13 research outputs found
Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers
We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[zeta] , i.e. the integers extended with zeta , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n^2) for n bit input. This is an improvement from the known results based on the Euclidian algorithm, and taking time O(n· M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols. The technique underlying our algorithms can be used to obtain equally fast algorithms for gcd and quartic residuosity in the ring of Gaussian integers, Z[i]
Norm-Euclidean Galois fields
Let K be a Galois number field of prime degree . Heilbronn showed that
for a given there are only finitely many such fields that are
norm-Euclidean. In the case of all such norm-Euclidean fields have
been identified, but for , little else is known. We give the first
upper bounds on the discriminants of such fields when . Our methods
lead to a simple algorithm which allows one to generate a list of candidate
norm-Euclidean fields up to a given discriminant, and we provide some
computational results
A Note on Koblitz Curves over Prime Fields
Besides the well-known class of Koblitz curves over binary fields, the class of
Koblitz curves over prime fields with is also
of some practical interest. By refining a classical result of Rajwade for the cardinality of , we obtain a simple formula of in terms of the norm on the ring of Eisenstein integers, that is, for some with and some unit ,
holds. This establishes an interesting relation between the number of points on this class of curves and the number of elements of their underlying fields, they are given by the norm of two integers of whose difference is just a unit. It is also interesting to note that such relationship has already been derived for the case of Koblitz curves over binary fields. Some tools that are useful in the computation of cubic residues are also developed
The Thirteenth Power Residue Symbol
This paper presents an efficient deterministic algorithm for computing \textsuperscript{th}-power residue symbols in the cyclotomic field , where is a primitive \textsuperscript{th} root of unity.
The new algorithm finds applications in the implementation of certain cryptographic schemes and closes a gap in the \textsl{corpus} of algorithms for computing power residue symbols
Short undeniable signatures:design, analysis, and applications
Digital signatures are one of the main achievements of public-key cryptography and constitute a fundamental tool to ensure data authentication. Although their universal verifiability has the advantage to facilitate their verification by the recipient, this property may have undesirable consequences when dealing with sensitive and private information. Motivated by such considerations, undeniable signatures, whose verification requires the cooperation of the signer in an interactive way, were invented. This thesis is mainly devoted to the design and analysis of short undeniable signatures. Exploiting their online property, we can achieve signatures with a fully scalable size depending on the security requirements. To this end, we develop a general framework based on the interpolation of group elements by a group homomorphism, leading to the design of a generic undeniable signature scheme. On the one hand, this paradigm allows to consider some previous undeniable signature schemes in a unified setting. On the other hand, by selecting group homomorphisms with a small group range, we obtain very short signatures. After providing theoretical results related to the interpolation of group homomorphisms, we develop some interactive proofs in which the prover convinces a verifier of the interpolation (resp. non-interpolation) of some given points by a group homomorphism which he keeps secret. Based on these protocols, we devise our new undeniable signature scheme and prove its security in a formal way. We theoretically analyze the special class of group characters on Z*n. After studying algorithmic aspects of the homomorphism evaluation, we compare the efficiency of different homomorphisms and show that the Legendre symbol leads to the fastest signature generation. We investigate potential applications based on the specific properties of our signature scheme. Finally, in a topic closely related to undeniable signatures, we revisit the designated confirmer signature of Chaum and formally prove the security of a generalized version
New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field
In this paper, we present a new cube root algorithm in finite
field with a power of prime, which extends
the Cipolla-Lehmer type algorithms \cite{Cip,Leh}. Our cube root
method is inspired by the work of Müller \cite{Muller} on
quadratic case. For given cubic residue
with , we show that there is an irreducible
polynomial with root such that
is a cube root of . Consequently we find an efficient cube root
algorithm based on third order linear recurrence sequence arising
from . Complexity estimation shows that our algorithm is
better than previously proposed Cipolla-Lehmer type algorithms
Trace Expression of r-th Root over Finite Field
Efficient computation of -th root in has many
applications in computational number theory and many other related
areas. We present a new -th root formula which generalizes
Müller\u27s result on square root, and which provides a possible
improvement of the Cipolla-Lehmer algorithm for general case. More
precisely, for given -th power , we show that
there exists such that
where and is a root of certain irreducible
polynomial of degree over
The Eleventh Power Residue Symbol
This paper presents an efficient algorithm for computing -power residue symbols in the cyclotomic field , where is a primitive root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the -power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes
Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers
AbstractWe present simple and efficient algorithms for computing the gcd and cubic residuosity in the ring of Eisenstein integers, Z[ζ], i.e. the integers extended with ζ, a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n2) for n-bit input. For the cubic residuosity problem this is an improvement from the known results based on the Euclidean algorithm, and taking time O(n⋅M(n)), where M(n) denotes the complexity of multiplying n-bit integers. For the gcd problem our algorithm is simpler and faster than an earlier algorithm of complexity O(n2). The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols