Investigation of the distribution of the solutions of Diophantine equations.

Unit equations,
decomposable form equations and in more general, decomposable polynomial equations, together with
elliptic and superelliptic equations play central roles in the theory of Diophantine equations. They have
many applications in algebraic number theory, transcendence theory, criptography and also for
recurrence sequences. In 1988, Evertse and Győry showed that the finiteness theory of
decomposable form equations and unit equations are equivalent. In 1993, Győry described in
full generality the structure of the set of solutions of decomposable form equations.
In the 1970’s, it was Győry to obtain the first results which provided uniform bounds
independent of the coefficients for the number of solutions of unit equations. Later, these investigations
attracted many researchers. Győry together with coauthors derived such
uniform bounds in full generality also for decomposable polynomial equations and unit equations.
Furthermore, in case of infinitely many solutions, he described the asymptotic behaviour of the solutions
of bounded size. His results are common generalizations of many former classical theorems
which led to many important applications.

Effective finiteness results concerning Diophantine equations.

The effective theory of the polynomial
equations in two variables and the exponential Diophantine equations were developed by Baker in the
60’s and Tijdeman in the 70’s, respectively. With the help of Baker’s method, it became possible to give
effective upper bounds for the size of the solutions of certain important equations. This made it possible,
at least in principle, to solve the equations in question. Several mathematicians joined these investigations
worldwide that resulted in a number of publications. Developing a new method, Győry has extended Baker’s results to decomposable form equations in any number of unknowns,
including discriminant form and index form equations. This settled some classical problems in Diophantine number theory going back to the 19th century.

Numerical resolution of unit equations and decomposable form equations.

The explicit upper
bounds for the solutions of Diophantine equations are generally too huge, therefore they do not allow the
determination of all solutions in particular cases. In the last 30 years, new methods have been developed
yielding a substantial reduction in the bounds and making it possible to determine all solutions of a given
equation with the use of computers. Important numerical results have been obtained for unit equations in
two unknowns by Smart, Wildanger, Gaál and Győry, for index form equations
over number fields of degree at most 6 by Gaál, Pethő and Győry.

Decomposition of polynomials.

To apply some deep Diophantine methods including Baker’s Theory or Bilu-Tichy Theorem we
have to know the structure of zeros to certain polynomials or families of polynomials. More precisely,
for applying Baker’s Theory we give an upper bound for the number of simple zeros (or the number
of zeros with odd multiplicities) and for using the Bilu-Tichy Theorem we need some information on
the decomposition of the corresponding polynomials or the families of polynomials.
On the frame of our planned research we will focus on the last problem. In a recent article all the possible
decompositions of the Bernoulli polynomials are given. Recently, Kreso and Rakaczki obtained a
similar result for Euler polynomials. We would like to extend these theorems to general Appell-
polynomials (the most popular Appell polynomials are Bernoulli and Euler polynomials). Further,
we would like to continue our investigations on certain addition theorems. We
observed a connection between generalized Bernoulli and the classical Euler polynomials and we try
to give an analogue formula between two arbitrary Appell polynomials.

Pseudorandom generators.

Pseudorandom generators have many applications ranging from simulations to cryptographic protocols. In all of these applications the quality of these generators is of great importance. In 1997
Mauduit and Sárközy introduced measures of pseudorandomness. They also provided a binary
pseudorandom sequence construction with good pseudorandom measures. Another widespread measure for the quality of a pseudorandom generator is the linear complexity. In 2006 Nina Brandstätter
and Arne Winterhof proved a correspondence between the linear complexity profile and the correlation
measure.
In cryptography large families of good pseudorandom sequences are needed. Later Mauduit and
Sárközy, partly with other coauthors, presented large families of binary pseudorandom sequences also
with good pseudorandom measures. Regarding these sequences it is not enough if each of them has
good pseudorandom measures: they also have to satisfy certain criteria together as a family. In 2004
Goubin, Mauduit and Sárközy introduced a measure addressing this issue, the F-complexity. In 2007
Viktoria Tóth adapted the notion of avalanche effect to families of pseudorandom binary sequences

Combinatorial Diophantine problems.

Combinatorial numbers, e.g. binomial coefficients and
power sums occur in many interesting and deep problems in number theory. The investigation of equal
values of such numbers has a long history. For results concerning equations of these types we refer to the
famous book of Mordell. Beside several leading foreign mathematicians, the members of our
research group obtained many important related results, see e.g. the papers of Győry, Pethő, Pintér,
Hajdu, Kovács and Tengely. The proofs of these results involve several deep, classical and modern methods from Diophantine number theory.

Ternary equations and binomial Thue equations of unknown degree.

