Set Shaping Theory as a Complementary Payload-Shaping Layer for Steganography Aida Koch∗
Logan Lewis
Lily Scott
Agi Weber
arXiv:2605.19885v1 [eess.IV] 19 May 2026
May 20, 2026
Abstract This paper studies the use of Set Shaping Theory (SST) as a reversible payload-shaping layer for least significant bit (LSB) image steganography. The proposal is not intended to replace existing steganographic methods or compete with them as a new embedding scheme. Instead, SST is positioned as a complementary preprocessing stage that makes an existing embedding method easier to apply with lower statistical disturbance. The transformation that applies set shaping theory by increasing the message length by k symbols is performed using the algorithm developed by Glen Tankersley which perform the transformation in an approximate but very fast way. Although the embedded payload is lengthened from N to N + K bits, the selected representation can reduce DKL (P ∥Q) and therefore make the subsequent steganographic insertion less detectable under histogram-based criteria. Across 1,800 controlled simulations on four synthetic cover-image models, SST reduced DKL (P ∥Q) by an average of 25.16% relative to a fair N + K LSB baseline, with a 95% confidence interval of ±1.22%. For K = 8, the average reduction reached 42.81%. Additional robustness simulations with keyed random embedding paths confirmed the effect across several distances: at K = 8, SST reduced KL divergence by 42.44%, Jensen–Shannon divergence by 29.62%, total variation by 12.41%, and a symmetric chi-square distance by 28.30%. An additional image-based matrix-embedding/STC-like simulation showed that SST also reduces the minimum weighted insertion cost: relative to the unshaped K = 0 reference, K = 8 reduced the cost by 6.93%.
1
Introduction
Modern steganography already contains many embedding strategies, ranging from simple least significant bit (LSB) substitution to wet-paper coding, LSB matching, syndrome-trellis coding, and distortion-minimizing adaptive schemes [16, 17, 18, 19, 20, 21]. The purpose of this paper is not to introduce a replacement for those methods. Instead, it studies whether Set Shaping Theory (SST) can act as a payload-shaping layer placed before an existing embedder. In this role, SST does not decide where or how the cover is modified; it only chooses which reversible representation of the same message should be given to the underlying steganographic method. LSB steganography is used here as a transparent testbed because its statistical effect can be measured directly. It embeds a binary message by replacing the least significant bits of selected pixels. The method is simple and high-capacity, but its direct substitution mechanism perturbs the empirical distribution of pixel intensities. Such perturbations can be measured by comparing the original cover histogram P with the stego histogram Q. This work investigates whether SST can reduce that perturbation before the message reaches the embedder. Rather than embedding the message in its original form, the encoder constructs multiple reversible representations of the same message. It then passes each candidate through the same embedding rule, evaluates the induced divergence, and keeps the candidate whose ∗
Contact: [email protected]
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histogram is closest to the original image. The receiver uses the additional K index bits to invert the selected transformation and recover the original message. The apparent trade-off is unusual: the payload becomes longer, from N to N + K, but the distributional damage produced by the existing embedder can decrease. Therefore SST should be interpreted as a compatibility layer that can make established steganographic techniques easier to apply under a statistical distortion constraint, not as a competing steganographic system. This article participates to Set Shaping Theory simulator project available at https: //sst-simulator.github.io/Set-Shaping-Theory-Simulator/. Its steganography section provides an interactive implementation of the same payload-shaping idea studied in this article, allowing the reader to vary the message, the shaping parameter, and the embedding comparison in a direct visual environment.
