222 research outputs found
Integer Factoring Using Small Algebraic Dependencies
Integer factoring is a curious number theory problem with wide applications in complexity and cryptography. The best known algorithm to factor a number n takes time, roughly, exp(2*log^{1/3}(n)*log^{2/3}(log(n))) (number field sieve, 1989). One basic idea used is to find two squares, possibly in a number field, that are congruent modulo n. Several variants of this idea have been utilized to get other factoring algorithms in the last century. In this work we intend to explore new ideas towards integer factoring. In particular, we adapt the AKS primality test (2004) ideas for integer factoring.
In the motivating case of semiprimes n=pq, i.e. p<q are primes, we exploit the difference in the two Frobenius morphisms (one over F_p and the other over F_q) to factor n in special cases. Specifically, our algorithm is polynomial time (on number theoretic conjectures) if we know a small algebraic dependence between p,q. We discuss families of n where our algorithm is significantly faster than the algorithms based on known techniques
On algebraic dependencies between Poincar\'e functions
Let be a rational function of one complex variable, and its
repelling fixed point with the multiplier A Poincar\'e function
associated with is a function meromorphic
on such that ,
and
In this paper, we investigate the following problem: given Poincar\'e functions
and , find
out if there is an algebraic relation
between
them and, if such a relation exists, describe the corresponding algebraic
curve. We provide a solution, which can be viewed as a refinement of the
classical Ritt theorem about commuting rational functions. We also complement
previous results concerning algebraic dependencies between B\"ottcher
functions
Invariant Generation for Multi-Path Loops with Polynomial Assignments
Program analysis requires the generation of program properties expressing
conditions to hold at intermediate program locations. When it comes to programs
with loops, these properties are typically expressed as loop invariants. In
this paper we study a class of multi-path program loops with numeric variables,
in particular nested loops with conditionals, where assignments to program
variables are polynomial expressions over program variables. We call this class
of loops extended P-solvable and introduce an algorithm for generating all
polynomial invariants of such loops. By an iterative procedure employing
Gr\"obner basis computation, our approach computes the polynomial ideal of the
polynomial invariants of each program path and combines these ideals
sequentially until a fixed point is reached. This fixed point represents the
polynomial ideal of all polynomial invariants of the given extended P-solvable
loop. We prove termination of our method and show that the maximal number of
iterations for reaching the fixed point depends linearly on the number of
program variables and the number of inner loops. In particular, for a loop with
m program variables and r conditional branches we prove an upper bound of m*r
iterations. We implemented our approach in the Aligator software package.
Furthermore, we evaluated it on 18 programs with polynomial arithmetic and
compared it to existing methods in invariant generation. The results show the
efficiency of our approach
Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity
Testing whether a set of polynomials has an algebraic dependence
is a basic problem with several applications. The polynomials are given as
algebraic circuits. Algebraic independence testing question is wide open over
finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is
NP (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we
put the problem in AM coAM. In particular, dependence testing is
unlikely to be NP-hard and joins the league of problems of "intermediate"
complexity, eg. graph isomorphism & integer factoring. Our proof method is
algebro-geometric-- estimating the size of the image/preimage of the polynomial
map over the finite field. A gap in this size is utilized in the
AM protocols.
Next, we study the open question of testing whether every annihilator of
has zero constant term (Kayal, CCC'09). We give a geometric
characterization using Zariski closure of the image of ;
introducing a new problem called approximate polynomials satisfiability (APS).
We show that APS is NP-hard and, using projective algebraic-geometry ideas, we
put APS in PSPACE (prior best was EXPSPACE via Grobner basis computation). As
an unexpected application of this to approximative complexity theory we get--
Over any field, hitting-set for can be designed in PSPACE.
This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly
mitigating the GCT Chasm (exponentially in terms of space complexity)
Extensions of differential representations of SL(2) and tori
Linear differential algebraic groups (LDAGs) measure differential algebraic
dependencies among solutions of linear differential and difference equations
with parameters, for which LDAGs are Galois groups. The differential
representation theory is a key to developing algorithms computing these groups.
In the rational representation theory of algebraic groups, one starts with
SL(2) and tori to develop the rest of the theory. In this paper, we give an
explicit description of differential representations of tori and differential
extensions of irreducible representation of SL(2). In these extensions, the two
irreducible representations can be non-isomorphic. This is in contrast to
differential representations of tori, which turn out to be direct sums of
isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde
Zariski Closures of Reductive Linear Differential Algebraic Groups
Linear differential algebraic groups (LDAGs) appear as Galois groups of
systems of linear differential and difference equations with parameters. These
groups measure differential-algebraic dependencies among solutions of the
equations. LDAGs are now also used in factoring partial differential operators.
In this paper, we study Zariski closures of LDAGs. In particular, we give a
Tannakian characterization of algebraic groups that are Zariski closures of a
given LDAG. Moreover, we show that the Zariski closures that correspond to
representations of minimal dimension of a reductive LDAG are all isomorphic. In
addition, we give a Tannakian description of simple LDAGs. This substantially
extends the classical results of P. Cassidy and, we hope, will have an impact
on developing algorithms that compute differential Galois groups of the above
equations and factoring partial differential operators.Comment: 26 pages, more detailed proof of Proposition 4.
Automated Generation of Non-Linear Loop Invariants Utilizing Hypergeometric Sequences
Analyzing and reasoning about safety properties of software systems becomes
an especially challenging task for programs with complex flow and, in
particular, with loops or recursion. For such programs one needs additional
information, for example in the form of loop invariants, expressing properties
to hold at intermediate program points. In this paper we study program loops
with non-trivial arithmetic, implementing addition and multiplication among
numeric program variables. We present a new approach for automatically
generating all polynomial invariants of a class of such programs. Our approach
turns programs into linear ordinary recurrence equations and computes closed
form solutions of these equations. These closed forms express the most precise
inductive property, and hence invariant. We apply Gr\"obner basis computation
to obtain a basis of the polynomial invariant ideal, yielding thus a finite
representation of all polynomial invariants. Our work significantly extends the
class of so-called P-solvable loops by handling multiplication with the loop
counter variable. We implemented our method in the Mathematica package Aligator
and showcase the practical use of our approach.Comment: A revised version of this paper is published in the proceedings of
ISSAC 201
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