Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Q-curve construction for endomorphism-accelerated elliptic curves   We give a detailed account of the use of $\mathbb{Q}$-curve reductions to
construct elliptic curves over $\mathbb{F}\_{p^2}$ with efficiently computable
endomorphisms, which can be used to accelerate elliptic curve-based
cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and
Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case
of our construction), we offer the advantage over GLV of selecting from a much
wider range of curves, and thus finding secure group orders when \(p\) is fixed
for efficient implementation. Unlike GLS, we also offer the possibility of
constructing twist-secure curves. We construct several one-parameter families
of elliptic curves over $\mathbb{F}\_{p^2}$ equipped with efficient
endomorphisms for every $p \textgreater{} 3$, and exhibit examples of
twist-secure curves over $\mathbb{F}\_{p^2}$ for the efficient Mersenne prime
$p = 2^{127}-1$.

26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics