocation: We can divide the audio transmission into frames corresponding to each symbol (or group of symbols). For example, if we generated 100 symbols, we send them sequentially in 100 time slots. Each time slot could be a short duration (a few milliseconds) where we embed the symbol’s data. Within that slot, we can encode the symbol bits via multiple frequencies. We may also send multiple symbols in parallel if the channel allows (e.g., using stereo channels or multiple frequencies at once), but given the complexity of the channel we might keep it serial for now.
• Using AI-Generated or Cover Audio (Optional): The problem statement mentions using audio or AI-generated audio as a medium. In practice, we could either transmit the fingerprint in a dedicated audio signal (like a high-frequency watermark above human hearing in a piece of music or noise), or actually modulate an existing audio clip slightly to hide the data (steganography). For development, we’ll likely use a known waveform (e.g., a simple sine or an artificial noise burst) to carry the fingerprint. We can experiment with embedding in silence vs. embedding in content. A cover audio might provide realism (the channel’s effect on a complex signal). For now, assume we directly synthesize an audio waveform representing the watermark.
• Embedding with a Secret Key: To improve security (prevent unauthorized detection or tampering), incorporate a secret watermark key (as identified in Task 1). This key can seed a pseudo-random number generator that controls aspects like frequency hopping patterns, phase rotations, or the ordering of symbol transmission. For example, using the key, we can pseudo-randomly permute the sequence of symbol transmissions, or choose a unique frequency for each symbol based on a hash of the key and symbol index. This means an attacker without the key cannot easily detect or decode the watermark, adding an unforgeability aspect .
• Synthesis Implementation: We will use a sampling rate (e.g., 48 kHz) for audio. Using Python (with NumPy or SciPy), we can generate the time-domain signal. For each symbol:
• Determine its bit pattern or value.
• Map it to one or more frequencies or a waveform. For example, if symbol is 8 bits, we could split into two 4-bit halves and use two frequencies in that slot, each slightly detuned around a center frequency (with patterns defined by key).
• Synthesize those sine waves for the duration of the symbol slot, possibly windowed with a smooth envelope to avoid clicks.
• Accumulate these into the final waveform array.

Here’s a simplified pseudocode to illustrate embedding bits with two-tone FSK per bit (for concept only):

import numpy as np

fs = 48000 # sampling rate
slot_duration = 0.002 # 2 ms per symbol slot
N = int(fs * slot_duration)
t = np.linspace(0, slot_duration, N, endpoint=False)

# Two frequencies for FSK (could be arrays for more complex modulation)
f0 = 3000.0 # frequency for bit 0
f1 = 4000.0 # frequency for bit 1

# Assume we have a list of bits to send (from the encoded symbols)
watermark_bits = get_bits_from_symbols(encoded_blocks) # e.g., list of 800 bits for 100 symbols of 8 bits each
audio_signal = np.zeros(len(watermark_bits) * N)

for i, bit in enumerate(watermark_bits):
freq = f1 if bit == 1 else f0
# Optionally, apply a random phase or slight freq offset from key
phase_offset = pseudorandom_phase(i, key)
tone = np.sin(2 * np.pi * freq * t + phase_offset)
# Fade in/out to reduce spectral splatter
tone *= np.hanning(N)
audio_signal[i*N:(i+1)*N] = tone

In this pseudocode, get_bits_from_symbols would convert our encoded symbol packets into a sequence of bits for transmission. We use two fixed frequencies for clarity; a real implementation might use a wider range or more complex modulation per symbol. We apply a Hanning window (np.hanning(N)) to smooth each tone burst. pseudorandom_phase(i, key) stands in for generating a random phase for each symbol based on the secret key (ensuring the pattern is repeatable by someone with the key but not obvious otherwise).
• Transmit through Simulation: The generated audio_signal will then be used in the simulation from Task 2. In the FDTD loop (Task 2 code), we called generate_audio_signal() which now corresponds to this embedding process. We inject the signal at the source and run the simulation. The output recorded at the detector is effectively the received audio after the black-hole channel.
• Expectation: Most of the energy of each symbol will vanish into the absorbing region. The received signal will be a very attenuated and likely noisy version of the original watermark signal. It might also be further distorted (the channel could disperse or shift phases). The next task will deal with decoding it.
• Confirmation & Adjustment: We will test a simpler scenario first (like one symbol or a short known bit pattern) to ensure the simulation is capturing something at the detector. We might need to amplify the recorded signal or apply filtering to focus on the frequency band we used (since we know where our signal lies, e.g., around 3–4 kHz in the example). If the output is below numeric noise, we can increase the symbol amplitudes or reduce damping slightly. We will likely iterate on the modulation scheme for resilience: for example, if pure tones at fixed frequencies don’t come through well, we might try wider-band chirps (which could survive better if the channel selectively attenuates certain frequencies).

