Related papers: On the maximum order complexity of Thue-Morse and …
Automatic sequences such as the Thue-Morse sequence and the Rudin-Shapiro sequence are highly predictable and thus not suitable in cryptography. In particular, they have small expansion complexity. However, they still have a large maximum…
Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the $N$th maximum order…
Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly…
Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some…
The $N$th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large $N$th linear complexity. However, in this paper we show that for $q$-automatic sequences over $\mathbb{F}_q$ the converse…
In 2009, Grant, Shallit, and Stoll constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2…
The Schur square of linear codes over a finite field has emerged as a fundamental operation in both classical and quantum coding theory. In this paper, we investigate the Schur square problem of Hyperderivative Reed-Solomon (HRS) codes. By…
In this paper, we study the additive complexity $\rho^{+}_{\mathbf{t}}(n)$ of a Thue-Morse like sequence $\mathbf{t}=\sigma^{\infty}(0)$ with the morphism $\sigma: 0\to 01, 1\to 12, 2\to 20$. We show that…
We present a generalisation of Twisted Reed-Solomon codes containing a new large class of MDS codes. We prove that the code class contains a large subfamily that is closed under duality. Furthermore, we study the Schur squares of the new…
We show that the M\"{o}bius function is orthogonal to the Thue-Morse sequence $t(n)$ taken along the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ for any $1 < c < 2$. Previously this property was established for the subsequence along the…
The Thue-Morse sequence is generalized to the $TM_m$ sequences and two equivalent definitions are given. This generalization leads to transcendental numbers and has Queff\'elec's theorem on Thue-Morse continued fractions as a special case.…
Proving super-polynomial size lower bounds for $\textsf{TC}^0$, the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that $\textsf{NEXP}$…
Generalized Davenport-Schinzel sequences are sequences that avoid a forbidden subsequence and have a sparsity requirement on their letters. Upper bounds on the lengths of generalized Davenport-Schinzel sequences have been applied to a…
We consider a numeration system in the ring of integers ${\mathcal O}_K$ of a number field, which we assume to be principal. We prove that the property of being a prime in ${\mathcal O}_K$ is decorrelated from two fundamental examples of…
Twisted generalized Reed-Solomon (TGRS) codes constitute an interesting family of evaluation codes, containing a large class of maximum distance separable codes non-equivalent to generalized Reed-Solomon (GRS) ones. Moreover, the Schur…
We show that if a simplicial complex is a near-cone of sufficiently high depth, then the only maximum families of small pairwise intersecting faces are those with a common intersection. Thus, near-cones of sufficiently high depth satisfy…
We show that the 2-abelian complexity of the infinite Thue-Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for…
First we generalize the Thue-Morse sequence (the generalized Thue-Morse sequences) by a cyclic permutations and p-adic system, and consider the necessary-sufficient condition that it is non-periodic. Moreover if the generalized Thue-Morse…
Motivated by the known autocorrelation properties of the Rudin-Shapiro sequence, we study the discrete correlation among infinite sequences over a finite alphabet, where we just take into account whether two symbols are identical. We show…
Type-two constructions abound in cryptography: adversaries for encryption and authentication schemes, if active, are modeled as algorithms having access to oracles, i.e. as second-order algorithms. But how about making cryptographic schemes…