Computing Multiplicative Order and Primitive Root in Finite Cyclic Group

Multiplicative order of an element $a$ of group $G$ is the least positive integer $n$ such that $a^n=e$, where $e$ is the identity element of $G$. If the order of an element is equal to $|G|$, it is called generator or primitive root. This paper describes the algorithms for computing multiplicative order and primitive root in $\mathbb{Z}^*_{p}$, we also present a logarithmic improvement over classical algorithms.Comment: 8 page

On Gaps Between Primitive Roots in the Hamming Metric

We consider a modification of the classical number theoretic question about the gaps between consecutive primitive roots modulo a prime $p$, which by the well-known result of Burgess are known to be at most $p^{1/4+o(1)}$. Here we measure the distance in the Hamming metric and show that if $p$ is a sufficiently large $r$-bit prime, then for any integer $n \in [1,p]$ one can obtain a primitive root modulo $p$ by changing at most $0.11002786...r$ binary digits of $n$. This is stronger than what can be deduced from the Burgess result. Experimentally, the number of necessary bit changes is very small. We also show that each Hilbert cube contained in the complement of the primitive roots modulo $p$ has dimension at most $O(p^{1/5+\epsilon})$, improving on previous results of this kind.Comment: 16 pages; to appear in Q.J. Mat

Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity

On the complexity of integer matrix multiplication

Let M(n) denote the bit complexity of multiplying n-bit integers, let ω ∈ (2, 3] be an exponent for matrix multiplication, and let lg* n be the iterated logarithm. Assuming that log d = O(n) and that M(n) / (n log n) is increasing, we prove that d × d matrices with n-bit integer entries may be multiplied in O(d^2 M(n) + d^ω n 2^O(lg* n − lg* d) M(lg d) / lg d) bit operations. In particular, if n is large compared to d, say d = O(log n), then the complexity is only O(d^2 M(n))

Algebraic techniques for deterministic networks

We here summarize some recent advances in the study of linear deterministic networks, recently proposed as approximations for wireless channels. This work started by extending the algebraic framework developed for multicasting over graphs in [1] to include operations over matrices and to admit both graphs and linear deterministic networks as special cases. Our algorithms build on this generalized framework, and provide as special cases unicast and multicast algorithms for deterministic networks, as well as network code designs using structured matrices

Deterministic Sparse Pattern Matching via the Baur-Strassen Theorem

How fast can you test whether a constellation of stars appears in the night sky? This question can be modeled as the computational problem of testing whether a set of points $P$ can be moved into (or close to) another set $Q$ under some prescribed group of transformations. Consider, as a simple representative, the following problem: Given two sets of at most $n$ integers $P,Q\subseteq[N]$, determine whether there is some shift $s$ such that $P$ shifted by $s$ is a subset of $Q$, i.e., $P+s=\{p+s:p\in P\}\subseteq Q$. This problem, to which we refer as the Constellation problem, can be solved in near-linear time $O(n\log n)$ by a Monte Carlo randomized algorithm [Cardoze, Schulman; FOCS'98] and time $O(n\log^2 N)$ by a Las Vegas randomized algorithm [Cole, Hariharan; STOC'02]. Moreover, there is a deterministic algorithm running in time $n\cdot2^{O(\sqrt{\log n\log\log N})}$ [Chan, Lewenstein; STOC'15]. An interesting question left open by these previous works is whether Constellation is in deterministic near-linear time (i.e., with only polylogarithmic overhead). We answer this question positively by giving an $n\cdot(\log N)^{O(1)}$-time deterministic algorithm for the Constellation problem. Our algorithm extends to various more complex Point Pattern Matching problems in higher dimensions, under translations and rigid motions, and possibly with mismatches, and also to a near-linear-time derandomization of the Sparse Wildcard Matching problem on strings. We find it particularly interesting how we obtain our deterministic algorithm. All previous algorithms are based on the same baseline idea, using additive hashing and the Fast Fourier Transform. In contrast, our algorithms are based on new ideas, involving a surprising blend of combinatorial and algebraic techniques. At the heart lies an innovative application of the Baur-Strassen theorem from algebraic complexity theory.Comment: Abstract shortened to fit arxiv requirement