The number of decomposable univariate polynomials. extended abstract

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if n has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different components (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all collisions of two such pairs.We present a normal form for collisions of any number of decompositions with any number of components and describe exactly the (non)uniqueness of the parameters in the tame case. This yields an exact formula for the number of such collisions over a finite field. We conclude with a fast algorithm for the exact number of decomposable polynomials of degree n over a finite field Fq of characteristic coprime to n.

Lower bounds for decomposable univariate wild polynomials

Compositions and collisions at degree [formula omitted

The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).

Factorization of a class of polynomials over finite fields

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

Concretely efficient interactive oracle proofs (IOPs) are of interest due to their applications to scaling blockchains, their minimal security assumptions, and their potential future-proof resistance to quantum attacks.Scalable IOPs, in which prover time scales quasilinearly with the computation size and verifier time scales poly-logarithmically with it, have been known to exist thus far only over a set of finite fields of negligible density, namely, over “FFT-friendly” fields that contain a sub-group of size \(2^\mathsf{{k}} \).Our main result is to show that scalable IOPs can be constructed over any sufficiently large finite field, of size that is at least quadratic in the length of computation whose integrity is proved by the IOP. This result has practical applications as well, because it reduces the proving and verification complexity of cryptographic statements that are naturally stated over pre-defined finite fields which are not “FFT-friendly”. Prior state-of-the-art scalable IOPs relied heavily on arithmetization via univariate polynomials and Reed–Solomon codes over FFT-friendly fields. To prove our main result and extend scalability to all large finite fields, we generalize the prior techniques and use new algebraic geometry codes evaluated on sub-groups of elliptic curves (elliptic curve codes). We also show a new arithmetization scheme that uses the rich and well-understood group structure of elliptic curves to reduce statements of computational integrity to other statements about the proximity of functions evaluated on the elliptic curve to the new family of elliptic curve codes.

An asymptotic formula for the number of primitive polynomials of the form f (x) + a, a = 1, ..., h, where f (x)Fp[x], is obtained, on the average over all polynomials f of fixed degree, and an estimate for the number of sparse factorable polynomials is also obtained.

We use the generating function approach to derive simple expressions for the factorial moments of the distance distribution over Reed-Solomon codes. We obtain better upper bounds for the error term of a counting formula given by Li and Wan, which gives nontrivial estimates on the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This improvement leads to new results on the classification of deep holes of Reed-Solomon codes.

Fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials: computing all entries of the extended Euclidean scheme of two polynomials over an arbitrary field, gcd and 1cm of many polynomials, factoring polynomials over finite fields, and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields. For the following estimates, assume that the input polynomials have degree at most n, and the finite field has $p^d $ elements. The Euclidean algorithm is deterministic and runs in parallel time $O(\log ^2 n)$. All the other algorithms are probabilistic (Las Vegas) in the general case, but when applicable to ${\bf Q}$ or ${\bf R}$, they can be implemented deterministically over these fields. The algorithms for gcd and lcm use parallel time $O(\log ^2 n)$. The factoring algorithm runs in parallel time $O(\log ^2 n\log ^2 (d + 1)\log p)$. The algorithm for squarefree decomposition runs in parallel time $O(\log ^2 n)$ for characteristic zero, and in parallel time $O(\log ^2 n+(d - 1)\log p)$ for finite fields. All Las Vegas algorithms have failure probability less than $2^{ - n} $. For all algorithms, the number of processors is polynomial in n.

The topic of permutation polynomials over finite fields is important both in theory and in engineering application. Let p be an odd prime number and q a power of p. A necessary and sufficient condition for a family of polynomials over F q to be permutation polynomials is presented. Moreover, the number of permutation polynomials of this type is obtained. The polynomials discussed in this note are generalizations and improvement of those discussed by Xi-Wang Cao(2009).

We present a deterministic $2^{O(t)}q^{\frac{t-2}{t-1}+o(1)}$ algorithm to decide whether a univariate polynomial $f$, with $t$ monomial terms and degree $<q$, has a root in the finite field $\mathbb{F}_q$. Our method is the first with complexity sublinear in $q$ when $t$ is fixed. We also prove a structural property for the nonzero roots in $\mathbb{F}_q$ of any $t$-nomial: The nonzero roots always admit a partition into no more than $2(q-1)^{\frac{t-2}{t-1}}$ cosets, each associated with one of two subgroups $S_1\subseteq S_2$ of $\mathbb{F}^*_q$. This can be thought of as a finite field analogue of Descartes' rule. A corollary of our results is the first deterministic sublinear algorithm for detecting common degree-one factors of $k$-tuples of $t$-nomials in $\mathbb{F}_q[x]$ when $k$ and $t$ are fixed. When $t$ is not fixed we show that, for $p$ prime, detecting roots in $\mathbb{F}_p$ for $f$ is $\mathbf{NP}$-hard with respect to $\mathbf{BPP}$-reductions. Finally, we prove that if the complexity of root detection is sublinear (in a refined sense), relative to the straight-line program encoding, then $\mathbf{NEXP}\nsubseteq\mathbf{P/poly}$.

Sparse univariate polynomials with many roots over finite fields

This paper gives an algorithm for finding the irreducible factors of any multivariate polynomial with integer coefficients. The algorithm begins by making substitutions for all but one of the variable. This univariate polynomial is then factored by a known method, which uses an algorithm of Berlekamp for factoring univariate polynomials over finite fields. After this factorization is done, the multivariate factors are recovered from the univariate ones by a kind of Hensel algorithm. A number of ideas are given which greatly speed the computation in some special cases.