Covering three-tori with cubes

Let $\mu(\varepsilon)$ be the minimum number of cubes of side $\varepsilon$ needed to cover the unit three-torus $[\mathbb{R}/\mathbb{Z}]^3$. We prove new lower and upper bounds for $\mu(\varepsilon)$ and find the exact value for all $\varepsilon\geq\frac{7}{15}$ and all $\varepsilon\in\left[\frac{1}{r+1/(r^2+r+1)},\frac{1}{r-1/(r^2-1)}\right)$ for any integer $r\geq 2$

Beyond the Limits of DPA: Combined Side-Channel Collision Attacks

The fundamental problem of extracting the highest possible amount of key-related information using the lowest possible number of measurements is central to side-channel attacks against embedded implementations of cryptographic algorithms. To address it, this work proposes a novel framework enhancing side-channel collision attacks with divide-and-conquer attacks such as differential power analysis (DPA). An information-theoretical metric is introduced for the evaluation of collision detection efficiency. Improved methods of dimension reduction for side-channel traces are developed based on a statistical model of Euclidean distance. The theoretical and experimental results of this work confirm that DPA-combined collision attacks are superior to both DPA-only and collision-only attacks. The new methods of dimension reduction lead to further complexity improvements. All attacks are treated for the case of AES-128 and are practically validated on a wide-spread 8-bit RISC microcontroller whose architecture is similar to that of many smart cards