References: Prior art in watermark embedding: e.g., robust audio watermarking often uses spread spectrum or echo-hiding techniques that are resilient to filtering. The audiowmark tool suggests dividing the watermark into blocks (A and B) and even using time diversity (patterns at different times) ; our slot-based approach aligns with that idea. No specific external library is needed for tone synthesis aside from standard math libraries. We also draw on communication theory for modulation techniques and may refer to texts on DSSS (Direct Sequence Spread Spectrum) for guidance.

Task 5: Recover the Fingerprint from Audio and Apply Error Correction

Goal: Process the output audio from the analogue simulation to detect the embedded fingerprint symbols, then use the FEC decoder to reconstruct the original fingerprint bits. This step is critical to achieve high-probability detection of the information despite the severe channel losses.
• Pre-processing the Received Signal: The recorded audio from the simulation is likely very weak and noisy. We may first apply band-pass filtering around the frequencies of interest (e.g., 3–4 kHz in the example) to improve SNR. If multiple frequency bands or a spread-spectrum approach was used, a more complex matched filtering or correlation will be needed. For each symbol time slot, we need to decide which symbol (or bits) was sent based on the received waveform segment:
• For an FSK example, we could compute the FFT of the segment and see if there’s energy at f0 or f1.
• More reliably, we could correlate with reference signals: correlate the segment with a reference tone at f0 and f1 (or with the expected PN sequence, if using DSSS). The higher correlation indicates which bit was likely.
• If using spread-spectrum or multi-frequency encoding, we might perform a cross-correlation with the known spreading sequence.
• If symbols were encoded as complex chirps or blocks, possibly run a small decoder (e.g., if we treated it as an OFDM symbol, do an FFT for subcarriers).
• Symbol/Bits Detection: Implement a detection algorithm to retrieve bits or symbol values. For instance:

import numpy.fft as fft

slot_len = int(fs * slot_duration)
recovered_bits = []
for j in range(num_slots):
segment = recorded_signal[j*slot_len:(j+1)*slot_len]
# Go to frequency domain
spectrum = np.abs(fft.rfft(segment * np.hanning(len(segment))))
f_axis = fft.rfftfreq(len(segment), d=1/fs)
# Measure energy around f0 and f1
energy_f0 = spectrum[(f_axis > f0-50) & (f_axis < f0+50)].sum()
energy_f1 = spectrum[(f_axis > f1-50) & (f_axis < f1+50)].sum()
bit = 1 if energy_f1 > energy_f0 else 0
recovered_bits.append(bit)

The above is a simplistic approach for an FSK scheme: it sums the spectral energy in a narrow band around the expected frequencies and chooses the bit based on which frequency had more energy. We use a Hanning window on the segment to reduce spectral leakage.

For more advanced schemes, we would replace this with the appropriate demodulation. If we used a pseudo-random spread sequence, we would correlate segment with the same sequence (mixed with the carrier) to detect the embedded chip pattern. If using multiple frequencies, we might detect each one similarly.
• Reassemble Symbols: Once we have a sequence of bits (or symbol values), we group them back into the original symbol packets. For example, if each symbol was 8 bits, take recovered_bits in chunks of 8. Each chunk should correspond to the encoded symbol payload. We also must retrieve symbol IDs if the code requires (fountain codes usually need to know which symbol is which). Depending on how we embedded, the symbol ID might be implicit from the position (e.g., the j-th slot corresponds to symbol ID j, if we sent in order). If we randomized order via key, we can invert that permutation now (using the same key to know which slot was which ID).
• Error Correction Decoding (RaptorQ): Now feed the collection of recovered symbols into the RaptorQ decoder. For instance, using libRaptorQ’s decoder:

from raptorq import Decoder

# Initialize decoder with expected original length (32 bytes) and symbol size used
decoder = Decoder(expected_length=32, symbol_size=2)
for pkt in recovered_symbol_packets:
decoder.add_symbol(pkt)
# Try to decode
if decoder.is_decodable():
output_data = decoder.decode()
else:
output_data = None

