[2014] [2013]
[2012] [2011]
[2010] [2009]
[2008] [2007]
[2006] [2005]
[2004] [2003]
[2002] [2001]
[2000] [1999]
[1998] [1997]
[1996]
[1994] [1993]
[1991]
[
INRIA Research Reports]
[publications from HAL]
Note: the years below correspond to the first preprint, not to the final
publication dates.
Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice, with David Adrian, Karthikeyan Bhargavan, Zakir Durumeric, Pierrick Gaudry, Matthew Green, J. Alex Halderman, Nadia Heninger, Drew Springall, Emmanuel Thomé, Luke Valenta, Benjamin VanderSloot, Eric Wustrow, Santiago Zanella-Beguelin, preprint, May 2015, 15 pages.
We investigate the security of Diffie-Hellman key exchange as
used in popular Internet protocols and find it to be less secure
than widely believed. First, we present a novel flaw in TLS
that allows a man-in-the-middle to downgrade connections
to “export-grade” Diffie-Hellman. To carry out this attack,
we implement the number field sieve discrete log algorithm.
After a week-long precomputation for a specified 512-bit
group, we can compute arbitrary discrete logs in this group
in minutes. We find that 82% of vulnerable servers use a
single 512-bit group, allowing us to compromise connections
to 7% of Alexa Top Million HTTPS sites. In response, major
browsers are being changed to reject short groups.
More details here.
Automatic Analysis, 5 pages. This is a preliminary
version of a chapter for a book about collected works of Philippe Flajolet.
I wrote that chapter in March to July 2012. While the book is not yet
published, I put this available here.
Magic Squares of Squares, with Paul Pierrat and
François Thiriet, 2015.
We give modular properties and new classes of potential solutions for
magic squares of squares and similar problems.
2014:
Beyond Double Precision, research
project submitted as Advanced Grant
to the European Research
Council, October 2014.
This project was judged ``of high quality but not sufficient to
pass to Step 2 of the evaluation''.
Better Polynomials for GNFS,
with Shi Bai, Cyril Bouvier, and Alexander Kruppa, preprint, September 2014,
to appear in Mathematics of Computation.
[HAL]
The general number field sieve (GNFS) is the most efficient algorithm known
for factoring large integers. It consists of several stages, the first
one being polynomial selection. The quality of the selected polynomials can
be modelled in terms of size and root properties. We propose a new kind of
polynomials for GNFS: with a new degree of freedom, we further improve the
size property. We demonstrate the efficiency of our algorithm by exhibiting a
better polynomial than the one used for the factorization of RSA-768, and a
polynomial for RSA-1024 that outperforms the best published one.
Note: the size-optimized RSA-1024 polynomial B_{1024} can be
reproduced using CADO-NFS using the command sopt -n 135...563 -f rsa1024.poly1 -d 6 with that rsa1024.poly1 file.
Division-Free Binary-to-Decimal Conversion,
with Cyril Bouvier, IEEE Transactions on Computers, volume 63, number 8,
pages 1895-1901, August 2014.
[HAL]
This article presents algorithms that convert multiple precision
integer or floating-point numbers from radix
2 to radix 10 (or to any radix b > 2).
Those algorithms, based on the ``scaled remainder tree'' technique,
use multiplications instead of divisions in their critical part.
Both quadratic and subquadratic algorithms are detailed, with
proofs of correctness.
Experimental results show that our implementation of those algorithms
outperforms the GMP library by up to 50% (using the same low-level routines).
Discrete logarithm in GF(2^{809}) with FFS, with Razvan Barbulescu, Cyril Bouvier, Jérémie Detrey, Pierrick Gaudry, Hamza Jeljeli, Emmanuel Thomé and Marion Videau,
proceedings of PKC 2014,
Lecture Notes in Computer Science Volume 8383, pages 221-238, 2014.
[HAL]
We give details on solving the discrete logarithm problem in the 202-bit
prime order subgroup of GF(2^{809}) using the Function Field
Sieve algorithm (FFS).
Factorization of
RSA-704 with CADO-NFS, with Shi Bai and Emmanuel Thomé, preprint, July
2012.
[HAL]
We give details of the factorization of RSA-704 with CADO-NFS. This is a
record computation with publicly available software tools.
The aim of
this experiment was to stress CADO-NFS --- which was originally designed
for 512-bit factorizations --- for larger inputs, and to identify possible
rooms of improvement.
Size Optimization of Sextic Polynomials in the Number
Field Sieve, with Shi Bai, preprint, March 2012, revised June 2013.
[HAL]
The general number field sieve (GNFS) is the most efficient algorithm known for
factoring large integers. It consists of several stages, the first one being
polynomial selection. The quality of the chosen polynomials in polynomial
selection can be modelled in terms of size and root properties. In this paper,
we describe some methods to optimize the size properties of sextic polynomials.
To reproduce the example on page 12, see this Sage file.
Note added January 23, 2015: part of the results of this preprint are given
in Better Polynomials for GNFS (see above).
Avoiding adjustments in modular computations,
preprint, March 2012.
We consider a sequence of operations
(additions, subtractions, multiplications) modulo a fixed
integer N, where only the final value is needed,
therefore intermediate computations might use any representation.
This kind of computation appears for example in number theoretic transforms
(NTT), in stage 1 of the elliptic
curve method for integer factorization,
in modular exponentiation, ...
Our aim is to avoid as much as possible adjustment steps,
which consist in adding or subtracting N, since those steps are useless
in the mathematical sense.
Note added July 17, 2012: the fact of using residues larger than N in
Montgomery multiplication is well known. See for example the article
"Software Implementation of Modular Exponentiation Using Advanced Vector
Instructions Architectures" by Shay Gueron and Vlad Krasnov in the proceedings
of WAIFI 2012, where it is called "Non Reduced Montgomery Multiplication"
(NRMM).
Finding Optimal Formulae for
Bilinear Maps, with Razvan Barbulescu, Jérémie Detrey and Nicolas Estibals,
proceedings of
WAIFI 2012, Bochum, Germany, July 16-19,
LNCS 7369, pages 168-186, 2012.
2011:
Maximal Determinants and Saturated D-optimal
Designs of Orders 19 and 37, with Richard P. Brent, William Orrick,
and Judy-anne Osborn, 28 pages.
A saturated D-optimal design is a {+1,-1} square matrix of given order with maximal determinant. We search for saturated D-optimal designs of orders 19 and 37, and find that known matrices due to Smith, Cohn, Orrick and Solomon are optimal. For order 19 we find all inequivalent saturated D-optimal designs with maximal determinant, 2^{30} * 7^{2} * 17, and confirm that the three known designs comprise a complete set. For order 37 we prove that the maximal determinant is 2^{39} * 3^{36}, and find a sample of inequivalent saturated D-optimal designs. Our method is an extension of that used by Orrick to resolve the previously smallest unknown order of 15; and by Chadjipantelis, Kounias and Moyssiadis to resolve orders 17 and 21. The method is a two-step computation which first searches for candidate Gram matrices and then attempts to decompose them. Using a similar method, we also find the complete spectrum of determinant values for {+1,-1} matrices of order 13.
Will Orrick compiles known results about
The Hadamard maximal determinant problem.
Related integer sequences are
A003432 and
A003433.
Note: Richard Brent wrote a paper
showing how to generate many Hadamard
equivalence classes of solutions from a given Gram matrix.
Numerical Approximation of the Masser-Gramain Constant to
Four Decimal Digits: delta=1.819..., with Guillaume Melquiond and W. Georg
Nowak, Mathematics of Computation, volume 82, number 282, pages 1235-1246, 2013.
[HAL entry]
We prove that the constant studied by Masser, Gramain, and Weber, satisfies
1.819776 < delta < 1.819833, and disprove a conjecture of Gramain.
This constant is a two-dimensional analogue of the Euler-Mascheroni constant;
it is obtained by computing the radius r_{k} of the smallest disk of
the plane containing k Gaussian integers. While we have used the original
algorithm for smaller values of k, the bounds above come from methods
we developed to obtain guaranteed enclosures for larger values of k.
