Gradient-free Profiling for Deep Learning based Side-Channel Analysis against Masking: the Solution for the Plateau Effect?

Neural networks trained against masked implementations stall before they learn: the loss sits at the value of a blind guess, then drops sharply, and the stall grows exponentially with the masking order $\sym{d}{d}$. The figure in the block is a sketch. The experiment below is real, but small: it trains a tiny network in your browser, on the same kind of task as block 5.

Run $\sym{d}{d} = 0$, then $1$, $2$, $3$. Each extra share stretches the flat part, by a lot more each time. During the plateau the network isn’t slowly improving: its loss sits at $\sym{entropy}{H(Y)}$. Then it leaves and falls to its floor, the dashed line: with noise-free Hamming weights, even a perfect model can’t tell every value of the secret apart.

In Scoop we blamed gradient-based training for the plateau. Gradients shrink exponentially with $\sym{d}{d}$, a saddle point sits near where training starts, and on XOR-like targets a gradient carries almost no information about the label. Scoop changes the optimizer accordingly, and it does shorten the plateau.

That makes a testable prediction. If the gradient causes the plateau, a learner that never computes one shouldn’t have it. So we need a learner that trains a neural network with no gradient at all (block 3), and a fair way to compare it with the others (block 4).

TAGI (tractable approximate Gaussian inference, Goulet et al.) trains a neural network without backpropagating a gradient. Every weight is a Gaussian random variable with a mean and a variance. The forward pass pushes means and variances through the network in closed form. Learning is Bayes’ rule: condition on the label, then pass the correction back layer by layer, the way backpropagation would, but with covariances instead of derivatives.

The prediction error $\sym{err}{\sym{y}{y} - \sym{muz}{\mu_z}}$ is scaled by how much the weight $\sym{wt}{\theta}$ covaries with the output $\sym{out}{z}$, relative to the output’s total uncertainty. Because every weight carries a variance, the network doesn’t output one function but a distribution over functions, and it knows where it is unsure. Below, a tiny network of this kind (one input, 32 hidden units, one output) fits a curve. The band is its prediction ± 2σ; the thin lines are five networks drawn from its weight distributions.

Three things to notice. Each label is one closed-form update, and the step size comes out of the variances: nothing plays the role of a learning rate. Every update also shrinks variances, so the band closes where labels fall and stays open where none did. And the prediction is a distribution, so “I don’t know” is part of the answer.

TAGI is one corner of a two-by-two grid: gradient-based or not, point estimate or Bayesian. The other corners are filled by Adam and Scoop (backpropagation), an evolution strategy, and variational inference. If the plateau came from the gradient, the two gradient-free learners would escape it.

Each one gets the same tuning budget. The plateau is measured in training samples seen until the loss falls below $\sym{entropy}{H(Y)} - \sym{tol}{\varepsilon}$, with $\sym{tol}{\varepsilon}$ = [placeholder: value of ε]. Click a learner in the grid, or in any legend, to follow it through blocks 5 and 6.

The cleanest setting: a 4-bit secret masked at order $\sym{d}{d}$, every combination of share values in the dataset ($16^{\sym{d}{d}+1}$ traces), noise-free Hamming-weight leakage, the same small MLP for every learner, 20 seeds. Median samples seen before the loss breaks:

TAGI never computes a gradient, and its plateau still grows exponentially with $\sym{d}{d}$, at the same rate as Adam’s. At some orders it escapes sooner, at others later; the trend is the same. The dotted line is one epoch, $16^{\sym{d}{d}+1}$ samples: every learner needs to see the whole dataset many times over before it escapes. A toy version of this experiment runs live in block 1.

Noisy leakage: a single masked bit plus dummy samples, with additive Gaussian noise, 10 seeds, and a budget of two million samples. Median samples until escape, at $\sym{d}{d} = 0, 1, 2$:

All five plateau, and all five plateau longer as $\sym{d}{d}$ grows: gradient or not, Bayesian or not. At the higher noise level and $\sym{d}{d} = 2$, only Adam and TAGI escape within the budget, and only six of the ten TAGI runs do. Flip the noise switch on the figure to watch every learner climb.

Every method seems to suffer from masking in the same way. This lets us conjecture that the plateau effect comes from some sort of hardness inherent to masking, not from the way a network is trained.

TAGI’s updates are easy to instrument, so we looked at what changes with $\sym{d}{d}$. Two quantities, measured on the last layer at $\sym{sigma}{\sigma} = 1.0$ and expressed relative to $\sym{d}{d} = 0$ (median over 10 seeds):

The updates don’t shrink. TAGI keeps moving its weights as much at $\sym{d}{d} = 2$ as at $\sym{d}{d} = 0$. What collapses is the covariance between the network’s output and the label, about five times smaller at $\sym{d}{d} = 2$, and Adam shows the same drop. The learner isn’t stuck; its updates have stopped pointing at the secret. That’s the same symptom Shalev-Shwartz et al. describe for gradients, and that we saw in Scoop, now without any gradient involved.

In the update of block 3, that covariance is the factor $\sym{cov}{\operatorname{Cov}}(\sym{wt}{\theta}, \sym{out}{z})$ in front of the error: the step keeps its size, but it is spent in directions that say little about $Y$.

One thing did come out in TAGI’s favour. On synthetic first-order masked bytes ($\sym{sigma}{\sigma} = 0.5$, 5 seeds), TAGI needs far fewer profiling traces than a tuned Adam baseline:

With few traces, Adam is overconfident: its perceived information is negative, so its predictions are worse than a uniform guess. TAGI is positive from 500 traces. At 2,000 profiling traces, TAGI’s model recovers the key in 5 attack traces and Adam’s in 3,070. With 10,000 or more, the two meet. Drag the slider under the figure to read both panels at one $\sym{Np}{N_p}$.

This is synthetic data only; whether TAGI stays this frugal on real traces is the next thing to test.