CurveMark: Detecting AI-Generated Text via Probabilistic Curvature and Dynamic Semantic Watermarking

. 2025 Jul 24;27(8):784. doi: 10.3390/e27080784

Algorithm 1 Information-Theoretic Watermark Embedding

Require: prompt, LM, EMB,

γ

ρ

τ

, NGram, k,

α

Ensure: watermarked_text

1:
Initialize watermarked_text = “”, bits_embedded = 0
2:
$H_{t o t a l} \leftarrow 0$ ▹ Track total entropy manipulation
3:
$P \leftarrow LM (p r o m p t)$ ▹ Initial probability distribution
4:
while not termination condition do
5:
$t_{0} \leftarrow {argmax}_{t^{'}} P (t^{'} | p r o m p t)$ ▹ Maximum likelihood token
6:
$H_{o r i g i n a l} \leftarrow - \sum_{t} P (t) log P (t)$ ▹ Original entropy
7:
$e_{t_{0}} \leftarrow EMB (t_{0})$ ▹ Token embedding
8:
$N_{τ} (t_{0}) \leftarrow {t : cos (e_{t_{0}}, EMB (t)) > τ}$ ▹ Semantic neighborhood
9:
$P_{n g r a m} \leftarrow NGram (watermarked_text)$ ▹ N-gram probabilities from context
10:
$r \leftarrow Uniform (0, 1)$
11:
if bits_embedded / len(watermarked_text) < $ρ$ then
12:
$GreenList, RedList \leftarrow Partition (N_{τ} (t_{0}), h)$
13:
$P_{w a t e r m a r k} \leftarrow ModifyDistribution (P, GreenList, RedList, γ)$ ▹ Boost green tokens by $γ$ , zero red tokens
14:
$H_{w a t e r m a r k} \leftarrow - \sum_{t} P_{w a t e r m a r k} (t) log P_{w a t e r m a r k} (t)$
15:
$Δ H \leftarrow | H_{o r i g i n a l} - H_{w a t e r m a r k} |$ ▹ Entropy change
16:
if $Δ H < δ_{e n t r o p y}$ then ▹ Entropy constraint
17:
$t^{'} \leftarrow Sample (P_{w a t e r m a r k}, P_{n g r a m}, α)$ ▹ Equation (6)
18:
watermarked_text += $t^{'}$
19:
bits_embedded += $1_{t^{'} \in GreenList}$
20:
$H_{t o t a l} \leftarrow H_{t o t a l} + Δ H$
21:
else
22:
$t^{'} \leftarrow Sample (P, P_{n g r a m}, α)$ ▹ Fallback to original
23:
watermarked_text += $t^{'}$
24:
end if
25:
else
26:
$t^{'} \leftarrow Sample (P, P_{n g r a m}, α)$
27:
watermarked_text += $t^{'}$
28:
end if
29:
prompt ← Update(prompt, $t^{'}$ )
30:
$P \leftarrow LM (p r o m p t)$ ▹ Update distribution
31:
end while
32:
return watermarked_text, $H_{t o t a l}$