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Set Shaping Theory
Classical information theory and source coding usually begin from the problem of representing a source sequence with minimum expected length or minimum redundancy [1, 2, 3]. In this framework, the central object is the source model: if the probability law is known, one designs a code adapted to that law; if the law is not known, universal methods attempt to approach the same limit without assuming complete prior knowledge [5, 6]. Model-selection principles such as minimum description length similarly connect representation quality to the regularity that can be extracted from the data and from the model used to describe it [4]. In all these approaches, the dominant question is therefore dimensional: how many bits or symbols are needed to represent the data? Set Shaping Theory (SST) starts from a different but compatible question. Instead of changing only the code used to describe a fixed sequence, SST studies transformations that change the combinatorial set in which the sequence is represented. In its original form, SST was introduced as the study of bijective maps that transform a set of strings of length N into a subset of strings of greater length N + K, while preserving the number of admissible messages [7]. If the alphabet is denoted by A, with |A| = A, the basic object can be written as f : AN −→ YN +K ,
YN +K ⊂ AN +K ,
|YN +K | = |AN | = AN .
(1)
Thus, SST does not simply append arbitrary redundant symbols. It selects a structured subset YN +K of the larger ambient space and creates a one-to-one correspondence between the original messages and the admissible longer representations. In the compression-oriented formulation, the selected subset is chosen so that the average information content, or equivalently the empirical entropy–length product, is reduced. If H0 (x) denotes the zero-order empirical entropy of a sequence x, one of the characteristic SST objectives is ⟨(N + K)H0 (f (s))⟩ < ⟨N H0 (s)⟩ , s ∈ AN . (2) This inequality expresses the central intuition of SST: a longer representation can still be more structured, and therefore more favorable for a later coding or processing operation, than the original shorter representation. The relevant quantity is not length alone, but the interaction between length and structure. The same idea was later explored in several related directions. In locally testable codes, SST was used to introduce dependencies before the final coding operation, so that inadmissible decoded sequences can reveal errors through the fact that they do not belong to the shaped image of the transformation [8]. In Huffman coding, SST was studied as a practical preprocessing operation, with particular attention to the difficulty of implementing the full correspondence table, whose size grows as AN [9]. Subsequent work emphasized the role of the shaping order K, which controls the size of the candidate representation family, and investigated the relation 2
between SST and empirical entropy coding limits [10]. More recent work on redundancy-free testable codes further develops the idea that sequence extension can create structural constraints without necessarily behaving like conventional explicit redundancy [12]. For the present paper, the most important point is that SST should not be interpreted only as a compression technique. Compression is one application in which the objective function is related to H0 , coding length, or empirical information content. More generally, SST can be viewed as a data-conditioning principle: given many reversible representations of the same underlying message, choose the representation that is most compatible with the downstream operation. In this broader interpretation, the objective function need not be entropy. It can be any measurable cost induced by the system that will process the shaped data. This distinction is essential in steganography. The goal of the present article is not to reduce the number of embedded bits and not to replace the steganographic embedder. The goal is to choose, among several reversible payload representations, the one that causes the smallest statistical disturbance when passed through the same embedding method. Therefore, the shaped set is evaluated with respect to the cover stego relationship rather than with respect to compression alone. In this sense, SST acts as a complementary payload-shaping layer. It does not decide where the cover should be modified, and it does not define a new extraction rule. Those tasks remain the responsibility of the underlying steganographic method. SST only changes the representation of the payload before embedding, so that the same method can operate on an input whose structure is more favorable under the chosen distortion criterion. This makes SST cooperative rather than competitive with existing steganographic systems: it is an upstream reversible preprocessing stage designed to reduce the statistical cost paid by a downstream embedder.
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Method
Let x = (x1 , . . . , xM ) be a grayscale cover image represented as pixel values in {0, . . . , 255}, and let s ∈ {0, 1}N be the secret message. Let E denote an existing steganographic embedder. In the experiments, E is the direct LSB rule, which produces a stego image by replacing the least significant bit of the first N pixels: yi = LSBSet(xi , si ),
i = 1, . . . , N.
(3)
The SST component is inserted before E and leaves the embedding rule itself unchanged. The Set Shaping Theory will be applied using the developed method by Glen Tankersley[11] . For SST-assisted embedding, a shaping overhead K defines a family of H = 2K reversible binary transformations. For each index h ∈ {0, . . . , H − 1}, a deterministic binary mask rh ∈ {0, 1}N is generated and the transformed message is s(h) = s ⊕ rh .
