Fast algorithms for l-adic towers over finite fields

@inproceedings{Feo2013FastAF,
  title={Fast algorithms for l-adic towers over finite fields},
  author={Luca De Feo and Javad Doliskani and {\'E}ric Schost},
  booktitle={ISSAC '13},
  year={2013}
}
Inspired by previous work of Shoup, Lenstra-De Smit and Couveignes-Lercier, we give fast algorithms to compute in the first levels of) the l-adic closure of a finite field. In many cases, our algorithms have quasi-linear complexity. 

Figures and Tables from this paper

Algorithms for Finite Field Arithmetic
TLDR
Algorithms to construct finite fields and perform operations such as field embedding and techniques that originate from algorithms for computing with triangular sets can be useful in such a context are shown.
Accelerated tower arithmetic
Fast arithmetic for the algebraic closure of finite fields
TLDR
This work presents algorithms to construct and do arithmetic operations in the algebraic closure of the finite field Fp, inspired by algorithms for constructing irreducible polynomials, to give efficient algorithms for embeddings and isomorphisms.
Computing in Algebraic Closures of Finite Fields
TLDR
Inspired by algorithms for constructing irreducible polynomials, this work presents algorithms to construct and perform computations in algebraic closures of finite fields based on polynomial arithmetic, rather than linear algebra.
Directed evaluation
Let K be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure ofK is to compute modulo its minimal polynomial. The determination of a minimal
Standard Lattices of Compatibly Embedded Finite Fields
TLDR
This work generalizes Conway polynomials and Bosma--Cannon--Steel lattices by leveraging the theory of the Lenstra-Allombert isomorphism algorithm, and shows that the construction in indeed practical.
Explicit isogenies in quadratic time in any characteristic
TLDR
The approach is inspired by a previous algorithm due to Couveignes, that involved computations using the $p$-torsion on the curves to determine $\psi$ from the knowledge of E, $E'$ and of its degree $r by using the structure of the $\ell$- torsion of the curves.
Directed evaluation
Computing isomorphisms and embeddings of finite fields
TLDR
The detailed complexity analysis shows that the newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better.
Mathematics of Isogeny Based Cryptography
  • L. D. Feo
  • Computer Science, Mathematics
    ArXiv
  • 2017
These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thi\`es, Senegal. They try to provide a guide for Masters' students to get through the vast
...
...

References

SHOWING 1-10 OF 46 REFERENCES
Isomorphisms between Artin-Schreier towers
TLDR
It is found that isomorphisms between towers of degree p over a fixed field Fq can be computed, composed, and inverted in time essentially linear in p n .
Conics - a Poor Man's Elliptic Curves
The aim of this article is to show that the arithmetic of Pell conics admits a description which is completely analogous to that of elliptic curves: there is a theory of 2-descent with associated
Fast arithmetics in artin-schreier towers over finite fields
TLDR
This work presents an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the <i>p</i>-torsion and gives algorithms with quasi-linear time complexity for arithmetic operations in such towers.
Fast construction of irreducible polynomials over finite fields
  • V. Shoup
  • Computer Science, Mathematics
    SODA '93
  • 1993
The main result of this paper a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field Fq . The algorithm is probabilistic, and is asymptotically faster
Computing in degree $$2^k$$2k-extensions of finite fields of odd characteristic
TLDR
It is shown how to perform basic operations (arithmetic, square roots, computing isomorphisms) over finite fields of the form Q2k in essentially linear time.
Speeding the Pollard and elliptic curve methods of factorization
Since 1974, several algorithms have been developed that attempt to factor a large number N by doing extensive computations module N and occasionally taking GCDs with N. These began with Pollard's p 1
Efficient Computation of Chebyshev Polynomials in Computer Algebra
TLDR
This article considers the builtin implementations of the Chebyshev polynomials of the first kind in the general purpose computer algebra systems Axiom, Macsyma, Maple, Mathematica, MuPAD and REDUCE.
Torus-Based Cryptography
TLDR
The concept of torus-based cryptography is introduced, a new public key system called CEILIDH is given, and other discrete log based systems including Lucas-based systems and XTR are compared.
Sylvester-Habicht Sequences and Fast Cauchy Index Computation
TLDR
A new exact divisibility for sub-resultants in the defective case is presented which extends the formulae for the non-defective situation in a natural way and the size of coefficients in the ordinary remainder sequence is quadratic in d.
...
...