Locating regions in a sequence under density constraints

Several biological problems require the identification of regions in a sequence where some feature occurs within a target density range: examples including the location of GC-rich regions, identification of CpG islands, and sequence matching. Mathematically, this corresponds to searching a string of 0s and 1s for a substring whose relative proportion of 1s lies between given lower and upper bounds. We consider the algorithmic problem of locating the longest such substring, as well as other related problems (such as finding the shortest substring or a maximal set of disjoint substrings). For locating the longest such substring, we develop an algorithm that runs in O(n) time, improving upon the previous best-known O(n log n) result. For the related problems we develop O(n log log n) algorithms, again improving upon the best-known O(n log n) results. Practical testing verifies that our new algorithms enjoy significantly smaller time and memory footprints, and can process sequences that are orders of magnitude longer as a result.Comment: 17 pages, 8 figures; v2: minor revisions, additional explanations; to appear in SIAM Journal on Computin

Guessing with lies

A practical algorithm was obtained for directly generating an optimal guessing sequence for guessing under lies. An optimal guessing strategy was defined as one which minimizes the number of average number of guesses in determining the correct value of a random variable. The information-theoretic bounds on the average number of guesses for optimal strategies were also derived

Searching a bitstream in linear time for the longest substring of any given density

Given an arbitrary bitstream, we consider the problem of finding the longest substring whose ratio of ones to zeroes equals a given value. The central result of this paper is an algorithm that solves this problem in linear time. The method involves (i) reformulating the problem as a constrained walk through a sparse matrix, and then (ii) developing a data structure for this sparse matrix that allows us to perform each step of the walk in amortised constant time. We also give a linear time algorithm to find the longest substring whose ratio of ones to zeroes is bounded below by a given value. Both problems have practical relevance to cryptography and bioinformatics.Comment: 22 pages, 19 figures; v2: minor edits and enhancement

Back to Massey: Impressively fast, scalable and tight security evaluation tools

None of the existing rank estimation algorithms can scale to large cryptographic keys, such as 4096-bit (512 bytes) RSA keys. In this paper, we present the first solution to estimate the guessing entropy of arbitrarily large keys, based on mathematical bounds, resulting in the fastest and most scalable security evaluation tool to date. Our bounds can be computed within a fraction of a second, with no memory overhead, and provide a margin of only a few bits for a full 128-bit AES key

On plateaued functions, linear structures and permutation polynomials

We obtain concrete upper bounds on the algebraic immunity of a class of highly nonlinear plateaued functions without linear structures than the one was given recently in 2017, Cusick. Moreover, we extend Cusick’s class to a much bigger explicit class and we show that our class has better algebraic immunity by an explicit example. We also give a new notion of linear translator, which includes the Frobenius linear translator given in 2018, Cepak, Pasalic and Muratović-Ribić as a special case. We find some applications of our new notion of linear translator to the construction of permutation polynomials. Furthermore, we give explicit classes of permutation polynomials over Fqn using some properties of Fq and some conditions of 2011, Akbary, Ghioca and Wang