(4)
The embedded payload is the concatenation z (h) = binK (h)∥s(h) ,
(5)
where binK (h) is the K-bit binary representation of the chosen index. This payload is then passed to the same underlying embedder E. The decoder first extracts the payload by the ordinary decoding procedure of that embedder, reads the index, regenerates the corresponding mask, and recovers s = s(h) ⊕ rh . (6) For each candidate h, the encoder computes the empirical histogram Qh of the corresponding stego image and minimizes the Kullback-Leibler divergence DKL (P ∥Qh ) =
255 X v=0
3
P (v) log2
P (v) . Qh (v)
(7)
Small additive smoothing is used only to avoid undefined logarithms when a bin has zero empirical mass. This formulation separates payload shaping from steganographic embedding. Any method that can embed and recover a binary payload could, in principle, receive an SST-shaped payload. LSB is therefore not the proposed final system; it is the controlled carrier used to quantify the shaping effect.
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Experimental Design
The simulation campaign uses 100 × 100 grayscale images, binary messages of length N ∈ {1000, 2500, 4000}, and shaping overhead K ∈ {0, 2, 4, 6, 8}. Four synthetic cover-image models are tested: uniform random images, smoothed random images, noisy gradients, and bimodal mixtures. Each configuration is repeated 30 times, yielding 1,800 runs. The goal is to isolate whether the shaping layer reduces the distortion caused by a fixed embedder, not to claim that LSB itself is the strongest steganographic choice. The primary baseline is a fair LSB insertion of N + K bits: the first K bits are a zero index and the remaining N bits are the unshaped message. This is the correct comparison because the SST-shaped payload also has length N + K and is embedded by the same LSB rule. The main reported metric is the relative gain G=
DKL (P ∥Qbase ) − DKL (P ∥QSST ) . DKL (P ∥Qbase )
(8)
Positive values indicate that SST produced a stego histogram closer to the original cover histogram while using the same underlying embedding mechanism. This is aligned with information-theoretic steganography, where relative entropy can bound the distinguishability between cover and stego distributions [15]. Two additional simulation blocks were then added to test robustness. First, the embedding path was randomized by a deterministic key rather than fixed to the first N +K pixels. The same fair N + K baseline was used, and the selected SST candidate was still chosen by minimizing KL divergence. The resulting stego images were then evaluated with P KL divergence, Jensen–Shannon divergence, total variation, symmetric chi-square distance v (P (v) − Q(v))2 /(P (v) + Q(v)), and the L1 distance between horizontal pixel-pair co-occurrence matrices. Second, a smaller timing study measured the cost of exhaustive candidate search for K ∈ {0, 4, 8, 10, 12}. A third simulation tested the same complementarity claim with a matrix-embedding/STC-like syndrome encoder applied to generated grayscale images. Pixel positions were selected from the image by a keyed random path, their LSBs formed the binary cover sequence, and local image texture determined the weighted cost of changing each selected pixel. The selected positions were then divided into syndrome blocks, and dynamic programming was used to compute the minimum weighted flip cost required to realize a target syndrome. SST tested the equivalent payloads binK (h)∥(s ⊕ rh ) and selected the one with minimum syndrome-embedding cost. The case K = 0 has no shaping choice and is used as the reference. The decoded message is therefore unchanged; only the representation given to the matrix embedder changes.
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Results
The central effect of the shaping overhead is reported in Table 1. Table 1: Mean reduction in DKL relative to the fair N + K LSB baseline as a function of shaping overhead. K Runs Mean gain 95% CI Success 0 2 4 6 8
360 360 360 360 360
0.00% 17.69% 28.68% 36.61% 42.81%
± 0.00% ± 3.08% ± 2.45% ± 2.25% ± 2.15%
0.00% 81.11% 91.94% 98.06% 100.00%
Gain vs. fair LSB (%)
With K = 0, SST has no additional choice and matches the fair baseline. Once K > 0, the average gain becomes positive and increases monotonically with the size of the candidate set. At K = 8, the encoder tests 256 reversible representations and obtains a 42.81% mean reduction in DKL (P ∥Q). The same monotone trend is shown visually in Figure 1.