Ballot stuffing in a postal voting system,
with Véronique Cortier, Jérémie Detrey, Pierrick Gaudry, Frédéric Sur,
Emmanuel Thomé and Mathieu Turuani, proceedings of
REVOTE 2011, International Workshop on
Requirements Engineering for Electronic Voting Systems, Trento, Italy,
August 29, 2011, pages 27-36.
We review a postal voting system used in spring 2011 by the French research institute CNRS and designed by a French company (Tagg Informatique).
We explain how the structure of the material can be easily understood out of a few samples of voting material (distributed to the voters), without any prior knowledge of the system.
Taking advantage of some flaws in the design of the system, we show how to perform major ballot stuffing, making possible to change the outcome of the election.
Our attack has been tested and confirmed by the CNRS. A fixed postal voting system has been quickly proposed by Tagg Informatique in collaboration with the CNRS, preventing this attack for the next elections.
Short Division of Long Integers,
with David Harvey, proceedings of the
20th IEEE Symposium on Computer
Arithmetic (ARITH 20), Tuebingen, July 25-27, 2011, pages 7-14.
[HAL entry,
DOI]
We consider the problem of short division --- i.e., approximate quotient
--- of multiple-precision integers.
We present ready-to-implement algorithms that yield an approximation
of the quotient, with tight and rigorous error bounds.
We exhibit speedups of up to 30% with respect to GMP division with remainder,
and up to 10% with respect to GMP short division, with room for further
improvements.
This work enables one to implement fast correctly rounded division routines
in multiple-precision software tools.
Note added July 27, 2011: the algorithm used by GMP 5 (implemented in the
mpn_div_q function) was presented by Torbjörn Granlund in his invited talk
at the ICMS 2006 conference.
Non-Linear
Polynomial Selection for the Number Field
Sieve
[doi],
with Thomas Prest, Journal of Symbolic
Computation, special issue in the honour of Joachim von zur Gathen,
volume 47, number 4, pages 401-409, 2012.
[HAL]
We present an algorithm to find two non-linear polynomials
for the Number Field Sieve integer factorization method.
This algorithm extends Montgomery's "two quadratics" method;
for degree 3, it gives two skewed polynomials with resultant
O(N^{5/4}), which improves on Williams O(N^{4/3}) result.
Note added in June 2011:
Namhun Koo, Gooc Hwa Jo and Soonhak Kwon extended our algorithm in
this preprint.
Note added on September 30, 2011: Nicholas Coxon extended and analysed our
algorithm in
this preprint.
Calcul mathématique avec
Sage [in french] with Alexandre Casamayou, Guillaume Connan, Thierry
Dumont, Laurent Fousse, François Maltey, Matthias Meulien, Marc Mezzarobba,
Clément Pernet and Nicolas M. Thiéry, 2010.
This book shows how to solve several problems in mathematics, and gives
examples using the Sage computer algebra
system.
Note added December 21st, 2011: a preliminary revised version (1.0.9)
is available here.
Modern Computer Arithmetic,
with Richard Brent, Cambridge University Press, 2010
[HAL entry].
This book collects in the same document all state-of-the-art
algorithms in multiple precision arithmetic (integers, integers modulo n,
floating-point numbers). The best current reference on that topic is
volume 2 from Knuth's The art of computer programming,
which misses some new important algorithms (divide and conquer division,
other variants of FFT multiplication, floating-point algorithms, ...)
Our aim is to give detailed algorithms:
- for all operations (not just multiplication as many text books),
- for all size ranges (not just schoolbook methods or FFT-based methods),
- and including all details (for example how to properly deal with carries
for integer algorithms, or a rigorous analysis of roundoff errors for
floating-point algorithms).
The book would be useful for graduate students in computer science and
mathematics (perhaps too specialized for most undergraduates, at least
in its present state), researchers in discrete mathematics, computer
algebra, number theory, cryptography, and developers of multiple-precision
libraries.
Reliable Computing with GNU MPFR,
proceedings of the 3rd
International Congress on Mathematical Software (ICMS 2010), June 2010,
pages 42-45, LNCS 6327, Springer.
The original publication is (or will be) available on www.springerlink.com.
This article presents a few applications where reliable computations are
obtained using the GNU MPFR library.
Why and how to use arbitrary precision, with
Kaveh R. Ghazi, Vincent Lefèvre and Philippe Théveny, March 2010,
Computing in Science and Engineering, volume 12, number 3, pages 62-65,
2010 (© IEEE).
Most nowadays floating-point computations are
done in double precision, i.e., with a significand (or
mantissa) of 53 bits.
However, some applications require more precision:
double-extended (64 bits or more), quadruple precision (113 bits) or even more.
In an article published in The Astronomical Journal in 2001, Toshio
Fukushima says: In the days of powerful computers,
the errors of numerical integration are the main limitation
in the research of complex dynamical systems,
such as the long-term stability of our solar system
and of some exoplanets [...] and gives an example
where using double precision leads to an accumulated
round-off error of more than 1 radian for the solar
system! Another example where arbitrary precision
is useful is static analysis of floating-point programs
running in electronic control units of aircrafts or in
nuclear reactors.
An O(M(n) log n) algorithm for the
Jacobi symbol, with Richard Brent, January 2010,
Proceedings of the Ninth Algorithmic Number Theory Symposium (ANTS-IX),
Nancy, France, July 19-23, 2010, LNCS 6197, pages 83-95, Springer Verlag
[the original publication is or will be available at www.springerlink.com].
Factorization of a 768-bit RSA
modulus, with Thorsten Kleinjung, Kazumaro Aoki, Jens Franke, Arjen K.
Lenstra, Emmanuel Thomé,
Joppe W. Bos, Pierrick Gaudry, Alexander Kruppa, Peter L. Montgomery,
Dag Arne Osvik, Herman te Riele and Andrey Timofeev, Proceedings of
Crypto'2010, Santa Barbara, USA, LNCS 6223, pages 333-350, 2010
[technical announcement].
This paper reports on the factorization of the 768-bit number
RSA-768 by the
number field sieve factoring method and discusses some implications for RSA
[more details here]
Note added 21 September 2012: in Section 2.3 (Sieving) we report 47 762 243 404
unique relations (including free relations). It appears the correct number
should be about 10^{9} less, i.e., 46 762 246 508 including free
relations, and 46 705 023 046 without free relations.
Note added 09 October 2012: in Section 2.3 (Sieving) the number of remaining
prime ideals we report during the filtering (initially 35.3G, then 14.5G and
10G) are most probably underestimated by about 5G.
The Great Trinomial Hunt,
with Richard Brent, Notices of the American Mathematical
Society, volume 58, number 2, pages 233-239, February 2011.
The glibc bug #10709, September 2009.
[bugzilla
entry]
On computers without double-extended precision,
the GNU libc 2.10.1 incorrectly rounds the sine
of (the double-precision closest to) 0.2522464. This is a bug in
IBM's Accurate Mathematical Library, which claims correct rounding,
as recommended by IEEE 754-2008.
We analyze this bug and propose a fix.
Calcul formel : mode d'emploi.
Exemples en Maple, with Philippe Dumas, Claude Gomez, Bruno Salvy, March 2009 (in french) [HAL entry].
This book is a free version of the book of the same name published by Masson in
1995. The examples use an obsolete version of Maple (V.3), but most of the text
still applies to Maple and other modern computer algebra systems.
Computing predecessor and successor in rounding to nearest,
with Siegfried
Rump, Sylvie Boldo and Guillaume Melquiond, BIT
Numerical Mathematics, volume 49, number 2, pages 419-431, 2009.
We give simple and efficient methods to compute and/or estimate the
predecessor and successor of a floating-point number using only floating-point
operations in rounding to nearest. This may be used to simulate interval
operations, in which case the
quality in terms of the diameter of the result is
significantly improved compared to existing approaches.
