Recently, Demenkov et al [22] presented a circuit of size 4.5n using which, the binary representation of the sum of n input bits can be computed. Note that, there, the size refers to 2-input **Boolean** functions from the set {∨, ∧, ⊕}. The overall complexity for a **symmetric** **Boolean** **function** is presented as 4.5n+O(n), where the O(n) is contributed by the comparator circuit, as discussed in the previous construction. Hence, we concentrate on the Hamming weight construction part and assume the comparator part to be implemented with exactly the same complexity as in construction I.

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In this paper, we study the balancedness of **symmetric** **Boolean** functions. We prove the conjecture that all balanced **symmetric** **Boolean** functions of fixed degree are trivial when the number n of variables is sufficient large. We also present the form of trivial balanced elementary **symmetric** **Boolean** functions. In addition, we estimate the lower bound of n with which Cusick’s conjecture for elementary **symmetric** **Boolean** functions is validated. But the bound is rough. We show some properties of the balancedness of a **symmetric** **Boolean** **function** in the view of the distance from a balanced one, which can be a new and clear way to proof some former results. Although an equivalence of Cusick’s conjecture is established, it is till an open problem. It would be helpful to remove the dependence on the degree from the bound, since then the complete proof of the conjecture would be reduced to a finite computation.

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Memoryless computation is a new topic, which is arguably not well known so far. It significantly differs from the many branches of “traditional” theoretical computer science, its peculiar setup yielding new phenomena, for instance the fact that binary instructions cannot always be used to compute a given **function** (shown for **Boolean** functions in [12] and developed in Theorem 4.1 of this paper). In fact, memoryless computation can be best studied via algebraic methods, especially as it can be completely recast in terms of permutation groups (when computing bijective functions) or transformation semigroups (in the general case).

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This paper provides a major example of a phase transition. A sort of “holy grail” or highly desirable property in algebraic cryptanalysis at large, which researchers have been aiming at for decades and rarely found any, cf. [16]. Phase transitions also happen with SAT solvers [14, 15, 41]. There is double phase transition ob- served in this paper as follows. When the polynomial P is of degree 1, we get linear cryptanalysis which is not excessively powerful, cf. [33] and where the complexity is very quickly degraded as the number of rounds grows. Then when the degree is 2, we already get some properties true with probability 1 working for any number of rounds, first transition. At this stage however we get attacks working only for a handful of very weak or degenerate **Boolean** functions, if at all, and the proportion of **Boolean** functions for which the attack works is neg- ligible or even somewhat double-negligible 8 . Then if we increase the degree to say 4, we can start “playing” with multiple solutions, for example some simple attacks of degrees say 1,2 can be “removed” yet an invariant property of degree 4 will remain, see [19] and Section 10 in [23]. Then at degree 8 a second phase transition happens: due to the structure of the ring of the **Boolean** functions with numerous divisors of zero, the attack becomes almost inevitable: it works for any **Boolean** **function** with some probability (this paper).

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Hash functions that map bit-strings of arbitrary finite length into strings of fixed length. This output is commonly called a hash value, a message digest, or a fingerprint. Given h with an input x, computing h(x) must be easy. A one-way hash **function** must satisfy the following properties.

Rather like scratch-building a model sailing ship out of matchsticks, all mathematical model-building approaches start from first principles. To help get off the ground, most make some basic assumptions about the universe of values. Primitive sets, such as **Boolean**, Natural and Integer are assumed to exist (although we could go back further and construct them from first principles, in the same way as we did the Ordinal type [1]; this is quite a fascinating exercise in the λ-calculus [2]). All other kinds of concept have to be defined using rules to say how the concept is formed, and how it is used. We shall assume that there are:

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Conclusions ❏ Classical channel equalisation has been revisited ✰ Inherently odd symmetry property of optimal Bayesian equaliser has been highlighted ❏ A novel symmetric radial basis fun[r]

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Some Schur-convex functions of the complete elementary symmetric function are given here.. Some analytic inequalities are established.[r]

In this paper, we study some the relationship and mutual influence, including: the correlation immunity of the sum **function** and the product **function**; the correlation immunity of **Boolean** functions, and the correlation immunity of derivative part and e-derivative; the correlation immunity of the sum functions and product functions, and the correlation immunity of derivative part and e-derivative of product functions and so on, and gives the corresponding results.

4.4 THE EKTARE STRUCTURE The Ektare algorithm seeks to realise a **Boolean** **function** with a minimum number of multiplexers for partitions where n > q + 1. If in table 3, we use the data input variables as data select and the data select variables as the data input variables, we obtain ten new partitions. However we do not need to draw additional Ashenhurst decomposition charts as the ten charts of section 4.3 can be used to obtain the desired residue functions by reading the chart horizontally rather than vertically. Examination of the Ashenhurst charts shows that the partition AD as the data select variables and BCE as the data input variables gives the cheapest multiplexer realisation of the **function** as it requires the minimum number of additional gates to implement the residue functions. Fig. l0(a) shows the Ashenhurst decomposition chart for this partition while Fig. l0(b) shows the resulting multiplexer implementation.

