arXiv subjects

Daniel Bernstein

Publications and source records attributed to Daniel Bernstein.

3 recordsLinked to original sources

arXiv2020

Faster computation of isogenies of large prime degree

Let $\mathcal{E}/\mathbb{F}_q$ be an elliptic curve, and $P$ a point in $\mathcal{E}(\mathbb{F}_q)$ of prime order $\ell$. V\'elu's formulae let us compute a quotient curve $\mathcal{E}' = \mathcal{E}/\langle{P}\rangle$ and rational maps defining a quotient isogeny $\phi: \mathcal{E} \to \mathcal{E}'$ in $\tilde{O}(\ell)$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is uniform in $q$.This article shows how to compute $\mathcal{E}'$, and $\phi(Q)$ for $Q$ in $\mathcal{E}(\mathbb{F}_q)$, using only $\tilde{O}(\sqrt{\ell})$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is again uniform in $q$.As an application, this article speeds up some computations used in the isogeny-based cryptosystems CSIDH and CSURF.

cs.CR↗

arXiv2020

On Alignment in Deep Linear Neural Networks

We study the properties of alignment, a form of implicit regularization, in linear neural networks under gradient descent. We define alignment for fully connected networks with multidimensional outputs and show that it is a natural extension of alignment in networks with 1-dimensional outputs as defined by Ji and Telgarsky, 2018. While in fully connected networks, there always exists a global minimum corresponding to an aligned solution, we analyze alignment as it relates to the training process. Namely, we characterize when alignment is an invariant of training under gradient descent by providing necessary and sufficient conditions for this invariant to hold. In such settings, the dynamics of gradient descent simplify, thereby allowing us to provide an explicit learning rate under which the network converges linearly to a global minimum. We then analyze networks with layer constraints such as convolutional networks. In this setting, we prove that gradient descent is equivalent to projected gradient descent, and that alignment is impossible with sufficiently large datasets.

cs.LG↗

arXiv2014

On Three Sets with Nondecreasing Diameter

Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \subseteq \mathbb{Z}$, let the diameter of $X$ be equal to $\max(X)-\min(X)$. We write $X<_p\,Y$ provided $\max(X)<\min(Y)$. For a positive integer $m$, let $f(m,m,m;2)$ be the least integer $N$ such that any $2$-coloring $\Delta: [1, N]\rightarrow \{0,1\}$ has three monochromatic $m$-sets $B_1, B_2, B_3 \subseteq [1,N]$ (not necessarily of the same color) with $B_1<_p\, B_2 <_p\, B_3$ and $diam(B_1)\leq diam(B_2)\leq diam(B_3)$. Improving upon upper and lower bounds of Bialostocki, Erd\H os and Lefmann, we show that $f(m,m,m;2)=8m-5+\lfloor\frac{2m-2}{3}\rfloor+\delta$ for $m\geq 2$, where $\delta=1$ if $m\in \{2,5\}$ and $\delta=0$ otherwise.

math.CO↗