2008:
Worst Cases for the Exponential Function in the
IEEE 754r decimal64 Format, with Vincent Lefèvre and Damien
Stehlé, LNCS volume 5045, pages 114-126,
special LNCS issue
following the Dagstuhl seminar 06021:
Reliable Implementation of Real Number Algorithms: Theory and Practice,
August 2008,
We searched for the worst cases for correct rounding of the exponential
function in the IEEE 754r decimal64 format, and computed all the bad cases
whose distance from a breakpoint (for all rounding modes) is less than
10^{-15} ulp, and we give the worst ones. In particular, the worst case
for |x| ≥ 3 * 10^{-11} is exp(9.407822313572878 * 10^{-2})
= 1.09864568206633850000000000000000278... This work can be
extended to other elementary functions in the decimal64 format and allows
the design of reasonably fast routines that will evaluate these functions
with correct rounding, at least in some domains.
[Complete lists of worst cases for the exponential are available
for the IEEE 754r
decimal32 and
decimal64
formats.]
Ten New Primitive Binary Trinomials,
with Richard Brent, Mathematics of Computation 78 (2009), pages 1197-1199
[Brent's web
page].
We exhibit ten new primitive
trinomials over GF(2)
of record degrees 24036583, 25964951, 30402457, and 32582657.
This completes the search for the currently known Mersenne prime exponents.
Implementation of the reciprocal square root in
MPFR, March 2008 (extended abstract),
Dagstuhl Seminar Proceedings following
Dagstuhl seminar 08021 (Numerical validation in current hardware
architectures), January 06-11, 2008.
We describe the implementation of the reciprocal square root --- also called
inverse square root --- as a native function
in the MPFR library. The difficulty is to implement
Newton's iteration for the reciprocal square root on top's of
GNU MP's
mpn layer, while guaranteeing a rigorous 1/2 ulp bound on the
roundoff error.
Landau's function
for one million billions,
with Marc Deléglise and Jean-Louis Nicolas,
Journal de Théorie des Nombres de Bordeaux, volume 20, number 3, pages 625-671,
2008.
A Maple program implementing the algorithm described in this paper is
available from
Jean-Louis Nicolas web page.
Let S_{n} denote the symmetric group
with n letters, and g(n) the maximal order of an element
of S_{n}. If the standard factorization of M into primes
is M=q_{1}^{a1}
q_{2}^{a2} ...
q_{k}^{ak}, we define l(M)
to be q_{1}^{a1} +
q_{2}^{a2} + ... +
q_{k}^{ak};
one century ago, E. Landau proved that g(n)=max_{l(M) ≤ n} M
and that,
when n goes to infinity, log g(n) ~ sqrt(n log(n)).
There exists a basic algorithm to compute g(n) for 1 ≤ n
≤ N; its running time is
O(N^{3/2} / sqrt(log N))
and the needed memory is O(N);
it allows computing g(n) up to, say, one million. We describe an
algorithm to calculate g(n) for n up to 10^{15}.
The main idea is to use
the so-called l-superchampion numbers. Similar numbers, the
superior highly composite numbers, were introduced by S. Ramanujan to
study large values of the divisor function tau(n)=sum_{d divides n} 1.
Faster Multiplication in
GF(2)[x], with Richard P. Brent, Pierrick Gaudry and Emmanuel Thomé,
Proceedings of the
Eighth Algorithmic Number Theory
Symposium (ANTS-VIII), May 17-22, 2008, Banff Centre, Banff, Alberta
(Canada), A. J. van der Poorten and A. Stein, editors, pages 153--166,
LNCS 5011, 2008.
A preliminary version appeared as INRIA Research Report, November 2007.
In this paper, we discuss an implementation of various algorithms for
multiplying polynomials in GF(2)[x]: variants of the window methods,
Karatsuba's, Toom-Cook's, Schönhage's and Cantor's algorithms. For most
of them, we propose improvements that lead to practical speedups.
The code that we developed for this paper is contained in the gf2x
package, available under the GNU General Public License from
http://wwwmaths.anu.edu.au/~brent/gf2x.html.
Note (added 20 May 2008): part of this paper will be soon obsolete,
namely the base case section, with the
PCLMULQDQ
instruction from
Intel's new AVX instruction-set
(compiler intrinsics,
simulator,
compiler support).
Note (added 26 Jan 2009): Gao and Mateer have found a theoretical
speedup in Cantor
multiplication, see http://cr.yp.to/f2mult.html.
Note (added 29 Nov 2011): Su and Fan analyze the use of PCLMULQDQ in
http://eprint.iacr.org/2011/589.
A Multi-level Blocking Distinct Degree Factorization
Algorithm, INRIA Research Report 6331,
with Richard P. Brent, 16 pages, October 2007.
This paper describes in detail the algorithm presented at the
8th International
Conference on Finite Fields and Applications (Fq8),
July 9-13, 2007, Melbourne, Australia
[extended abstract],
[Richard's slides].
A revised version appeared in a special issue of Contemporary Mathematics,
volume 461, pages 47-58, 2008.
We give a new algorithm for performing the distinct-degree factorization of
a polynomial P(x) over GF(2), using a multi-level
blocking strategy. The coarsest level of blocking replaces GCD computations
by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup
(1992), and others.
The novelty of our approach is that
a finer level of blocking replaces multiplications by squarings,
which speeds up the computation
in GF(2)[x]/P(x) of certain interval polynomials
when P(x) is sparse.
As an application we give a fast
algorithm to search for all irreducible trinomials
x^{r} + x^{s} + 1 of degree
r over GF(2), while producing a certificate that can be checked
in less time than the full search.
Naive algorithms cost O(r^{2}) per trinomial, thus O(r^{3}) to search
over all trinomials of given degree r.
Under a plausible assumption about the
distribution of factors of trinomials, the new algorithm has
complexity O(r^{2} (log r)^{3/2} (log log r)^{1/2})
for the search over all trinomials of degree r.
Our implementation achieves a speedup of greater than a factor
of 560 over the naive algorithm
in the case r = 24036583 (a Mersenne exponent).
Using our program, we have found two new primitive trinomials
of degree 24036583 over GF(2) (the previous record degree was 6972593).
A GMP-based implementation of
Schönhage-Strassen's large integer multiplication algorithm,
with Pierrick Gaudry and Alexander Kruppa,
Proceedings of the
International Symposium on
Symbolic and Algebraic Computation (ISSAC 2007),
Waterloo, Ontario, Canada, pages 167-174,
editor C.W.Brown, 2007.
Schönhage-Strassen's algorithm is one of the best known algorithms for
multiplying large integers. Implementing it efficiently is of utmost
importance, since
many other algorithms rely on it as a subroutine. We present here an improved
implementation, based on the one distributed within the GMP library.
The following ideas and
techniques were used or tried: faster arithmetic modulo 2^{n}+1,
improved cache
locality, Mersenne transforms, Chinese Remainder Reconstruction, the
sqrt(2) trick, Harley's and Granlund's tricks, improved tuning. We
also discuss some ideas we plan to try in the future.
Note: this paper was motivated by
Allan
Steel, and the corresponding code is available from
http://www.loria.fr/~zimmerma/software/.
Andrew Sutherland used our FFT code to set a new record for
elliptic curve point
counting.
Note added on April 29, 2011: Tsz-Wo Sze multiplied integers of 2^{40}
bits in about 2000 seconds using a cluster of 1350 cores :
preprint.
Time- and Space-Efficient
Evaluation of Some Hypergeometric Constants, with Howard Cheng,
Guillaume Hanrot, Emmanuel Thomé and Eugene Zima,
Proceedings of the International Symposium on
Symbolic and Algebraic Computation (ISSAC 2007),
Waterloo, Ontario, Canada, pages 85-91,
editor C.W.Brown, 2007.
The currently best known algorithms for the numerical evaluation of
hypergeometric constants such as zeta(3) to d decimal digits have time
complexity O(M(d) log^{2} d) and space complexity of O(d log d) or
O(d). Following work from Cheng, Gergel, Kim and Zima, we present a new
algorithm with the same asymptotic complexity, but more efficient in practice.
Our implementation of this algorithm improves over existing programs
for the computation of Pi, and we announce a new record of 2 billion digits
for zeta(3).
Worst Cases of a Periodic Function for Large Arguments,
with Guillaume Hanrot,
Vincent Lefèvre and Damien Stehlé,
Proceedings of the 18th IEEE Symposium on Computer Arithmetic
(ARITH'18), pages 133-140,
Montpellier, France, 2007.
A preliminary version appeared as
INRIA Research Report 6106,
January 2007.