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The paper investigates nonlinear equalisation using a novel **symmetric** radial basis **function** (RBF) network. By explicitly exploiting the inherently **symmetric** struc- ture of the optimal Bayesian equaliser, the proposed sym- metric RBF equaliser can be determined from the re- ceived noisy training data. Both a block-data based and a sample-by-sample adaptive algorithm are designed for this novel **symmetric** RBF equaliser. Simulation results are also provided to demonstrate the efficiency of the pro- posed **symmetric** RBF network equaliser.

Pebbling games were first used in cryptography by Dwork, Naor and Wee [9] in the context of proofs of work. They observed a strong connection between pebbling games and the random oracle model: Given a graph G and a hash **function** H, they design a **boolean** **function** whose computational complexity subject to a space constraint (in the RO model) is given by number of moves needed to pebble G subject to a contraint on the number of available pebbles. More specifically, each vertex of the graph is assigned a label of length w (the output size of the hash **function**). Input vertices are labeled using the function’s inputs, and other vertices are labeled by the hash of the labels of their predecessors. The output of the **function** is the concatenation of the labels of output vertices. This connection between pebbling and the RO model was developed further by DKW to design “one-time” pseudorandom functions [11] and leakage-resilient key evaluation schemes [10]. Assuming the availability of a random oracle, DKW showed that the computations in their model could be made to correspond to a variant of pebbling (see Section 4). We develop new techniques for analyzing such pebbling games in this work.

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The graphs of the symmetrical hyperbolic Fibonacci functions have a **symmetric** form and are similar to the graphs of the classical hyperbolic functions. Also, the symmetrical hy- perbolic Fibonacci functions sFs(x) and cFs(x) are increasing on (, +∞). The graphs of the symmetrical hyperbolic Fibonacci functions are given in []. The **symmetric** hyperbolic Fibonacci functions have properties that are similar to the classical hyperbolic functions. Some of them are []:

5.3.7 The NIL Function 5.3.8 The String Skip Indicator · 5.3.9 The QMARK Function • 5.3.10 The Bit Expression 5.3.11 The Restricted Boolean Expression 5.3.12 The Restricted Arithmetic Ex[r]

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Lattice is a very important structure in mathematics. In [14], we introduced the concept of soft sets on a complete atomic **Boolean** lattice B. We combine soft set and rough set by introducing the concept of soft rough set on B. Some shortcoming became the part of soft rough sets on B. In this paper, we introduced the concept of Modified soft rough sets(MSR) on a complete atomic **Boolean** lattice. Some important properties of MSR on B have been discussed. Similar results which require some strong conditions for their proof in soft rough sets on B can be proved in MSR sets without these conditions. Furthermore, we used the modified soft rough approximation operators to introduce the concept of modified soft rough topology and apply this concept in diabetes mellitus.

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Finite **Boolean** rings are widely used in cryptography. Many cryptographic problems such as hash-**function** inverting and computing the key of block or stream cipher under known plaintext/ciphertext take solving systems of **Boolean** equations. The set of variables contains the key bits and the bits of intermediate texts. Since a hash **function** and a ciphertext are easy to compute, the equations are sparse (each polynomial depends on relatively small number of variables comparatively to total number of variables).

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ANN application in cryptology can be categorized in two sub-fields, that is cryptanalysis and key-exchange. Neural cryptanalysis work was conducted by Ramzan [40]. The aim of the research is to introduce new cryptanalytic techniques based on principles from machine learning, particularly ANN. He used Unix Crypt cryptosystems as a test bed and the results showed that ANN can accurately predict many of the plaintext bits with high probability even though the transfer **function** chosen for the network was rather naive. This proved that ANN can be trained to do cryptanalytic attack.

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Fig. 3. The FEWS DBM module makes use of a directory and file naming convention. A catchment has a top level DBM <name> di- rectory. Inside this, the functions of each node are contained within a separate Folder<ID> directory. The Config directory con- tains the main executable script and common processing functions. A Work directory stores input, output and log files. The states directory stores each node’s internal state data between execution steps. Individual scripts are labelled with “F” for a **function**, “D” for a data file, “T” for a text file, or “M” for the main executable script.

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It is easy to see that, if U, V ∈ O n satisfy (4.7), then V > U ∈ O n λ(A) . Moreover, every matrix in O n λ(A) can be obtained in this way, that is, if W ∈ O n λ(A) and U ∈ O n satisfies (4.7), then V : = UW > also satisfies (4.7) and W = V > U . Thus, F(Diag λ(A)) is in the centralizer of O n,k λ(A) . Finding a representation theorem for k-tensor isotropic functions reduces to finding the centralizer of O n,k λ(A) . With that in mind, denote the centralizer and the **symmetric** centralizer of a collection A of n × n matrices by

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