A Performance-Guided Algorithm for Efficiently Generating Massive S-Boxes Using WGAN-GP

MA Yingjie; WEI Zongze; ZHAO Geng; ZHAO Mingjing; ZHANG Ziyan

doi:10.11999/JEIT260777

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MA Yingjie, WEI Zongze, ZHAO Geng, ZHAO Mingjing, ZHANG Ziyan. A Performance-Guided Algorithm for Efficiently Generating Massive S-Boxes Using WGAN-GP[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT260777

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MA Yingjie, WEI Zongze, ZHAO Geng, ZHAO Mingjing, ZHANG Ziyan. A Performance-Guided Algorithm for Efficiently Generating Massive S-Boxes Using WGAN-GP[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT260777

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A Performance-Guided Algorithm for Efficiently Generating Massive S-Boxes Using WGAN-GP

doi: 10.11999/JEIT260777 cstr: 32379.14.JEIT260777

Beijing Electronic Science and Technology Institute, Beijing 100070, China

Funds: Fundamental Research Funds for the Central Universities(No: 3282026046)；The National Natural Science Foundation of China (No: 62441208, 62476013, and 62501025)

Accepted Date: 2026-09-28
Rev Recd Date: 2026-09-28

Available Online: 2026-10-09

Abstract

Abstract

Objective In modern block ciphers, the substitution box (S-box) is the only nonlinear component, and its cryptographic properties directly determine the resistance of a cryptosystem against linear and differential cryptanalysis. Hence, the efficient generation of S-boxes with strong cryptographic strength remains a central challenge, in which a trade-off among cryptographic quality, generation quantity, and generation efficiency has to be achieved. Conventional methods are classified into algebraic construction, chaos-based construction, and heuristic search. Algebraic methods yield S-boxes with excellent and predictable properties, but their design procedure is complicated and the number and diversity of the generated S-boxes are severely limited. Chaos-based methods produce numerous samples rapidly, yet their nonlinearity is usually inferior and the strict avalanche criterion (SAC) and the bit independence criterion (BIC) are difficult to satisfy simultaneously. Heuristic search methods, including simulated annealing, particle swarm optimization (PSO), and hill climbing, obtain high-performance S-boxes at excessive computational cost, so real-time dynamic application is not supported. Recently, artificial intelligence has opened a new path. Meta-heuristic algorithms still rely on iterative search, reinforcement learning (RL) optimizes masking efficiency rather than direct cryptographic metrics, and generative adversarial network (GAN) based methods, despite incorporating cryptographic properties into the loss of the Wasserstein GAN with gradient penalty (WGAN-GP), achieve a maximum nonlinearity of only about 104. Moreover, the computational overhead of metric evaluation remains the core bottleneck, and training instability, mode collapse, and the difficulty of integrating non-differentiable cryptographic properties into the optimization objective have not been resolved. To address these issues, a performance-guided intelligent S-box generation algorithm based on WGAN-GP is proposed, by which stable training is inherited, non-differentiable cryptographic metrics are directly embedded into the optimization objective, and massive, efficient, and high-strength S-box supply is realized for dynamic cipher systems. Methods The framework follows the logic of "data driving, adversarial learning, performance guidance, and batch generation" and consists of three cooperating modules. In the dataset preparation module, a large-scale dataset is constructed by applying random affine transformations to the standard AES S-box, for which random 8x8 invertible matrices and random 8-bit constant vectors are generated over GF(2) and used to pre- and post-process the original S-box; after normalization, the resulting variants are fed into the discriminator as real samples (Fig. 1). In the adversarial training module, a 256-dimensional noise vector is mapped by the generator to a 256-dimensional floating-point vector through fully connected layers with batch normalization and LeakyReLU, while the discriminator outputs a real-valued score through fully connected layers with LayerNorm, LeakyReLU, and Dropout (Fig. 2). The Wasserstein distance is combined with a gradient penalty that enforces Lipschitz continuity by encouraging the gradient norm at sampled points to approach 1, and the penalty weight is set to 10. The adversarial loss of the generator is the negative expectation of the discriminator score. The nonlinearity (NL) of an S-box is defined as the mean of the nonlinearities of its coordinate Boolean functions computed by the Walsh-Hadamard transform (WHT), and the differential uniformity (DU) is the maximum value in the difference distribution table (DDT). The cryptographic loss is the mean squared error between the estimated metrics and the ideal targets, normalized by the maximum nonlinearity of 112 and by 252, the range above the ideal differential uniformity of 4 that corresponds to almost perfect nonlinear (APN) functions; the total loss is a weighted sum with weights of 0.8 and 1.5. During the first 30 warm-up epochs, only the adversarial loss is used, after