40 30 20 10 0
0
1
2
3
4 K
5
6
7
8
Figure 1: Mean KL-divergence reduction obtained by SST as the number of shaping index bits increases. Error bars show approximate 95% confidence intervals. The error bars indicate that the observed gain is not limited to a single configuration, and the increasing curve supports the interpretation that the additional SST choices are doing useful work.
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Figure 2 compares the absolute KL values of the fair baseline and the best SST-shaped payload. ·10−3
Mean DKL (P ∥Q)
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Fair LSB baseline Best SST
8 7 6 5 4 0
1
2
3
4 K
5
6
7
8
Figure 2: Average KL divergence for the fair N + K LSB baseline and for the best SST-shaped payload. The fair baseline remains roughly stable because it embeds the same length N + K without candidate selection, while the best SST candidate steadily lowers the final histogram divergence as K increases. The dependence on the cover-image model is summarized in Table 2. Table 2: Mean reduction in DKL by synthetic cover-image model. Cover model uniform smooth gradient bimodal
Runs
Mean gain
95% CI
Success
450 450 450 450
15.87% 36.73% 16.78% 31.25%
± 1.24% ± 3.04% ± 1.45% ± 2.87%
74.22% 74.67% 73.78% 74.22%
Smooth and bimodal images show larger average gains than uniform and gradient images. This is plausible because structured histograms provide more room for candidate selection to avoid visibly unbalanced bin changes.
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The same cover-model dependence is displayed in Figure 3.
Gain vs. fair LSB (%)
40 30 20 10 0 uniform
smooth
gradient
bimodal
Cover model Figure 3: Mean SST gain by cover-image model. The figure is useful as a practical reading of the table: SST is most effective when the cover distribution has structure that the candidate search can exploit. Table 3 reports the effect across the tested message lengths. Table 3: Mean reduction in DKL by embedded message length. N
Runs
Mean gain
95% CI
Success
1000 2500 4000
600 600 600
24.87% 25.98% 24.62%
± 2.40% ± 1.99% ± 1.94%
74.17% 74.33% 74.17%
The gain remains close to 25% for all three values of N , which suggests that the result is not caused by a single payload size. These results should be read as evidence that SST can reduce the statistical burden placed on an existing steganographic method. The robustness simulations with keyed random embedding paths are reported in Table 4. Table 4: Robustness check with keyed random embedding paths. Mean reduction relative to the fair N + K baseline is reported for several statistical distances. K
DKL
JS
TV
χ2
Cooc. L1
0 2 4 6 8
0.00% 16.72% 28.72% 37.06% 42.44%
0.00% 9.47% 17.15% 23.80% 29.62%
0.00% 2.53% 6.63% 9.05% 12.41%
0.00% 8.77% 16.03% 22.50% 28.30%
0.00% 0.26% 0.92% 1.31% 1.49%
The main KL result is preserved when the embedding locations are selected by a key rather than by the first pixels of the image. At K = 8, the KL reduction is 42.44%, very close to the sequential-path result. The same shaped payload also improves Jensen–Shannon divergence, total variation, and symmetric chi-square distance, although the improvement is smaller for co-occurrence statistics. This is expected: the SST objective in the present experiments is histogram-based, so spatial pair statistics are affected only indirectly. 7
The distribution of the selected shaping index is shown in Table 5. Table 5: Distribution of selected shaping index h. Values close to 0.5 for normalized h and small maximum-bucket shares indicate that gains are not caused by a single fixed index. K
Runs
Mean normalized h
Largest bucket share
2 4 6 8
240 240 240 240
0.500 0.463 0.518 0.517
28.33% 9.17% 3.33% 2.08%
The selected index is not concentrated on a single fixed value. For K = 8, the mean normalized index is 0.517 and the largest selected-index bucket accounts for only 2.08% of runs. Thus the gain comes from genuine candidate selection rather than from a particular mask dominating all covers. The computational cost of larger shaping orders is given in Table 6. Table 6: Computational cost for larger shaping orders. The timing measures exhaustive candidate search with keyed random paths, N = 1000, and two structured cover models. K
Configurations
Mean search time (ms)
Mean time/candidate (µs)
0 4 8 10 12
1 16 256 1024 4096
0.08 0.64 9.72 40.04 165.31
77.17 40.23 37.97 39.10 40.36
The table shows the expected exponential cost in K, but also that the per-candidate time remains roughly constant. In the tested Node.js implementation, exhaustive search with K = 12 requires about 165 ms for N = 1000 on the selected structured cover models. The image-based matrix-embedding/STC-like experiment is summarized in Table 7. Table 7: Matrix-embedding/STC-like minimum insertion cost on generated grayscale images after SST payload shaping. The reference is the unshaped case K = 0; larger K values give the encoder more equivalent payload representations to choose from. K