One considers the problem of finding hard to round cases of a periodic
function for large floating-point inputs, more precisely when the function
cannot be efficiently approximated by a polynomial.
This is one of the last few issues that prevents from guaranteeing an
efficient computation of correctly rounded transcendentals for the whole
IEEE-754 double precision format.
The first non-naive algorithm for that problem is presented, with an heuristic
complexity of O(2^{0.676 p}) for a precision of p bits.
The efficiency of the algorithm is shown on the largest IEEE-754 double
precision binade for the sine function, and some corresponding bad cases are
given.
We can hope that all the worst cases of the trigonometric functions
in their whole domain will be found within a few years, a task that
was considered out of reach until now.
2006:
Asymptotically Fast Division for GMP,
October 2005, revised August 2006, October 2006 and February 2015.
Until version 4.2.1, GNU MP
(GMP for short) division has complexity
O(M(n) log(n)), which is not asymptotically optimal.
We propose here some division algorithms that achieve O(M(n)) with small
constants, with corresponding GMP code.
Code is available too: invert.c computes an approximate
inverse within 1 ulp in 3M(n) [revised in February and May 2015].
A patch for the 2-limb case was proposed by
Marco
Bodrato: invert.diff.
Errors Bounds on Complex Floating-Point Multiplication, with Richard Brent
and Colin Percival,
Mathematics
of Computation volume 76 (2007), pages 1469-1481.
Some technical details are given
in
INRIA Research Report 6068, December 2006.
Given floating-point arithmetic with t-digit base-β significands
in which all arithmetic operations are performed as if calculated to infinite
precision and rounded to a nearest representable value, we prove that
the product of complex values z_{0} and z_{1}
can be computed with
maximum absolute error |z_{0}| |z_{1}| (1/2)
β^{1-t}
sqrt(5). In particular, this provides relative error bounds of 2^{-24}
sqrt(5) and 2^{-53} sqrt(5) for IEEE 754 single and double precision
arithmetic respectively, provided that overflow, underflow, and
denormals do not occur.
We also provide the numerical worst cases for IEEE 754 single and double
precision arithmetic.
20 years of ECM (©
Springer-Verlag),
with Bruce Dodson, Proceedings of
ANTS VII, July 2006.
A preliminary version appeared as
INRIA Research Report 5834,
February 2006.
The Elliptic Curve Method for integer factorization (ECM) was invented by
H. W. Lenstra, Jr., in 1985 [Lenstra87].
In the past 20 years, many improvements of ECM were proposed
on the mathematical, algorithmic, and implementation sides.
This paper summarizes the current state-of-the-art,
as implemented in the GMP-ECM software.
Erratum: on page 541 we write ``Computer experiments indicate that these curves
have, on average, 3.49 powers of 2 and 0.78 powers of 3, while Suyama's family
has 3.46 powers of 2 and 1.45 powers of 3''. As noticed by Romain Cosset,
those experiments done by the
first author are wrong, since he ran 1000 random curves with the same
prime input p=10^{10}+19, and the results differ according on the
congruence of p mod powers of 2 and 3. With 10000 random curves on the 10000
primes just above 10^{20}, we get an average of
2^{3.34}*3^{1.68} = 63.9 for Suyama's family, and
2^{3.36}*3^{0.67} = 21.5 for curves of the form
(16d + 18) y^{2} = x^{3} + (4d + 2)x^{2} + x.
MPFR:
A Multiple-Precision Binary Floating-Point
Library With Correct Rounding, with Laurent Fousse, Guillaume Hanrot,
Vincent Lefèvre, Patrick Pélissier,
INRIA Research Report RR-5753, November 2005.
A revised version appeared in ACM TOMS (Transactions on Mathematical
Software), volume 33, number 2, article 13, 2007.
This paper presents a multiple-precision binary
floating-point library,
written in the ISO C language, and based on the GNU MP library.
Its particularity is to extend ideas from the IEEE-754 standard
to arbitrary precision, by providing correct rounding
and exceptions.
We demonstrate how these strong semantics are achieved ---
with no significant slowdown with respect to other tools ---
and discuss a few applications where such a library can be useful.
Techniques algorithmiques et méthodes de
programmation (in french), 11 pages, July 2005, appeared in
Encyclopédie de
l'informatique et des systèmes
d'information, pages 929-935, Vuibert, 2006.
5,341,321, June 2005.
This short note shows the nasty effects of patents for the development of free
software, even for patents that were not written with software applications
in mind.
The Elliptic Curve Method,
November 2002, revised April 2003 and September
2010,
appeared in the
Encylopedia of Cryptography and Security, Springer, 2005
(old link).
Describes in two pages the history of ECM, how it works at high level,
improvements to the method, and some applications.
MPFR : vers un calcul flottant correct ? (in french),
Interstices, 2005.
Obtenir un seul résultat pour un calcul donné : à
première vue, cela semble une évidence ; c'est en fait un vaste
sujet de recherche auquel les chercheurs apportent petit à petit leurs
contributions. Une nouvelle étape est franchie aujourd'hui grâce
à MPFR, une bibliothèque de
calcul multi-précision sur les nombres flottants.
A primitive trinomial of degree 6972593,
with Richard Brent and Samuli Larvala,
Mathematics of Computation, volume 74, number 250, pages 1001-1002, 2005.
The only primitive trinomials of degree 6972593 over GF(2)
are x^{6972593} + x^{3037958} + 1 and its reciprocal.
An elementary digital plane recognition algorithm,
with Yan Gerard and Isabelle Debled-Rennesson,
appeared in Discrete Applied Mathematics, volume 151, issue 1-3,
pages 169-183, 2005.
A naive digital plane is a subset of
points (x,y,z) in Z^{3} verifying
h ≤ ax+by+cz < h+max{ | a|,|b|, |c| } where
(a,b,c,h) in Z^{4}. Given a finite unstructured subset
of Z^{3}, determine whether there
exists a naive digital plane containing it is called
digital plane recognition. This question is
rather classical in the field of digital geometry (also called
discrete geometry). We suggest in this paper a new algorithm to
solve it. Its asymptotic complexity
is bounded by O(n^{7}) but its behavior seems to be linear in
practice. It uses an original strategy of optimization in a set of
triangular facets (triangles). The code is short and elementary
(less than 300 lines) and available on
http://www.loria.fr/~debled/plane and here.
Searching Worst Cases of a One-Variable Function Using Lattice Reduction,
with Damien Stehlé and Vincent Lefèvre,
IEEE Transactions on Computers,
volume 54, number 3, pages 340-346, 2005.
A preliminary version appeared as
INRIA Research Report 4586.
Some results for the 2^{x} function double-extended precision
are available
here.
We propose a new algorithm to find worst cases for the correct rounding of a
mathematical function of one variable. We first reduce this problem to the
real small value problem--i.e., for polynomials with real coefficients. Then,
we show that this second problem can be solved efficiently by extending
Coppersmith's work on the integer small value problem--for polynomials with
integer coefficients--using lattice reduction. For floating-point numbers with
a mantissa less than N and a polynomial approximation of degree d, our
algorithm finds all worst cases at distance less than N^{-d^2/(2d+1)}
from a machine number in time O(N^{(d+1)/(2d+1)+epsilon}). For d=2, a
detailed study improves on the O(N^{2/3+epsilon}) complexity from
Lefèvre's algorithm to O(N^{4/7+epsilon}). For larger d, our
algorithm can be used to check that there exist no worst cases at distance
less than N^{-k} in time O(N^{1/2+epsilon}).
2004:
Gal's Accurate Tables Method Revisited,
with Damien Stehlé, INRIA Research Report RR-5359, October 2004.
An improved version
appeared in the Proceedings of
Arith'17.
Those ideas are demonstrated by an implementation of the
exp2 function
in double precision.
Erratum in the final version of the paper: in Section 4, the
simultaneous worst case for sin and cos is t0=1f09c0c6cde5e3
and not t0=31a93fddd45e3.
See also
my coauthor page.
