A study of monogenity of binomial composition

Conditions are derived which guarantee that products of linear recurring sequences attain maximum linear complexity. It is shown that the product of any number of maximum-length GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> sequences has maximum linear complexity, provided only the degrees of the corresponding minimal polynomials are distinct and greater than two. It is also shown that if the roots of any number of (not necessarily irreducible) minimal polynomials are simple and lie in extension fields of pairwise relatively prime degrees, then the product of the corresponding GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> sequences attains maximum linear complexity, provided only that no two roots of any minimal polynomial are linearly dependent over the groundfield GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> (which is automatically satisfied when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q = 2)</tex> . The results obtained for products are extended to arbitrary linear combinations of product sequences.

Three classes of binary sequences of period 4N with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period N , i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4N and optimal autocorrelation value/magnitude are discussed in the splitting field of xN-1 . As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on m-sequences instead of generalized GMW sequences.

Using easy coefficients conjecture for rotation symmetric Boolean functions

We show that the binary expansions of algebraic numbers do not form secure pseudorandom sequences; given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables us to devise a simple algorithm to factor polynomials with rational coefficients. All algorithms work in polynomial time. Introduction. Manuel Blum raised the following question: Suppose we are given an approximate root of an unknown polynomial with integral coefficients and a bound on the degree and size of the coefficients of the polynomial. Is it possible to infer the polynomial? We answer his question in the affirmative. We show that if a complex number a satisfies an irreducible polynomial h(X) of degree d with integral coefficients in absolute value at most H, then given 0(d? + d ■ logH) bits of the binary expansion of the real and complex parts of a, we can find h(X) in deterministic polynomial time (and then compute in polynomial time any further bits of a). Using the concept of secure pseudorandom sequences formulated by Shamir [23], Blum and Micali [3] and Yao [25], we then show that the binary (or m-ary for any m) expansions of algebraic numbers do not form secure sequences in a certain well-defined sense. We are able to extend our results with the same techniques to transcendental numbers of the form log(a),cos_1(a), etc., where a is algebraic. The technique is based on the lattice basis reduction algorithm from [16]. Our answer to Blum's question enables us to devise a simple polynomial-time algorithm to factor polynomials with rational coefficients: We find an approximate root of the polynomial and use our algorithm to find the irreducible polynomial satisfied by the exact root, which must then be a factor of the given polynomial. This is repeated until all the factors are found. This algorithm was found independently by Schonhage [22], and was already suggested in [16]. The technique of the paper also provides a natural, efficient method to compute with algebraic numbers. This paper is the final journal version of [13], which contains essentially the entire contents of this paper. 1. A Polynomial-Time Algorithm for Blum's Question. Throughout this paper, Z denotes the set of the integers, Q the set of the rationals, R the set of the reals, and C the set of the complex numbers. The ring of polynomials with integral Received December 23, 1985; revised October 13, 1986 and April 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 68Q15, 68Q25, 68Q40. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page

CHAPTER ELEVEN - ROOTS OF POLYNOMIALS

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton. WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.

The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form (t - 1)d and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated. In the general case the problem of computing the minimal polynomial of the image of a given element of order ps in a fixed irreducible representation of a classical group over a field of characteristic p > 2 can be reduced to a similar problem for certain s unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I - XII. The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from p-solvable linear groups to semisimple groups.

In this paper we study the hardness of some discrete logarithmlike problems defined in linear recurring sequences over finitefields from a point of view as general as possible. Theintractability of these problems plays a key role in the securityof the class of public key cryptographic constructions based onlinear recurring sequences. We define new discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems for any nontrivial linear recurring sequence in any finite field whose minimal polynomial is irreducible. Then, we prove that these problems are polynomially equivalent to the discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems in the subgroup generated by any root of the minimal polynomial of thesequence.

Taking as a model the completed theory of vector space endomorphisms, the present text aims at extending this theory to endomorphisms of finitely generated projective modules over a general commutative ring; now analogous results often require totally different methods of proof. The first important result is a structure theorem for such modules when the characteristic polynomial of the endomorphism is separable. The second topic deals with the minimal polynomial, whose mere existence is shown to require additional hypotheses, even over a domain. In the third topic we extend the classical notion of ‘cyclic modules’ as the modules which are invertible over the ring of polynomials modulo the characteristic polynomial. Regarding the diagonalization of endomorphisms, we show that a classical criterion of being diagonalizable over some extension of the base field can be transferred nearly verbatim to rings, provided that diagonalization is expected only after some faithfully flat base change. Many results that hold over a field, like the fact that commuting diagonalizable endomorphisms are simultaneously diagonalizable, hold over arbitrary rings, with this extended meaning of diagonalization. The Jordan-Chevalley-Dunford decomposition, shown as a particular case of the lifting property of étale algebras, also holds over rings. Finally, in several reasonable situations, the eigenspace associated with any root of the characteristic polynomial is shown to be given a more concrete description as the image of a map. In these situations the classical theory generalizes to rings.

As we saw in Chap. 8, when V is a finite-dimensional vector space over \({\mathbb {F}}\), then a linear mapping \(T:V\rightarrow V\) is semisimple if and only if its eigenvalues lie in \({\mathbb {F}}\) and its minimal polynomial has only simple roots. It would be useful to have a result that would allow one to predict that T is semisimple on the basis of a criterion that is simpler than finding the minimal polynomial, which, after all, requires knowing the roots of the characteristic polynomial.

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

On linear complexity of sequences over [formula omitted

We give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$ . We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$ . For $(1/2,1]$ , we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$ , where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$ .

A new algorithm to factorize univariate polynomials over an algebraic number field has been implemented in Algol-68 on a CDC-Cyber 170-750 computer. The algebraic number field is given as the field of rational numbers adjoined by a root of a prescribed minimal polynomial. Unlike other algorithms [1,2] the efficiency of our so-called lattice algorithm does not depend on the irreducibility of the minimal polynomial modulo some prime. The factorization of the polynomial to be factored is constructed from the factorization of that polynomial over a finite field determined by a prime p and an irreducible factor of the minimal polynomial modulo p. The algorithm is based on a theorem on integral lattices and a theorem giving a lower bound for the length of a shortest-length polynomial having modulo p k a non-trivial common divisor with the minimal polynomial. These theorems also enable us to formulate a new algorithm for factoring polynomials over the integers. A technical report describing the algorithms will soon be available from the Mathematisch Centrum, Amsterdam.

We investigate which numbers are expressible as differences of two conjugate algebraic integers. Our first main result shows that a cubic, whose minimal polynomial over the field of rational numbers has the form x3 + px + q, can be written in such a way if p is divisible by 9. We also prove that every root of an integer is a difference of two conjugate algebraic integers, and, more generally, so is every algebraic integer whose minimal polynomial is of the form f (xe) with an integer e ≥ 2.