17 research outputs found
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
Batching Cipolla-Lehmer-Müller\u27s square root algorithm with hashing to elliptic curves
The present article provides a novel hash function to any elliptic curve of -invariant over a finite field of large characteristic. The unique bottleneck of consists in extracting a square root in as well as for most hash functions. However, is designed in such a way that the root can be found by (Cipolla--Lehmer--)Müller\u27s algorithm in constant time. Violation of this security condition is known to be the only obstacle to applying the given algorithm in the cryptographic context. When the field is highly -adic and , the new batching technique is the state-of-the-art hashing solution except for some sporadic curves. Indeed, Müller\u27s algorithm costs multiplications in . In turn, original Tonelli--Shanks\u27s square root algorithm and all of its subsequent modifications have the asymptotic complexity , where is the -adicity of and a function . As an example, it is shown that Müller’s algorithm actually needs several times fewer multiplications in the field (whose ) of the standardized curve NIST P-224
On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a
2x2-block-matrix by its transpose using 5 block products (3 recursive calls and
2 general products) over C or any finite field.We use geometric considerations
on the space of bilinear forms describing 2x2 matrix products to obtain this
algorithm and we show how to reduce the number of involved additions.The
resulting algorithm for arbitrary dimensions is a reduction of multiplication
of a matrix by its transpose to general matrix product, improving by a constant
factor previously known reductions.Finally we propose schedules with low memory
footprint that support a fast and memory efficient practical implementation
over a finite field.To conclude, we show how to use our result in LDLT
factorization.Comment: ISSAC 2020, Jul 2020, Kalamata, Greec
An algorithms for finding the cube roots in finite fields
Let Fq be a finite field with q elements. Quadratic residues in number theory and finite fields is an important theory that has many applications in various aspects. The main problem of quadratic residues is to find the solution of the equation x2 = a, given an element a. It is interesting to find the solutions of x3 = a in Fq. If the solutions exist for a we say that a is a cubic residue of Fq and x is a cube root of a in Fq. In this paper we examine the solubility of x3 = a in general finite fields. Here, we give some results about the cube roots of cubic residue, and we propose an algorithm to find the cube roots using primitive elements
On fast multiplication of a matrix by its transpose
We present a non-commutative algorithm for the multiplication of a block-matrix by its transpose over C or any finite field using 5 recursive products. We use geometric considerations on the space of bilinear forms describing 2×2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose space and time efficient schedules that enable us to provide fast practical implementations for higher-dimensional matrix products