Gal's accurate tables algorithm aims at providing an efficient implementation of
elementary functions with correct rounding as often as possible. This method
requires an expensive pre-computation of a table made of the values taken by the
function or by several related functions at some distinguished points. Our
improvements of Gal's method are two-fold: on the one hand we describe what is
the arguably best set of distinguished values and how it improves the efficiency
and correctness of the implementation of the function, and on the other hand we
give an algorithm which drastically decreases the cost of the pre-computation.
These improvements are related to the worst cases for the correct rounding of
mathematical functions and to the algorithms for finding them. We show that the
whole method can be turned into practice by giving complete tables for
2^{x} and sin(x) for x in [1/2,1[, in double precision.
Newton iteration revisited,
with Guillaume Hanrot, March 2004.
On March 10, 2004, Dan Bernstein announced a revised draft of his paper
Removing redundancy in high-precision Newton iteration,
with algorithms that compute a reciprocal
of order n over C[[x]] 1.5+o(1) times longer than a product;
a quotient or logarithm 2.16666...+o(1) times longer;
a square root 1.83333...+o(1) times longer;
an exponential 2.83333...+o(1) times longer.
We give better algorithms.
Note added on March 24, 2004: the 1.5+o(1) reciprocal algorithm was already
published by Schönhage (Information Processing Letters 74, 2000, p. 41-46)
Note added on July 24, 2006: in a preprint
Newton's method and FFT trading, Joris van der Hoeven announces
better constants for the exponential (2.333...) and the quotient (1.666...).
[October 27, 2009: those constants need yet to be confirmed.]
Note added on April 20, 2009: as noticed by David Harvey, in Section 3,
in the Divide algorithm, Step 4 should read
q <- q_{0} + g_{0} (h_{1} - ε) x^{n}, where
h = h_{0} + x^{n} h_{1}.
Indeed, after Step 3 we have q_{0} f = h_{0} +
ε x^{n} + O(x^{2n}),
i.e., q_{0} = h_{0}/f + ε/f x^{n} + O(x^{2n}).
Thus h/f = q_{0} + (h_{1} - ε)/f x^{n}
+ O(x^{2n}) = q_{0} + g_{0} (h_{1} - ε)
x^{n} + O(x^{2n}).
Note added on September 11, 2009: as noticed by David Harvey, the
1.91666...M(n) cost for the square root (Section 4) is incorrect;
it should be 1.8333...M(n) instead. Indeed, Step 3 of Algorithm SquareRoot
costs only M(n)/3 to compute f_{0}^{2} mod x^{n}-1
instead of M(n)/2, since we only need two FFT transforms of size n.
David Harvey has improved the reciprocal to 1.444...M(n) and the square root
to 1.333...M(n) [preprint].
Arithmétique
flottante,
with Vincent Lefèvre, INRIA Research Report RR-5105,
February 2004 (in french).
Ce document rassemble des notes d'un cours donné en 2003 dans
la filière Algorithmique Numérique et
Symbolique du DEA d'Informatique de l'Université Henri Poincaré
Nancy 1.
Ces notes sont basées en grande partie sur le livre
Elementary Functions. Algorithms and Implementation de Jean-Michel
Muller.
Aussi disponible sur LibreCours.
A Formal Proof of Demmel and Hida's Accurate Summation
Algorithm, with Laurent Fousse, January 2004.
A new proof of the ``accurate summation'' algorithm proposed by Demmel and
Hida is presented.
The main part of that proof
has been written in the Coq language and verified by the Coq proof assistant.
The Middle Product Algorithm,
I. Speeding up the division and square root of
power series, with Guillaume Hanrot and Michel Quercia, AAECC, volume 14,
number 6, pages 415-438, 2004.
A preliminary version appeared as
INRIA Resarch Report 3973.
We present new algorithms for the inverse, division, and
square root of power series.
The key trick is a new algorithm --- MiddleProduct or, for
short, MP --- computing the n middle coefficients of a
(2n-1) * n full product in the same number of multiplications
as a full n * n product.
This improves previous work of Brent, Mulders, Karp and Markstein,
Burnikel and Ziegler.
These results apply both to series and polynomials.
Note added June 10, 2009: Part II of this work was planned to deal with
integer entries, but David Harvey was faster than us, see
The Karatsuba middle product for integers.
Proposal for a Standardization of Mathematical Function Implementation in Floating-Point Arithmetic, with David Defour,
Guillaume Hanrot, Vincent Lefèvre, Jean-Michel Muller and
Nathalie Revol, January 2003, Numerical Algorithms, volume 37, number 1-4,
pages 367-375, 2004.
Extended version appeared as
INRIA Research Report RR-5406.
Some aspects of what a standard for the implementation of the elementary
functions could be are presented. Firstly, the need for such a standard is
motivated. Then the proposed standard is given. The question of roundings
constitutes an important part of this paper: three levels are proposed,
ranging from a level relatively easy to attain (with fixed maximal relative
error) up to the best quality one, with correct rounding on the whole range
of every function.
We do not claim that we always suggest the right choices, or that we have
thought about all relevant issues. The mere goal of this paper is to raise
questions and to launch the discussion towards a standard.
A long note on Mulders'
short product, with Guillaume Hanrot, INRIA Research Report RR-4654,
November 2002. A revised version appeared in the
Journal of Symbolic Computation, volume 37, pages 391-401, 2004.
A corrigendum appeared in 2014 (pdf).
The short product of two power series is
the meaningful part of the product of these objects,
i.e., sum(a[i] b[j] x^{i+j}, i+j < n). In [Mulders00], Mulders gives an
algorithm to compute a short product faster than the full product in the
case of Karatsuba's multiplication [KaOf62].
This algorithm works by selecting a
cutoff point k and performing a full k x k product and two
(n-k) x (n-k) short products recursively. Mulders also gives a
heuristically optimal cutoff point beta n. In this paper, we determine
the optimal cutoff point in Mulders' algorithm. We also give a slightly
more general description of Mulders' method.
Note added November 24, 2011: Murat Cenk and Ferruh Ozbudak published a paper
"Multiplication of polynomials modulo x^{n}" in Theoretical Computer
Science, vol. 412, pages 3451-3462, 2011. The main result of this paper
(Theorem 3.1) is a direct consequence of Mulders' short product, which is not
cited in this paper. Blame to the authors, to the anonymous reviewers and to
the editor in charge, Victor Y. Pan.
Moreover the results in Table 1 of this paper are not optimal:
M̂(14) <= M(10) + 2*M̂(4) <= 39+2*8 = 55 (instead of 56);
M̂(16) <= M(12) + 2*M̂(4) <= 51+2*8 = 67 (instead of 70);
M̂(17) <= M(12) + 2*M̂(5) <= 51+2*11 = 73 (instead of 76);
M̂(18) <= M(12) + 2*M̂(6) <= 51+2*15 = 81 (instead of 85).
Note added February 10, 2014: Karim Belabas pointed out an error in Algorithm
ShortProduct (and in Theorem 2). Indeed, the result is not necessarily reduced
modulo x^{n}, since for example with f=2*x+1 and g=3*x+4 with n=2,
we have l=1*4=4, h=2*3=6, m=(1+2)*(4+3)-4-6=11, thus the result is
4 + 11*x + 6*x^{2}. To fix this, it suffices to zero out the
coefficient of x^{n} in the final result.
2003:
A Binary Recursive Gcd Algorithm, with Damien Stehlé,
INRIA Research Report RR-5050, December 2003.
A revised version (© Springer-Verlag)
is published in the Proceedings of the
Algorithmic Number Theory Symposium (ANTS VI).
[Erratum]
[implementation in GMP]
The binary algorithm is a variant of the Euclidean algorithm that
performs well in practice. We present a quasi-linear time
recursive algorithm that computes the greatest common divisor of
two integers by simulating a slightly modified version of the
binary algorithm. The structure of the
recursive algorithm is very close to the one of the well-known
Knuth-Schönhage fast gcd algorithm, but the description and the
proof of correctness are significantly simpler in our case. This
leads to a simplification of the implementation and to better
running times.
Note (added 14 Oct 2009): since that work, Niels Möller has designed a
classical left-to-right/MSB fast gcd algorithm, cf his paper
On Schönhage's
algorithm and subquadratic integer gcd computation (Mathematics of
Computation, volume 77, number 261, pages 589--607, 2008) and his
slides.
