59 research outputs found
Selmer groups and quadratic reciprocity
In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reciprocity law in F. Ā© Mathematische Seminar 2006
Properties of the series solution for PainlevƩ I
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first PainlevƩ equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented
On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations
In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves
Amicable pairs and aliquot cycles for elliptic curves
An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good
reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study
elliptic amicable pairs and analogously defined longer elliptic aliquot cycles.
We show that there exist elliptic curves with arbitrarily long aliqout cycles,
but that CM elliptic curves (with j not 0) have no aliqout cycles of length
greater than two. We give conjectural formulas for the frequency of amicable
pairs. For CM curves, the derivation of precise conjectural formulas involves a
detailed analysis of the values of the Grossencharacter evaluated at a prime
ideal P in End(E) having the property that #E(F_P) is prime. This is especially
intricate for the family of curves with j = 0.Comment: 53 page
On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations
New Assumptions and Efficient Cryptosystems from the -th Power Residue Symbol
The -th power residue symbol is a useful mathematical tool in cryptography, where is an integer, is a prime ideal in the prime factorization of with a large prime satisfying , and is an -th primitive root of unity. One famous case of the -th power symbol is the first semantic secure public key cryptosystem due to Goldwasser and Micali (at STOC 1982). In this paper, we revisit the -th power residue symbol and its applications. In particular, we prove that computing the -th power residue symbol is equivalent to solving the discrete logarithm problem. By this result, we give a natural extension of the Goldwasser-Micali cryptosystem, where is an integer only containing small prime factors. Compared to another extension of the Goldwasser-Micali cryptosystem due to Joye and Libert (at EUROCRYPT 2013), our proposal is more efficient in terms of bandwidth utilization and decryption cost. With a new complexity assumption naturally extended from the one used in the Goldwasser-Micali cryptosystem, our proposal is provable IND-CPA secure. Furthermore, we show that our results on the -th power residue symbol can also be used to construct lossy trapdoor functions and circular and leakage resilient public key encryptions with more efficiency and better bandwidth utilization
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