Configurations
Runs
Minimum cost
Reduction vs. K = 0
0 2 4 6 8
1 4 16 64 256
960 960 960 960 960
160.092 157.473 154.018 151.312 148.996
0.00% 1.64% 3.79% 5.48% 6.93%
Here the objective is no longer histogram divergence but the minimum weighted insertion cost needed to obtain the same recoverable message. The result is consistent with the main thesis: increasing K gives the encoder more equivalent message representations, and the minimum achievable embedding cost decreases as the candidate set grows. Relative to the K = 0 reference, K = 8 reduces the minimum insertion cost by 6.93% over 960 image-based runs.
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6
Discussion
The experiments support the central hypothesis of this work: Set Shaping Theory can serve as an effective complementary payload-shaping layer for steganography. The important point is that SST is not proposed as a replacement for existing steganographic methods. It does not compete with LSB embedding, matrix embedding, syndrome-trellis coding, adaptive embedding, or other advanced schemes. Instead, it acts before the embedding stage, selecting a more favorable reversible representation of the same message. In this sense, SST improves the input given to the embedder rather than changing the embedder itself. This distinction is crucial. The results show that even when the payload length is increased from N to N + K, the final statistical disturbance can decrease significantly. In the LSB experiments, the SST-shaped payload produced a substantial reduction in DKL (P ∥Q) relative to the fair N + K baseline. The gain increased systematically with K, reaching more than 40% reduction in KL divergence for K = 8. This means that the additional representation choices created by SST are not merely redundant overhead; they provide useful degrees of freedom that allow the encoder to select a payload representation more compatible with the cover image. The robustness tests reinforce this interpretation. When the embedding positions were selected by keyed random paths, the same positive trend remained visible across several statistical distances, including KL divergence, Jensen–Shannon divergence, total variation, and symmetric chi-square distance. This indicates that the effect is not limited to a single sequential embedding path or to a single metric. The shaped payload is generally less disruptive under several histogrambased criteria, which is precisely the desired behavior for a preprocessing layer intended to support steganographic embedding. The matrix-embedding/STC-like experiment is particularly important because it moves the interpretation beyond simple LSB substitution. In that setting, the objective is not only to reduce a histogram divergence after embedding, but to reduce the minimum weighted insertion cost required by an existing syndrome-based embedder. The observed decrease in insertion cost shows that SST can be useful even when the downstream method already performs an optimization. This supports the view that SST is not limited to basic steganographic schemes; rather, it can provide an additional optimization layer on top of more advanced methods. Although LSB steganography is used in this article as a transparent and easily measurable testbed, the principle is more general. Any steganographic method that embeds a binary payload can, in principle, receive multiple reversible representations of the same message and choose the one that minimizes a relevant distortion function. For LSB, this function can be a histogram divergence. For adaptive steganography, it could be a spatial distortion cost. For syndrome-trellis coding, it could be the weighted embedding cost. For more advanced systems, it could be a detector-based or model-based security score. The common structure is the same: SST increases the number of admissible payload representations and allows the encoder to select the one that is most compatible with the cover and with the embedding rule. Therefore, the main contribution of this work is not the proposal of a new steganographic algorithm in competition with existing ones, but the introduction of a general reversible shaping layer that can cooperate with them. The results suggest that important gains can be obtained simply by improving the representation of the payload before embedding. This makes SST a promising tool for steganographic systems in which the message can be preprocessed and where the encoder is allowed to choose among equivalent representations in order to reduce statistical detectability or embedding cost.