Note (added 1st April 2010): Robert Harley had published in his
ECDL code a similar
extended binary gcd algorithm (however only in the quadratic case).
Algorithms for finding almost irreducible and almost primitivive trinomials,
with Richard Brent, April 2003, Proceedings of a Conference in Honour of Professor H. C. Williams, Banff, Canada (May 2003), The Fields Institute, Toronto.
Consider polynomials over GF(2).
We describe efficient
algorithms for finding trinomials with
large irreducible (and possibly primitive) factors, and give examples
of trinomials
having a primitive factor of degree r for all Mersenne exponents
r = + 3 mod 8 in the range 5 < r < 2976221,
although there is no irreducible trinomial of degree r.
We also give
trinomials with a primitive factor of degree r = 2^{k}
for 3 <= k <= 12. These trinomials enable efficient
representations of the finite field GF(2^{r}).
We show how trinomials with large primitive factors can
be used efficiently in applications where primitive trinomials
would normally be used.
Note added April 22, 2009: this paper is mentioned in Divisibility of
Trinomials by Irreducible Polynomials over F_{2}, by Ryul Kim and
Wolfram Koepf, International Journal of Algebra, Vol. 3, 2009, no. 4, 189-197
(arxiv).
Accurate Summation: Towards a Simpler and Formal Proof,
with Laurent Fousse, March 2003, in Proc. of
RNC'5, pages 97-108.
This paper provides a simpler proof of the ``accurate summation'' algorithm
proposed by Demmel and Hida in
DeHi02.
It also gives improved bounds in some cases, and examples showing that those
new bounds are optimal.
This simpler proof will be used to obtain a computer-generated proof
of Demmel-Hida's algorithm, using a proof assistant like HOL, PVS or Coq.
10^{2098959} [in french], décembre 2002,
paru dans la
Gazette du Cines,
numéro 14, janvier 2003.
Cet article décrit les premiers résultats de la recherche
(avec Richard Brent et Samuli Larvala) de trinômes primitifs de
degré 6972593 sur GF(2), et indique quelques conséquences
amusantes de la loi de Moore.
A Fast Algorithm for Testing Reducibility of Trinomials mod 2
and Some New Primitive Trinomials of Degree 3021377,
with Richard Brent and Samuli Larvala, Mathematics of Computation,
volume 72, number 243, pages 1443-1452, 2003.
A preliminary version appeared as
Report
PRG TR-13-00, Oxford
University Computing Laboratory, December 2000.
The standard algorithm for testing irreducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory and of order r^{2}
bit-operations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and less bit-operations than the standard algorithm.
Using the algorithm, we have found several new irreducible trinomials of high degree.
If r is a Mersenne exponent (i.e. 2^{r} - 1 is a Mersenne prime), then an irreducible trinomial of degree r is necessarily primitive and can be used
to give a pseudo-random number generator with period at least 2^{r} - 1. We give examples of primitive trinomials for r = 756839, 859433, and
3021377. The results for r = 859433 extend and correct some computations of Kumada et al [Mathematics of Computation 69 (2000),
811-814]. The two results for r = 3021377 are primitive trinomials of the highest known degree.
2002:
Worst Cases and Lattice Reduction, with Damien Stehlé and Vincent Lefèvre,
INRIA Research Report RR-4586, October 2002.
Appeared in the proceedings of the 16th IEEE Symposium on Computer Arithmetic
(Arith'16), IEEE Computer Society, pages 142-147, 2003.
We propose a new algorithm to find worst cases for correct rounding
of an analytic function.
We first reduce this problem to the real small value problem
--- i.e. for polynomials with real coefficients.
Then we show that this second problem can be solved efficiently,
by extending Coppersmith's work
on the integer small value problem
--- for polynomials with integer coefficients ---
using lattice reduction
[Coppersmith96a,Coppersmith96b,Coppersmith01].
For floating-point numbers with a mantissa less than $N$,
and a polynomial approximation of degree $d$,
our algorithm finds all worst cases at distance $< N^{\frac{-d^2}{2d+1}}$
from a machine number in time $O(N^{\frac{d+1}{2d+1}+\varepsilon})$.
For $d=2$, this improves on the $O(N^{2/3+\varepsilon})$ complexity
from Lef\`evre's algorithm [Lefevre00,LeMu01] to $O(N^{3/5+\varepsilon})$.
We exhibit some new worst cases found using our algorithm,
for double-extended and quadruple precision.
For larger $d$, our algorithm can be used to check that there exist no
worst cases at distance $< N^{-k}$ in time $O(N^{\frac{1}{2}+O(\frac{1}{k})})$.
A Proof of GMP Square Root, with Yves Bertot and Nicolas Magaud,
Journal of Automated Reasoning, volume 29, 2002,
pages 225--252, Special Issue on Automating and Mechanising Mathematics: In
honour of N.G. de Bruijn.
A preliminary version appeared as
INRIA Research Report 4475.
We present a formal proof (at the implementation level) of an efficient
algorithm proposed by Paul Zimmermann
[Zimmermann00]
to compute square roots
of arbitrary large integers.
This program, which is part of the GNU Multiple Precision
Arithmetic Library (GMP) is completely proven within the
Coq system. Proofs are developed using the Correctness tool
to deal with imperative features of the program.
The formalization is rather large (more than 13000 lines) and requires
some advanced techniques for proof management and reuse.
Note: Vincent Lefèvre found a potential problem in the GMP implementation,
which is fixed by the following patch. This does not
contradicts our proof: the problem is due to the different C data types
(signed or not, different width), whereas our proof assumed a unique type.
Aliquot Sequence 3630 Ends After Reaching 100 Digits, with M. Benito,
W. Creyaufmüller and J. L. Varona, Experimental Mathematics, volume 11,
number 2, pages 201-206.
In this paper we present a new computational record: the aliquot sequence
starting at 3630 converges to 1 after reaching a hundred decimal digits.
Also, we show the current status of all the aliquot sequences starting with a
number smaller than 10,000; we have reached at leat 95 digits for all of them.
In particular, we have reached at least 112 digits for the so-called
"Lehmer five sequences," and 101 digits for the "Godwin twelve sequences."
Finally, we give a summary showing the number of aliquot sequences of unknown
end starting with a number less than or equal 10^{6}.
Note added 29 July 2008: in this paper, we say (page 203, middle of right
column) "It is curious to note that the driver 2^{9} * 3 * 11 * 31
has appeared in no place in any of the sequences given in Table 1";
Clifford Stern notes that this driver appears at index 215 of sequence 165744,
which gives 2^{9} * 3 * 7 * 11 * 31² * 37 * 10594304241173.
2001:
De l'algorithmique à l'arithmétique via le calcul formel, Habilitation à diriger des recherches,
novembre 2001. (Transparents de la soutenance.)
This document presents my research contributions from 1988 to 2001,
performed first at INRIA Rocquencourt within the Algo project (1988 to 1992),
then at INRIA Lorraine and LORIA within the projects Euréca (1993-1997),
PolKA (1998-2000), and Spaces (2001).
Three main periods can be roughly
distinguished: from 1988 to 1992 where my research
focused on analysis of algorithms and random generation,
from 1993 to 1997 where I worked on computer algebra and related algorithms,
finally from 1998 to 2001 where I was interested in arbitrary precision
floating-point arithmetic with well-defined semantics.
Arithmétique en précision
arbitraire, rapport de recherche INRIA 4272,
septembre 2001, paru dans la revue
"Réseaux et Systèmes Répartis, Calculateurs parallèles",
volume 13, numéro 4-5, pages 357-386, 2001 [in french].
This paper surveys the available algorithms for integer or
floating-point arbitrary precision calculations. After a brief
discussion about possible memory representations, known algorithms
for multiplication, division, square root, greatest common
divisor, input and output, are presented, together with their
complexity and usage. For each operation, we present the naïve
algorithm, the asymptotically optimal one, and also intermediate
«divide and conquer» algorithms, which often are very useful. For
floating-points computations, some general-purpose methods are
presented for algebraic, elementary, hypergeometric and special
functions.
Tuning and Generalizing Van Hoeij's Algorithm,
with Karim Belabas and Guillaume Hanrot, INRIA Research report 4124,
February 2001.