The recovered_symbol_packets would be constructed from the detected data. If our modulation preserved the symbol ordering and each symbol’s data, we can pair each recovered symbol value with its ID (e.g., if encoder.encode(sym_id) required ID, likely the packet contains it or we know it by sequence). We add all recovered packets to the decoder. RaptorQ can handle missing symbols; as long as enough are added, is_decodable() will return true. We then get the output_data, which should match the original fingerprint bytes.
• High-Probability Detection: Thanks to error correction, we don’t need 100% symbol recovery. For example, if we transmitted 100 symbols and only 50 arrive correctly, decoding might still succeed (given the nature of fountain codes, success probability is high when you have >= k symbols, here k would be (32 bytes)/(symbol_size) symbols). We overbuilt redundancy to account for this. The design target is that with the expected channel conditions, the decoder succeeds with high probability. We will verify this by simulations, perhaps running the whole pipeline multiple times (Monte Carlo runs with different noise seeds) to estimate the success rate. If it’s too low, we will adjust redundancy or encoding.
• Verification of Decoding: We should confirm that the output_data indeed equals the original hash (digest). To be thorough, we can recompute BLAKE3 on the content and compare, or simply compare to the stored original digest. This ensures no decoding error slipped through (though FEC decoding either gives correct data or fails; there’s a small chance of wrong data if an undetected error occurs, but cryptographic hashes are random enough that an incorrect decode would likely produce a hash not matching anything expected, which we would catch in the verification step).
• Partial Recovery and Merging Blocks: Our approach inherently merges information from multiple parts of the signal. The audiowmark results we studied show that combining pieces from multiple blocks yields the most reliable decoding . Similarly, our fountain code decoder is effectively combining all the partial “blocks” (symbols) to reconstruct the whole. This is the digital equivalent of summing up tiny Hawking radiation signals over time to reveal the hidden information from a black hole. If some symbols were especially hard to detect, we don’t necessarily need them all; any sufficient subset works . This property drastically improves our chances of recovery compared to naive encoding.
• Error Rates and Refinement: We will measure how many symbols (or bits) were incorrectly detected before decoding. If the error-correction fails (decoder not decodable), that implies too much loss. We can then refine the modulation or add more redundancy. Perhaps increase symbol count or use a stronger code (RaptorQ is already strong, but we could even do multiple layers e.g. small Reed-Solomon code on top if needed). Also, if the analogue channel is too destructive (maybe absolutely nothing gets through in current setup), we might dial it back or provide alternative paths (e.g., if no direct transmission, maybe rely on slight reflections or noises – akin to how information might still be preserved as correlations , but that’s advanced).

References: The Cloudflare blog on cryptographic watermarking underscores using ECC such that decoding succeeds if the watermark is present even with many bit flips  – our use of FEC accomplishes this. The audiowmark tool’s strategy of verifying decoded messages against expectations  is analogous to our verification of the hash and signature. Finally, the fundamental property of fountain codes from the Error Correction Zoo: the receiver can recover data from any subset of encoding symbols above a threshold , directly enabling our partial recovery approach.