7
Conclusion
This paper has presented Set Shaping Theory as a complementary payload-shaping layer for steganography. SST is not proposed as a replacement for existing embedding methods, but as a
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reversible preprocessing stage that improves the representation of the payload before embedding. The results show that this approach can produce significant gains. In the LSB experiments, SST reduced the KL divergence between the cover and stego histograms, with the improvement exceeding 40% for K = 8. The same positive trend appeared with keyed random embedding paths and with other statistical distances, confirming that the effect is not limited to one specific insertion path. The matrix-embedding/STC-like experiment further shows that SST can also reduce the minimum weighted insertion cost required by a more advanced embedding mechanism. This supports the main interpretation of the method: SST can cooperate with existing steganographic systems by giving them a more favorable payload representation. Although LSB is used here as the main testbed, the principle is more general. Any steganographic method that receives a binary payload can potentially benefit from a reversible shaping layer that selects the most compatible representation of the same message. For this reason, SST should be viewed as a general payload-conditioning method, suitable for both simple and advanced steganographic schemes.
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[14] G. J. Simmons, “The prisoners’ problem and the subliminal channel,” in Advances in Cryptology: Proceedings of CRYPTO ’83, D. Chaum, Ed. Springer, 1984, pp. 51–67. [15] C. Cachin, “An information-theoretic model for steganography,” Information and Computation, vol. 192, no. 1, pp. 41–56, 2004. [16] J. Fridrich, M. Goljan, P. Lisonek, and D. Soukal, “Writing on wet paper,” IEEE Transactions on Signal Processing, vol. 53, no. 10, pp. 3923–3935, 2005. [17] J. Mielikainen, “LSB matching revisited,” IEEE Signal Processing Letters, vol. 13, no. 5, pp. 285–287, 2006. [18] T. Filler and J. Fridrich, “Gibbs construction in steganography,” IEEE Transactions on Information Forensics and Security, vol. 5, no. 4, pp. 705–720, 2010. [19] T. Pevny, T. Filler, and P. Bas, “Using high-dimensional image models to perform highly undetectable steganography,” in Information Hiding, Lecture Notes in Computer Science, vol. 6387. Springer, 2010, pp. 161–177. [20] T. Filler, J. Judas, and J. Fridrich, “Minimizing additive distortion in steganography using syndrome-trellis codes,” IEEE Transactions on Information Forensics and Security, vol. 6, no. 3, pp. 920–935, 2011. [21] V. Holub, J. Fridrich, and T. Denemark, “Universal distortion function for steganography in an arbitrary domain,” EURASIP Journal on Information Security, vol. 2014, article 1, 2014.
A
Online Set Shaping Theory Simulator
This article is part of the Set Shaping Theory simulator project, available online at https: //sst-simulator.github.io/Set-Shaping-Theory-Simulator/. The project is not only a demonstration tool, but also a research site intended to collect examples, use cases, and problem instances in which the structure of the data is more important than its raw length. The steganography section of the simulator focuses on the same complementary role studied in this paper. The user provides or generates a binary message and selects a shaping parameter K. The simulator then applies Set Shaping Theory to the message producing a message with k more inputs but with a much smaller Kullback-Leibler divergence. In this way, the simulator shows that SST does not hide the message by itself and does not replace the underlying steganographic method. Instead, it changes the representation given to that method so that the subsequent embedding step can be easier, less statistically disturbing, or less costly. The online tool therefore provides a practical way to reproduce the central idea of the article: the relevant comparison is always against the unshaped K = 0 reference, while larger values of K give the encoder more equivalent payload representations from which to choose. The simulator is especially useful for visualizing why a longer shaped payload can still reduce divergence or insertion cost when the chosen representation is better matched to the cover data and to the embedding rule.
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