Recently, van Hoeij's published a new algorithm for factoring
polynomials over the rational integers. This algorithms rests
on the same principle as Berlekamp-Zassenhaus, but uses
lattice basis reduction to improve drastically on the
recombination phase. The efficiency of the LLL algorithm is very
dependent on fine tuning; in this paper, we present such tuning to
achieve better performance. Simultaneously, we describe a
generalization of van Hoeij's algorithm to factor polynomials over
number fields.
Efficient isolation of a polynomial real roots,
with Fabrice Rouillier, INRIA Research report 4113, February 2001.
Appeared in Journal of Computational and Applied Mathematics, volume 162,
number 1, pages 33-50, 2004.
This paper gives new results for the isolation of real roots of a univariate
polynomial using Descartes' rule of signs, following work of Vincent,
Uspensky, Collins and Akritas, Johnson, Krandick.
The first contribution is a generic algorithm which enables one to describe
all the existing strategies in a unified framework.
Using that framework, a new algorithm is presented, which is optimal
in terms of memory usage, while doing no more computations than other
algorithms based on Descartes' rule of signs.
We show that these critical optimizations have important consequences
by proposing a full efficient solution for isolating the real roots
of zero-dimensional polynomial systems.
Density results on floating-point invertible numbers, with Guillaume Hanrot, Joël Rivat and Gérald Tenenbaum,
Theoretical Computer Science, volume 291, number 2, 2003, pages 135-141.
(The slides of a related talk I gave in January 2002 at the workshop
"Number Theory and Applications" in Luminy are
here.)
Let F_k denote the k-bit mantissa
floating-point (FP) numbers.
We prove a conjecture of J.-M. Muller according to which
the proportion of numbers in F_k with no FP-reciprocal (for rounding to the
nearest element) approaches
1/2-3/2 log(4/3) i.e. about 0.06847689 as
k goes to infinity.
We investigate a similar question for the inverse square root.
2000:
Factorization of a 512-bit RSA
Modulus,
with Stefania Cavallar, Bruce Dodson, Arjen K. Lenstra, Walter Lioen,
Peter L. Montgomery, Brian Murphy, Herman te
Riele, Karen Aardal, Jeff Gilchrist,
Gérard Guillerm, Paul Leyland, Joël
Marchand, François Morain, Alec Muffett,
Chris Putnam, Craig Putnam, Proceedings of Eurocrypt'2000,
LNCS 1807, pages 1-18, 2000.
On August 22, 1999, we completed the factorization of the 512-bit 155-digit number RSA-155 with the help of the Number Field Sieve factoring method (NFS). This is a new record for factoring general numbers. Moreover, 512-bit RSA keys are frequently used for the protection of electronic commerce -- at least outside the USA -- so this factorization represents a breakthrough in research on RSA-based systems. The previous record, factoring the 140-digit number RSA-140, was established on February 2, 1999, also with the help of NFS, by a subset of the team which factored RSA-155. The amount of computing time spent on RSA-155 was about 8400 MIPS years, roughly four times that needed for RSA-140; this is about half of what could be expected from a straightforward extrapolation of the computing time spent on factoring RSA-140 and about a quarter of what would be expected from a straightforward extrapolation from the computing time spent on RSA-130. The speed-up is due to a new polynomial selection method for NFS of Murphy and Montgomery which was applied for the first time to RSA-140 and now, with improvements, to RSA-155.
A proof of GMP fast division and square root implementations,
September 2000. This short note gives a detailed correctness proof of
fast (i.e. subquadratic) versions of the
GNU MP
mpn_bz_divrem_n and mpn_sqrtrem functions, together with
complete GMP code.
The mpn_bz_divrem_n function divides (with remainder)
a number of 2n limbs by a divisor of n limbs in 2K(n), where K(n) is the
time spent in a (n times n) multiplication, using the
Moenck-Borodin-Jebelean-Burnikel-Ziegler algorithm.
The mpn_sqrtrem computes the square root and the remainder
of a number of 2n limbs (square root and remainder have about n limbs each)
in time 3K(n)/2; it uses Karatsuba Square Root.
Speeding up the Division and Square Root of Power Series, with Guillaume Hanrot and Michel Quercia,
INRIA Research Report 3973, July 2000.
We present new algorithms for the inverse, quotient, or
square root of power series.
The key trick is a new algorithm --- RecursiveMiddleProduct or RMP
--- computing the n middle coefficients of a
(2n * n) product in essentially the same number of operations --- K(n)
--- than a full (n * n) product with Karatsuba's method.
This improves previous work of Mulders, Karp and Markstein,
Burnikel and Ziegler.
These results apply both to series, polynomials, and multiple precision
floating-point numbers.
A Maple implementation is available here,
together with slides of a talk given at ENS Paris
in January 2004.
Factorization in Z[x]: the searching phase, with John Abbott and Victor Shoup, April 2000,
Proceedings of ISSAC'2000
[HAL entry].
In this paper we describe ideas used to accelerate the Searching Phase
of the Berlekamp-Zassenhaus algorithm, the algorithm most widely used
for computing factorizations in Z[x]. Our ideas do not alter the
theoretical worst-case complexity, but they do have a significant effect in
practice: especially in those cases where the cost of the Searching
Phase completely dominates the rest of the algorithm. A complete
implementation of the ideas in this paper is publicly available in the
library NTL. We give timings of this implementation on
some difficult factorization problems.
1999:
Karatsuba Square
Root, INRIA Research Report 3905, November 1999.
We exhibit an algorithm to compute the square-root with remainder of a n-word
number in (3/2) K(n) word operations, where K(n) is the number of words
operations to multiply two n-word numbers using Karatsuba's algorithm. If the
remainder is not needed, the cost can be reduced to K(n) on average. This
algorithm can be used for floating-point or polynomial computations too;
although not optimal asymptotically, its simplicity gives a wide range of use,
from about 50 to 1,000,000 digits, as shown by computer experiments.
On Sums of Seven Cubes,
with Francois Bertault and Olivier
Ramaré, Mathematics of Computation, volume 68, number 227,
pages 1303-1310, 1999.
We show that every integer between 1290741 and 3.375 × 10^{12} is
a sum of 5 nonnegative cubes, from which we deduce that every integer which
is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum
of 7 nonnegative cubes.
Uniform Random Generation of
Decomposable Structures Using Floating-Point Arithmetic with Alain
Denise, Theoretical Computer Science, volume 218, number 2, 219--232, 1999.
A preliminary version appeared as
INRIA Research Report 3242, September 1997.
The recursive method formalized by Nijenhuis and Wilf [NiWi78] and
systematized by Flajolet, Van Cutsem and Zimmermann [FlZiVa94], is extended
here to floating-point arithmetic. The resulting ADZ method enables one to
generate decomposable data structures --- both labelled or unlabelled ---
uniformly at random, in expected O(n^{1+ε}) time and space, after a
preprocessing phase of O(n^{2+ε}) time, which reduces to
O(n^{1+ε}) for context-free grammars.
Estimations asymptotiques
du nombre de chemins Nord-Est de
pente fixée et de largeur bornée, avec Isabelle Dutour et
Laurent Habsieger, INRIA Research Report RR-3585, décembre 1998
[in french].
We study here a quantity related to the number of walks with North and East steps staying under the line of slope d starting from the origin. We give an asymptotic analysis of this quantity with respect to both the width n and the slope d, answering to a question asked by Bernard Mourrain.
1997:
Calcul formel : ce qu'il y a dans la boîte,
journées X-UPS, octobre 1997.
Cinq algorithmes de calcul
symbolique, INRIA Technical Report RT-0206,
notes de cours d'un module de spécialisation du DEA d'informatique
de l'Université Henri Poincaré Nancy 1, 1997 [in french].
These are lecture notes of a course entitled «Some computer algebra
algorithms» given by the author at the University of Nancy 1 in 1997. Five
fundamental algorithms used by computer algebra systems are briefly described:
Gosper's algorithm for computing indefinite sums, Zeilberger's algorithm for
definite sums, Berlekamp's algorithm for factoring polynomials over finite
fields, Zassenhaus' algorithm for factoring polynomials with integer
coefficients, and Lenstra's integer factorization algorithm using elliptic
curves. All these algorithms were implemented --- or improved --- by the
author in the computer algebra system MuPAD.