Task 6: Cryptographic Verification and Ownership Proof Integration

Goal: Incorporate cryptographic signing to provide verifiable proof of ownership or origin for the recovered fingerprint. This adds an additional layer so that even after recovering the fingerprint, one can prove who embedded it or that it matches a known identity.
• Selecting Cryptographic Primitive: We choose the secp256k1 elliptic curve and ECDSA (Elliptic Curve Digital Signature Algorithm) as used in Bitcoin/Ethereum for signing. This gives us a 256-bit private key and a 512-bit signature (64 bytes) typically, with a 33-byte public key (compressed form). The reason to use secp256k1 ECDSA is its wide adoption and availability of optimized libraries . The same approach would work with any strong signature scheme (even Ed25519 or RSA), but using secp256k1 keeps the theme consistent with blockchain tech and ensures we can verify with existing tools if needed.
• Generating Keys: We will generate a keypair for the “owner”. In practice, this could be the user who creates the watermark or the AI model vendor, etc. The private key will sign the fingerprint, and the public key (or its hash) will be used to verify. We might derive an identity from the public key (e.g., a Bitcoin address or Ethereum address) to label the watermark.
• Signing the Fingerprint: After obtaining the fingerprint (hash) of the content in Task 3, we will sign that digest with ECDSA. We do this before encoding it into the audio (i.e., at embedding time, the signature is known). There are two ways to include the signature:
1. Out-of-band verification: The signature is not embedded in the audio (to save capacity) but can be provided alongside the content or looked up in a registry/blockchain. The recovered fingerprint (hash) can then be verified against the signature using the signer’s public key. This is simpler and avoids adding a lot of bits to the watermark payload.
2. In-band embedding: Alternatively, if we wanted the audio itself to carry proof without external data, we could try to embed the signature bits as well. However, a 64-byte signature would drastically increase the payload (we’d then be embedding ~96 bytes total: 32-byte hash + 64-byte sig). This may not be feasible given our channel limitations, so we lean towards out-of-band or partial in-band (maybe embed a shorter verification tag).
• Recommended Approach: We will take the approach of embedding the 32-byte hash (already being done) and not embed the 64-byte signature, but instead make the signature available through another mechanism. For example, the signature could be stored on a blockchain or in a database keyed by the content’s ID, or even in the metadata of the content file. The important part is that the fingerprint links to the signature. Anyone who recovers the fingerprint from the audio can then retrieve the signer’s public key and verify the ECDSA signature, confirming the watermark’s legitimacy. This method is analogous to C2PA provenance metadata but with the robust watermark as the carrier of the ID  .
• Using Libraries: We can use the coincurve Python library (which wraps libsecp256k1) for signing, as it is optimized and widely used . Alternatively, Python’s ecdsa library can do secp256k1 as well. Coincurve usage example:

from coincurve import PrivateKey

# Owner's private key (could be loaded or generated)
priv_key = PrivateKey() # generate new random key
pub_key = priv_key.public_key # to distribute or publish

# Sign the 32-byte fingerprint
signature = priv_key.sign(digest)
print("Signature length:", len(signature), "bytes")

# Later, verification (to be done by verifier with pub_key)
assert pub_key.verify(signature, digest)
print("Signature verified OK.")

This will produce a DER-encoded signature by default; we might want a compact 64-byte signature (coincurve provides sign_recoverable or similar for compact forms). In any case, we obtain signature.
• Integrating with the Pipeline: When embedding, we might include a small identifier in the watermark payload indicating whose key to use. For example, we might embed a short key ID or the last few bytes of the public key, so that a verifier knows which public key corresponds. However, if only one owner is expected, this is not needed; the verifier will be given the correct public key out-of-band. In a multi-owner scenario, embedding an ID (which could be as small as 4 bytes if collision-resistant enough or an index in a registry) would help. We must account for that in the fingerprint design if used.
• Post-Recovery Verification: After Task 5 yields the output_data (the recovered fingerprint bytes), the final step is to verify the signature. The verifier (which could be the system itself or an external agent) will do:

if output_data is not None:
if pub_key.verify(signature, output_data):
print("Watermark ownership verified: signature matches.")
else:
print("Verification failed: fingerprint not signed by expected key!")

If the verification passes, we have cryptographic proof that the holder of the private key (the owner) indeed generated that fingerprint. This connects the recovered data to an identity or ownership claim. If the verification fails, it indicates either an inauthentic watermark or a decoding error (which should be extremely unlikely if error correction succeeded and the signature is correct – any error in output_data would cause signature mismatch, which is a good fail-safe).
• Security Considerations: By using a strong signature, even if someone could fully recover the watermark from the audio, they cannot forge a new valid watermark without the private key. They also cannot alter the content and have it produce the same fingerprint unless they have a hash collision (BLAKE3 is collision-resistant). This ensures that our watermark not only survives analogue destruction but also resists forgery. It addresses the unforgeability goal often cited in watermarking schemes.
• Optional Ownership Proof Extensions: In a real deployment, we might log the fingerprint and signature on a blockchain as well, so there’s a public ledger of the claim (tying into concepts like NFT or content provenance frameworks like C2PA). That is beyond the scope of the dev roadmap, but this cryptographic backbone would allow such integration easily.

