Generalized Middle-Square Method

Interior components of a tile associated to a quadratic canonical number system

Signed-digit number system is an unconventional number system which has been found to cause greater efficacy and efficiency in performing some arithmetic operations. Obviously, it seems interesting to investigate signed-digit number system in wider context, including designing magnitude comparator. Although magnitude comparator is an integral part of every standard arithmetic processor, no significant work on magnitude comparison has been reported yet even for the more propitious categories of signed-digit number system, including canonical signed-digit number system. In this paper, an algorithm for magnitude comparison of canonical signed-digit number system is proposed in terms of comparing the magnitudes and signs of some counterpart portions of the inputs. The proposed algorithm can be immediately implemented at hardware level employing some ordinary logic gates only and in every execution stage of the proposed algorithm, all operations may be completed in a fairly small number of logic levels.KeywordsComputer ArithmeticCanonical Signed-Digit NumberMagnitude ComparisonRecurrence RelationPropagation Delay

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

Integral domains with canonical number systems

The topological structure of fractal tilings generated by quadratic number systems

Letbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base”x. In particular, we define and characterize digit representations for the ringS:=((x−1,y−1))/f((x−1,y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.

The sum of digits of squares in [formula omitted

The sum-of-digits function of canonical number systems: Distribution in residue classes

We investigate the existence and length of canonical expansion of integers in polynomial bases where the bases are taken from some infinite families of polynomials which we call roofline polynomials. A roofline polynomial's coefficients form a sequence that is first increasing, then constant, then decreasing, with an additional symmetry condition. We prove necessary conditions for the CNS property, and in special cases we also prove that the condition is sufficient. We provide results on the length of the expansion of -1 as a function of the degree of the base. We also formulate some open problems about the canonical number system property of these families.

Parallel implementations of Brunotte’s algorithm

Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution

Multimedia data security has become a noticeable problem in the continuous development of multimedia technology. Traditional chaotic systems usually require larger bit-width iterative calculations to affect the generation rate of chaotic systems when generating. To solve this problem, we proposed a general Residual Chaotic System (RCS) for image security in heterogeneous multimedia signal processing environments. Based on the Residue Number System (RNS) commonly used in VLSI, we generate the general chaotic sequences by iterating unary polynomial function through it in Galois Field (GF). Moreover, we make use of the Box-Muller algorithm to generate RCS, which improved the speed of operation by a large margin when doing R/B backward transformation. Experiments show that the key sensitivity and plaintext sensitivity of the proposed encryption system are good, and the system can encrypt pictures efficiently and safely. Then, we proposed a watermark embedding and extraction algorithm with block selection. Based on the proposed new judgment conditions, the $4\times 4$ brightness blocks are divided into five categories, then the appropriate types of blocks are selected accordingly. After testing the performance indexes of cypher-text images, including statistical and sensitivity-related tests, all results indicate that our algorithm has good randomness and security. The invisibility is good, the watermark capacity has been significantly improved, and the video bit-rate growth rate is in a good range. The watermark can resist attacks such as noise and the bit error rate under re-encoding and Gaussian low-pass filtering attacks. Compared with other algorithms, the algorithm that we proposed is faster, much more efficient, and more secure. Therefore, it is a practical alternative scheme for heterogeneous communication environments.