Progress Report on
Parallelism in MuPAD, with
Christian Heckler and Torsten Metzner,
Inria Research Report 3154, April 1997.
MuPAD is a general purpose computer algebra system with two programming concepts for parallel processing: micro-parallelism for shared-memory machines and macro-parallelism for distributed architectures. This article describes language instructions for both concepts, the current state of implementation, together with some examples.
1996:
Polynomial Factorization Challenges,
with L. Bernardin and M. Monagan, poster presented at
the International Symposium on Symbolic and Algebraic Computation
(ISSAC), July 1996, 4 pages.
Joachim von zur Gathen has proposed a challenge for factoring
univariate polynomials over finite fields to evaluate the practicability
of current factorization algorithms ("A Factorization Challenge",
SIGSAM Bulletin 26(2):22-24, 1992).
More recently, Victor Shoup has proposed an alternate family of polynomials
with a similar goal in mind.
Our effort is to take these challenges on using the general purpose computer
algebra systems Maple and MuPAD.
The result of our work are the factorizations of the von zur Gathen polynomials
f_{n} and of the Shoup polynomials F_{n} for n from 1 to 500.
We also present
the factorization of the degree 1000 von zur Gathen polynomial
f_{1000}.
1994:
GFUN: a Maple package for the manipulation of generating and holonomic
functions in one variable, with Bruno Salvy, ACM Transactions on
Mathematical Software, volume 20, number 2, 1994.
A preliminary version appeared as
INRIA Technical Report 143, October 1992.
We describe the GFUN package which contains functions for manipulating sequences, linear recurrences or differential equations and generating functions of various types. This document is intended both as an elementary introduction to the subject and as a reference manual for the package.
Random walks, heat equation and distributed algorithms, with Guy
Louchard, René Schott and Michael Tolley, Journal of Computational and
Applied Mathematics, volume 53, pages 243-274, 1994.
New results are obtained concerning the analysis of the storage allocation
algorithm which permits one to maintain two stacks inside a shared (continuous)
memory area of fixed size m and of the banker's algorithm (a deadlock
avoidance policy).
The formulation of these problems is in terms of random walks inside polygonal
domains in a two-dimensional lattice space with several reflecting barriers
and one absorbing barrier.
For the two-stacks problem, the return time to the origin, the time to
absorption, the last leaving time from the origin and the number of returns
to the origin before absorption are investigated. For the banker's algorithm,
the trend-free absorbed random walk is analyses with numerical methods.
We finally analyse the average excursion along one axis for the classical
random walk: an analytic method enables us to deduce asymptotic results for
this average excursion.
The n-Queens Problem, with Igor Rivin and Ilan Vardi, American
Mathematical Monthly, volume 101, number 7, pages 629-639.
We give several lower and upper bounds for the number Q(n) of ways to put n
queens on an nxn chessboard, and the number T(n) to put n queens on a toroidal
chessboard (i.e., with n upper diagonals instead of 2n-1).
We also conjecture that (log T(n))/(n log n) and (log Q(n))/(n log n) tend to
two positive constants.
Note added 11 September 2012:
in Remark 1 page 636, the primes p = 2q+1 are called
safe primes (and the primes q are called Sophie-Germain primes).
Moreover Warren Smith pointed the paper "The n-queens problem" by Bruen and
Dixon (Discrete Mathematics, 1975) which gives a construction yielding
(if I am correct) at least 8 * binomial(floor((p-3)/8),2) * p different
toroidal solutions for any prime p >= 13.
Also on page 635 we say "Remark. Corollary 1 gives the first example of a set
of n's for which Q(n) grows faster than a polynomial in n". This is wrong,
since T. Kløve gives in "The modular n-queens problem" (Discrete Mathematics,
1977) a construction yielding for n=p*p at least n*(p-3)*p^{p-1}
(thanks again to Warren Smith for pointing us this).
1993:
A Calculus of Random
Generation, with Philippe Flajolet and Bernard Van Cutsem, Proceedings
of European Symposium on Algorithms (ESA'93), LNCS 726, pages 169-180, 1993.
A systematic approach to the random generation of labelled combinatorial objects is presented. It applies to structures that are decomposable, i.e., formally specifiable by grammars involving union, product, set, sequence, and cycle constructions. A general strategy is developed for solving the random generation problem with two closely related types of methods: for structures of size n, the boustrophedonic algorithms exhibit a worst-case behaviour of the form
O(n log n); the sequential algorithms have worst case
O(n^{2}), while offering good potential for optimizations in the
average case. (Both methods appeal to precomputed numerical tables of linear size).
A companion calculus permits to systematically compute the average case cost of the sequential generation algorithm associated to a given specification. Using optimizations dictated by the cost calculus, several random generation algorithms are developed, based on the sequential principle; most of them have expected complexity 1/2 n log n, thus being only slightly superlinear. The approach is exemplified by the random generation of a number of classical combinatorial structures including Cayley trees, hierarchies, the cycle decomposition of permutations, binary trees, functional graphs, surjections, and set partitions.
Epelle : un logiciel de
détection de fautes
d'orthographe, INRIA Research Report 2030, September 1993.
This report describes the algorithm used by the epelle program, together with its implementation in the C language. This program is able to check about 30.000 words every second on modern computers, without any error, contrary to the Unix spell program which makes use of hashing methods and could thus accept wrong words. The main principle of epelle is to use digital trees (also called dictionary trees), which in addition reduces the space needed to store the list of words (by a factor of about 5 for the french dictionary). Creating a new digital tree for the franch language (about 240.000 words) takes only a dozen of seconds. The same program is directly usable for other languages and more generally for any list of alphanumeric keys.
1991:
Automatic Average-case Analysis of
Algorithms, with Ph. Flajolet and B. Salvy, Theoretical Computer Science,
volume 79, number 1, pages 37-109, 1991.
Many probabilistic properties of elementary discrete combinatorial structures of interest for the average-case analysis of algorithms prove to be decidable. This paper presents a general framework in which such decision procedures can be developed. It is based on a combination of generating function techniques for counting, and complex analysis techniques for asymptotic estimations.
We expose here the theory of exact analysis in terms of generating functions for four different domains: the iterative/recursive and unlabelled/labelled data type domains. We then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed.
A fair fragment of this theory is also incorporated into a system called Lambda-Upsilon-Omega. In this way, using computer algebra, one can produce automatically non-trivial average-case analyses of algorithms operating over a variety of “decomposable” combinatorial structures.
At a fundamental level, this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions. In several cases, it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary “special” functions and classes of asymptotic forms relative to counting, probabilities, or average-case complexity.
Séries
génératrices et analyse automatique d'algorithmes,
PhD thesis (in french), École Polytechnique, Palaiseau, 1991
[in french].
Also available in
postscript.
This thesis studies systematic methods to determine automatically the
average-case cost of an algorithm. Those methods apply generally to descent
schemes in decomposable data structures, which enables one to model a large
class of problems.
More precisely, this thesis focuses on the first stage of the analysis of
an algorithm, namely the algebraic analysis, which translates the program
into mathematical objects, whereas the second stage extracts from those
mathematical objects the desired information about the average-case cost.
We define a language to describe decomposable data-structures and descent
schemes on them. When one uses generating functions as mathematical objects
(counting generating functions for data-structures, and cost generating
functions for programs), we show that the algorithms described in this language
directly translate into systems of equations for the corresponding generating
functions, moreover using simple rules. From those equations, we can then
determine in polynomial time the exact average-case cost for a given size of
the input data-structures. We can also use those equations to compute using
asymptotic analysis the average-case cost when the input data size tends to
infinity, since we know that the asymptotic average cost is directly related
to the behaviour of those generating functions around their singularities.
Therefore, we show that to a given class of algorithms corresponds a
well-defined class of generating functions, and in turn a given class of
formulae for the asymptotic average cost.
Those algebraic analysis rules were included in a software tool for the
average-case analysis of algorithms, called Lambda-Upsilon-Omega, which proved
useful for experiments and research.