Wiles’ proof in 1995 for Fermat’s conjecture induced several significant results obtained by Serre, Darmon,
Merel, Ribet, Kraus, Bennett, Skinner, Vatsal, Yazdani, Siksek for Diophantine equations of the shape \[ax^n + by^n = cz^m,\] where \(a,b,c\) are certain fixed nonzero integers, \(x,y,z,n>2\) and \(m\in\{2,3,n\}\) are unknown
integers. In the extremely important case \(m=n, c=1,\) the equation was solved by Darmon, Merel and Ribet
in the more general situation when \(a,b\) are also unknowns, namely unknown \(S\)-units with \(S=\{2\}\) (i.e. \(a,b\)
are divisible by 2 only). Recently, Pintér and Győry, partly with foreign coauthors, generalized
significantly the above mentioned result. Namely, they considered equations in unknown integers \(x,y,z\)
and unknown prime \(n>13,\) and solved completely the equations in all such cases when \(a,b\) are \(S\)-units
corresponding to \(S=\{p,q\}, 2\leq p < q \leq 30, p,q\) primes. Their result in case of \(n>13, a=b=1,\) includes Wiles’
famous theorem as a special case. In their proof, they used a generalized version of Wiles’ method.

Powers in products of consecutive integers.

A celebrated result of Erdős and Selfridge
states that the product of consecutive integers is never a perfect power. Erdős conjectured 70 years ago
that apart from some certain exceptions, binomial coefficients cannot be perfect powers. Important related
results have been obtained by, among others, Erdős and Tijdeman. In 1997, Győry
gave a positive answer to Erdős’ conjecture. Later, he gave a common generalization of this result
and the theorem of Erdős and Selfridge. An old, hard conjecture is that the product of \(k\geq 3\) consecutive
terms of an arithmetic progression with common difference \(d\geq 1\) cannot be a perfect power. After several
partial results, recently, Bennett, Bruin, Hajdu, Pintér and Győry achieved a
significant breakthrough, namely, they proved the conjecture when \(12\leq k\leq 35.\)

Arithmetic progressions on varieties.

An arithmetic progression on a curve \(y^2=F(x),\) is an arithmetic progression in either the \(x\) or \(y\) coordinates. One can pose the following natural question: What is the longest arithmetic progression
in the \(x\) coordinates. In case of linear polynomials, Fermat claimed and Euler proved that four distinct squares cannot form an arithmetic progression.
In case of quadratic polynomials, Allison found an infinite family of curves which contains an integral arithmetic progression of length eight. Arithmetic progressions on Pellian equations were studied by many authors.
Pethő and Ziegler proved that for the four-term arithmetic progression 1, 3, 5, 7 there exists a Pellian equation \(x^2 - dy^2 = m,\) where \(d\) is not a square, such that 1, 3, 5, 7 are \(y\)-components of solutions of this equation.
If \(F\) is a cubic polynomial, then the problem is to determine arithmetic progressions on elliptic curves. Bremner and Campbell found distinct infinite families of elliptic curves, with arithmetic progression of length eight.
Other models of elliptic curves (Edwards curves, Huff curves, quartic curves) were investigated by many researchers. There are also interesting results related to higher genus curves. A. Bérczes and A. Pethő investigated those solutions of norm
form equations which form arithmetic progressions. Their results initiated a fruitful research on the topic of describing solutions of different type of Diophantine equations with special arithmetic properties. All solutions of
parametric families of norm form equations have been determined. There are many open problems in this field. Finding long arithmetic progressions is a difficult question. There are known infinite families of curves containing arithmetic
progressions of some length, however to extend or to prove that extension is not possible.

Hash functions.

Hash functions are essential building blocks for many cryptographic protocols, and play an important role in digital signature and password based authentication schemes.
In the classical hash functions, due to the limited computation power, the speed of the hash function was a primal attribute. Nowadays a typical adversary has an access to a huge and cheap computing power.
Thomas Roth demonstrated a powerful attack against SHA1 hash using the Amazon EC2 service breaking approximately in 4 minutes and 45 cents each.
In 2004 Bérczes, Ködmön and Folláth presented a new hash function, that was implemented and brought to commercial use under the name Codefish by Crypto Ltd. In 2008 A. Bérczes and I. Járási extended the concept to index forms.
The codefish was cryptanalysed by Aumasson in 2009.
In 2010 Bérczes, Folláth, and Pethő presented a new construction with better operand length and without the weakness of Codefish. This construction was implemented under the name UDHash.
Its fastest implementation is also relatively slow and because of this it is more resistant to the attack mentioned above than the fast hash functions. Later Folláth studied the avalanche effect of the
UDHash function and it turned out that it does not possess the strict avalanche criterion, but a special case of it has it in a bit looser asymptotic way. The practical investigations confirm that it
shows a very similar behavior to the strict avalanche criterion.

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