References: Bitcoin Wiki on secp256k1 and ECDSA usage  (gives context that secp256k1 is the curve used for Bitcoin’s signatures), coincurve documentation (Python bindings for libsecp256k1) indicating it’s a safe and fast choice . Also, the Cloudflare blog’s discussion of watermark provenance ties into our use of signatures as an orthogonal method (they mention C2PA’s chain of digital signatures as a complement to watermarks ). Our approach essentially embeds a cryptographic hash in the content and uses a signature to link it to an identity, aligning with those best practices.

Task 7: System Integration, Testing, and Iteration

Goal: Integrate all components into a unified pipeline, then rigorously test and refine the system. We will verify that each part works individually and in concert, using both synthetic tests and realistic scenarios (e.g., different audio content or channel conditions).
• Integration Pipeline: Assemble the full system as follows:
1. Input Preparation: Start with an original piece of content or an ID (for example, a piece of audio or an AI-generated image’s binary data). Compute its BLAKE3 hash (digest). Sign the digest with the owner’s private ECDSA key to produce signature (retain this for verification).
2. Encoding: Use RaptorQ (Encoder) to generate redundant symbols from digest. Optionally, add a short identifier for key or content if needed and include that in the data before encoding.
3. Modulation: Generate an audio waveform carrying those symbols using the chosen scheme (FSK/PSK/Chirp, etc., with secret-key-based patterning). This results in watermark_signal array.
4. Analogue Channel Simulation: Feed watermark_signal into the wave simulation (Task 2 environment). Run the simulation to get received_signal at the detector. If implementing physically, this step could instead be playing sound through an actual environment or device configured as an acoustic black hole, then recording with a microphone – for now, simulation suffices.
5. Demodulation: Process received_signal to detect symbol data (filtering, FFT, correlation as needed). Reconstruct the list of symbols or bits that were successfully received.
6. FEC Decoding: Run the RaptorQ Decoder with the received symbols to recover the fingerprint output_data (32 bytes). Check if decode succeeded; if not, this attempt failed (in testing, we then adjust parameters and retry).
7. Verification: If output_data is obtained, verify the ECDSA signature using the stored signature and the public key. If it matches, report success: the information has been recovered and ownership is verified. If decode or verify fails, report failure.
We will implement this pipeline as a script or program, making use of the libraries for each part. Careful attention will be paid to data formats (e.g., converting between byte arrays and bits, symbol IDs, endianness, etc., especially when interfacing with the FEC library).
• Testing Strategy:
• Unit Tests: Test each component in isolation. For instance, ensure that encoding a message and decoding it via RaptorQ (without channel) returns the original. Test that the modulation-demodulation on a clean channel (no black hole, just direct) can correctly transmit bits. Verify signature generation and verification with known test vectors (perhaps use a known secp256k1 test or compare with another library).
• Simulated Channel Tests: Run the entire pipeline through the simulation with different random seeds or slight variations. Measure the fraction of symbols that get through. For example, log how many symbols were correctly detected vs. sent. If the number is consistently above the threshold needed for decoding, that’s good. If it’s marginal, increase redundancy or symbol energy.
• Parameter Tuning: Adjust slot durations, frequencies, amplitude of the watermark signal, etc., to improve recovery. There is a trade-off: too large changes might break the analogy (e.g., if we make the signal extremely loud or long, we’re effectively defeating the black hole absorption by brute force). We want minimal but sufficient signals to get through.
• Robustness: Introduce additional distortions to test robustness – for example, add background noise to the audio, or slightly vary the speed of audio playback (the audiowmark tool addresses replay speed changes with detection algorithms  – while we may not implement that, it’s good to be aware). Our focus is mainly the extreme case of heavy absorption, but robustness to minor timing or frequency shifts can be considered.
• Multiple Runs: Use Monte Carlo testing: run the full embed-and-recover cycle 100+ times with random content hashes (or random noise input as content), and record the success rate of recovery and verification. This will statistically validate high-probability detection. Ideally, we aim for a success rate near 100% under controlled conditions. If we see failures, analyze where (e.g., detection missed too many bits, or decoding failed due to not enough symbols).
• Analysis of Results: If the system works, even a small portion of the watermark should be enough to get the message. We will document cases like: “Out of 100 symbols, typically ~30 were detected; RaptorQ successfully reconstructed the 32-byte fingerprint in 95% of trials. The signature verified in all reconstructed cases. False positives (wrong fingerprint decoding) were not observed since any incorrect decoding yields a hash that fails signature verification, effectively filtering them out.” This aligns with the idea from audiowmark that the user should check if a decoded message is valid (e.g., look up the hash in a database or verify signature) and discard spurious results .
• Iteration: Based on test outcomes, refine each component:
• If certain frequency tones are consistently not getting through, try different frequencies or using chirps (which might pass through better).
• If time slots interfere (one symbol’s tail overlaps next), increase slot spacing or use better windowing.
• If the simulation is too slow for long signals, optimize or simplify (maybe simulate in 1D for faster iteration when tuning parameters).
• Ensure that the system can be generalized: e.g., try using a piece of music as cover audio where the watermark is hidden in the ultrasonic range. The same detection pipeline should apply, possibly needing only to isolate that frequency band at the receiver.
• Tooling and Automation: We will use Python’s capabilities for automation: writing scripts to run batches of simulation with varying parameters, storing the results, and analyzing (perhaps with matplotlib to visualize a spectrogram of the signal pre- and post-channel, to see the effect). We will also version-control the code (e.g., in a Git repository) and possibly integrate with continuous integration to run tests on code changes.
• Documentation and Examples: As we finalize the development, create clear documentation and usage examples:
• How to embed a watermark into audio (with command-line or function interface).
• How to detect a watermark from audio.
• Provide example audio files (watermarked vs. not watermarked) so others can experiment with the detector.
• Provide references to the research that inspired design choices (many of which we cited above).

