Goldwasser et al. showed that undetectable backdoors can be planted in machine learning models trained with the Random Fourier Features (RFF) algorithm, under a hardness assumption tied to the Continuous Learning With Errors (CLWE) problem. Under standard cryptographic assumptions, even a full white-box audit of a model's weights cannot detect this class of backdoor. The construction is stated in terms of cryptographic reductions and probabilistic lemmas, without a reference implementation, and relies on secondary machinery such as the Sparse Gaussian Pancakes distribution and a homogeneous CLWE conditional density. Its realizability in ordinary numerical code is not obvious from the paper alone. This paper implements the white-box CLWE-RFF backdoor construction end to end using only numpy and scipy, to test whether this threat is realizable with commodity scientific-computing tools or requires specialized cryptographic infrastructure. We give two samplers for the core GPd(bk) distribution. The first is a rejection-sampling proxy. The second is an exact closed-form sampler derived from the homogeneous CLWE density and verified against its own analytic form. Using this implementation, we run statistical indistinguishability tests, covering both weight-space and functional black-box comparisons. We find no evidence of detectable difference between backdoored and clean models across a range of sparsity ratios ρ=dsparse/D. We report which parts of the construction were straightforward to realize, which required derivation not spelled out in the paper. We also highlight which parts we did not attempt to reproduce, including the underlying lattice hardness reduction. We see this work as a contribution to understanding the practical realizability of the Goldwasser white-box CLWE core, not as a new theoretical result.
We show how an adversarial model trainer can plant backdoors in a large class of deep, feedforward neural networks. These backdoors are statistically undetectable in the white-box setting, meaning that the backdoored and honestly trained models are close in total variation distance, even given the full descriptions of the models (e.g., all of the weights). The backdoor provides access to invariance-based adversarial examples for every input, mapping distant inputs to unusually close outputs. However, without the backdoor, it is provably impossible (under standard cryptographic assumptions) to generate any such adversarial examples in polynomial time. Our theoretical and preliminary empirical findings demonstrate a fundamental power asymmetry between model trainers and model users.
We present Sparse Backdoor, a supply-chain attack that plants a provably undetectable backdoor in pre-trained image classifiers, including convolutional networks and Vision Transformers. The attack injects a structured sparse perturbation along a randomly chosen direction into a small subset of columns at each fully connected layer, propagating a trigger signal to an adversary-chosen target class, and masks the perturbation with an independent isotropic Gaussian dither. The dither serves a single technical purpose: it induces a clean reference distribution anchored at the pre-trained weights, against which undetectability can be formalized. Under a mild margin condition on the pre-trained classifier, we show that the dithered reference is functionally equivalent to the original classifier. We prove that distinguishing the backdoor-injected model from this reference is at least as hard as Sparse PCA detection, which is computationally infeasible under standard hardness assumptions. The guarantee holds against any probabilistic polynomial-time distinguisher with white-box access to the parameters.
Sarthak Choudhary, Atharv Singh Patlan, Nils Palumbo +3
Recent cryptographic results establish that neural networks can be backdoored such that no efficient algorithm can distinguish them from a clean model. These guarantees, however, have been confined to stylised architectures of limited practical relevance, leaving open whether comparable undetectability extends to modern, end-to-end trained networks. We construct such an attack mechanism for state-of-the-art architectures, closely aligned to the cryptographic notion of undetectability, by identifying backdoor channels as learned latent directions, and show that the question of undetectability reduces to a hypothesis test between two unknown distributions over model parameters, which we conjecture to be intractable in practice. The consequence of this reframing is significant: if exploitable channels within a network's latent space are statistically indistinguishable from naturally learned directions, an attacker need not introduce foreign structure but can instead exploit the geometry the network already possesses. Demonstrating the approach on ResNet and Vision Transformer architectures trained on standard image classification datasets, the attack achieves both consistently high success rates with negligible clean accuracy degradation, and resists a comprehensive suite of post-training defences, none of which neutralise the backdoor without rendering the model unusable. Our results establish that cryptographic backdoors need not be artefacts requiring exotic architectures or artificial constructions, but identifiable as latent properties inherent to the geometry of learned representations.