Elements of Modern Algebra
... that could be used to solve problems of the same type, and treatments were generalized to
deal with whole classes of problems rather than individual ones.
In our study of abstract algebra, we shall make use of our knowledge of the various
number systems. At the same time, in many cases we wish to ex ...
Notes on Galois Theory
... Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group
under function composition, which we denote by Aut E. Let E be a finite
extension of a field F . Define the Galois group Gal(E/F ) to be the subset
of Aut ...
Information Protection Based on Extraction of Square Roots of
... Hence, on the decryption stage, if Bob gets (2755,
3466) and (3466, 2755), there is no way to decide which
of two Gaussians is authentic. In general, for p = 6221
there are several hundreds Gaussians that have this property, which creates ambiguity. Indeed, consider a Gaussian with three-digit compo ...
CParrish - Mathematics
... (GCD) of a and b. We will denote a GCD of a and b by gcd(a, b). (You may be somewhat
mystified by what seems to be an effort to make a simple concept appear more complex.
Please be patient; you will see that this definition of a GCD will generalize readily to other
mathematical entities, which in so ...
Results on Some Generalizations of Interval Graphs
... v ∈ V (G), v is not in one of N [X], N [Y ], or N [Z]. Thus N [X] ∩ N [Y ] ∩ N [Z] = ∅.
⇐ Suppose G is an n − star graph with middle clique C. Let (P, <) be the
middle clique partial order for middle clique C. Suppose (P, <) has a covering
with three chains with top elements X, Y, Z such that N [X] ...
Lecture 5 Message Authentication and Hash Functions
... • if x is composite it outputs yes with probability at most ¼.
Probability is taken only over the internal randomness of the algorithm,
so we can iterate!
The error goes to zero exponentially fast.
This algorithm is fast and practical!
, Elementary Number Theory
... where a,b, and m are large numbers
All computations will involve repeated modular
multiplications (one multiplication followed by one
division by the modulus m, to get the remainder)
If modulus m is a n-bit number the largest possible value
after a multiplication is guaranteed to be less than 2nbits ...
Algebraic Number Theory, a Computational Approach
... groups, and such a presentation can be reinterpreted in terms of matrices over the
integers. Next we describe how to use row and column operations over the integers
to show that every matrix over the integers is equivalent to one in a canonical
diagonal form, called the Smith normal form. We obtain ...
Revision 2 - Electronic Colloquium on Computational Complexity
... codes and PCPs of proximity). We define a concrete-efficiency threshold that indicates the
smallest problem size beyond which the PCP becomes “useful”, in the sense that using it
actually pays off relative to naive verification by simply rerunning the computation; our
definition takes into account b ...
Polynomial greatest common divisor
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by Euclid's algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.The similarity between the integer GCD and the polynomial GCD allows us to extend to univariate polynomials all the properties that may be deduced from Euclid's algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this allows to get information on the roots without computing them. For example, the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow to compute the square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity.The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of Euclid's algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need of efficiency of computer algebra systems.