By the end of Task 7, we expect to have a fully functional prototype demonstrating the combination of analogue simulation and digital cryptographic techniques. We will have proven the concept: information can be encoded in a sound wave, pass through a black-hole-like analogue channel, and be recovered with high probability using error correction, then verified with a digital signature.

Conclusion

This development roadmap outlined a comprehensive plan to merge cutting-edge concepts from physics (analogue sonic black holes), information theory (error-correcting fountain codes), and cryptography (hashes and signatures on secp256k1) into a single project. Each task progressively builds the system: from theoretical groundwork and simulation of an absorbing audio channel, through robust watermark encoding with BLAKE3 and RaptorQ, to final cryptographic verification of recovered data. Throughout, we referenced prior research in analogue gravity , AI and audio watermarking  , and used open-source tools where possible  .

Upon implementation, this system will allow experimental exploration of information encoding and retrieval in extreme channels. It can be seen as an analogy to the black hole information paradox: even when almost all signal is lost, a carefully encoded fingerprint (like quantum hair) can still be recovered from the slightest output, and verified to prove its origin . Potential applications include robust watermarking of AI-generated media, covert channel communications, and educational demos linking physics and information science. We have focused on clarity, modularity, and verifiability in this plan, ensuring each component can be tested and understood in isolation before integration.

References:
• Visser, M. (1998). Acoustic black holes: horizons, ergospheres and Hawking radiation – Classical and Quantum Gravity . (Introduces sonic black hole concept where phonons are trapped by a horizon, analogous to light in a gravitational black hole).
• Ricci, et al. (2023). Analogue simulations of quantum gravity with fluids – arXiv preprint . (Discusses how infalling particles encode information in a black hole’s field, introducing the notion of a “quantum fingerprint” or hair for black holes).
• Wikipedia – Acoustic Black Hole (Engineering) . (Describes the acoustic black hole effect in structures: gradual reduction of wave velocity leading to near-total absorption of waves).
• GitHub: libRaptorQ RFC6330 implementation  . (Documentation excerpt explaining fountain codes and RaptorQ’s efficiency – recovery of data from any subset of packets above a threshold).
• GitHub: BLAKE3 official implementation README  . (Highlights BLAKE3’s speed, security, and 256-bit output, which we use for hashing fingerprints).
• Cloudflare Blog (2025). Cryptographic watermarks for AI content . (Shows how error-correcting codes are used to embed robust watermarks that survive distortions, informing our use of ECC for robustness).
• Westerfeld, S. audiowmark project (2018)  . (Demonstrates an audio watermark approach using convolutional codes and multiple block merging to improve error correction and detection reliability).
• Bitcoin Wiki – Secp256k1  and coincurve PyPI . (Provide details on the secp256k1 curve used for ECDSA and a tool for using it in Python, which we adopt for signing/verification of the fingerprint).
• (Additional references are inlined above in the respective tasks where applicable.)