which the cryptographic loss is smoothly introduced. A post-processing step based on the argsort operation converts the continuous output into a permutation of the integers from 0 to 255, by which bijectivity is guaranteed; the resulting continuous-discrete gap is alleviated by the gradient penalty and by the cryptographic losses computed directly on discrete S-boxes. A GPU-accelerated parallel estimation method based on PyTorch tensor operations is developed to incorporate non-differentiable metrics into training: for NL, an eighth-order Hadamard matrix is precomputed, 1024 rows are randomly sampled, and all WHT coefficients are computed concurrently by torch.einsum; for DU, the DDT is constructed by broadcasting, advanced indexing, and torch.scatter_add, and its maximum is taken as the estimate. Python loops are eliminated in both procedures. After training, a two-stage strategy is adopted, in which ten million candidates are generated and rapidly screened by the GPU estimator against thresholds of NL greater than 100 and DU no more than 10, and the exact metrics of the top 1000 elite candidates are then computed by a NumPy-based CPU function (Fig. 3). Results and Discussions The effectiveness of the algorithm is verified in terms of efficiency, quality, and structural diversity. During training, the loss curves converge smoothly without the severe oscillation common in conventional GAN training, and the estimated cryptographic metrics increase steadily once the performance guidance mechanism takes effect after the warm-up period (Fig. 4). In terms of efficiency, only 382 minutes are required to generate ten million S-boxes, with an average of 0.0023 seconds per S-box, whereas 6520 minutes are needed without the estimation strategy, so an improvement of 94.2% is achieved (Table 1). Scalability is verified on three GPU platforms, on which the per-S-box times are 0.0023, 0.0011, and 0.0004 seconds and the total times are 382, 142, and 67 minutes, confirming that the efficiency originates from the algorithm rather than the hardware (Table 1). In terms of quality, the top 1000 candidates achieve an average NL of 104.88 with a maximum of 107.5 close to the theoretical maximum of 112, while all values are no less than 100; the mean per-S-box minimum nonlinearity is 102.10 with the worst case of 100.0, the average DU is 9.50 with a minimum of 8, the average SAC deviation is 0.0085, the average BIC-SAC is 0.4979, and the average BIC-nonlinearity is 102.26, indicating excellent diffusion and output bit independence (Table 2). The frequency distributions show that a large number of S-boxes are concentrated in high-performance regions, which verifies both effectiveness and mode coverage (Fig. 5). Bijectivity is satisfied by all ten million S-boxes, and the estimator is validated on 1000 samples, for which the NL error is below 0.5% and the DU estimate matches the exact value for 98% of the samples (Table 2). The continuous-discrete gap is examined, and the estimated NL of 104.7 differs from the exact value of 104.88 by only 0.17% while the estimated DU of 9.6 differs from the exact value of 9.50 by only 1.04%, so no inconsistency between training and inference is observed (Fig. 4). Unlike AES affine-equivalent S-boxes whose DU is always 4 and whose NL is always 112, the generated S-boxes exhibit an average DU of 9.50 with a minimum of 8 and an average NL of 104.88 with a minimum of 100.0, and their Walsh spectral statistics confirm that genuine exploration rather than memorization is performed by the model (Table 3). Existing methods fall into an objective-oriented paradigm and a data-driven paradigm, whose merits cannot be judged solely by identical metrics; the value of this work lies in the massive supply scale and the order-of-magnitude gain in efficiency obtained at the cost of only a minor concession in individual metrics (Table 2). Conclusions A performance-guided WGAN-GP framework for massive S-box generation is proposed, by which stable generation is realized, non-differentiable cryptographic properties are directly embedded into the optimization objective, and the bottleneck of metric evaluation is overcome by a GPU-accelerated estimation strategy that improves generation efficiency by 94.2%. A dynamic supply scheme compatible with standard key update protocols is also presented. Ten million S-boxes are generated within 382 minutes with an average of 0.0023 seconds per S-box, and the top 1000 exactly evaluated S-boxes achieve an average NL of 104.88 with a maximum of 107.5 and an average DU of 9.50 with a minimum of 8, meeting or exceeding the level of existing dynamic S-box generation algorithms. Meanwhile, training instability, mode collapse, and the difficulty of integrating non-differentiable properties are effectively alleviated. The potential of deep generative learning for cryptography is demonstrated, and the algorithm can be extended to other cryptographic primitives, with applications in lightweight cipher design for the Internet of Things and mobile platforms and in high-security systems where S-boxes are frequently updated to resist side-channel attacks.
- S-box,
- WGAN-GP,
- Nonlinearity,
- Differential Uniformity,
- Estimation Strategy

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References(39)

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