introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

affine equivalence of quartic homogeneous rotation symmetric boolean functions homogeneous rotation symmetric boolean functions have been extensively studied in recent years because of their applications in cryptography little is known about the basic question of when two such functions in n variables are affine equivalent the simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009 and the first substantial progress on the much more complicated cubic case came in 2010 in this paper we show that much of the work on the cubic case can be extended to the quartic case we also prove an exact formula for the number and sizes of the affine equivalence classes when n is a prime

large sets of disjoint spectra plateaued functions inequivalent to partially linear functions in this paper we give an efficient method for constructing a large set of disjoint spectra functions without linear structures which are not equivalent to partially linear functions this positively answers the open problem how to construct a large set of disjoint spectra functions which are not linearly equivalent to partially linear functions raised by zhang and xiao at the same time this significantly extends a recent result of zhang where a method of specifying four disjoint spectra functions was given it is demonstrated that such sets can be utilized in the design of highly nonlinear resilient functions in the second part based on a generalization of the indirect sum method we give an alternative approach for designing sets of disjoint spectra functions of even larger cardinality than already given ones but these functions then admit linear structures in addition it is shown that using suitable initial functions in the generalized indirect sum method we can specify highly nonlinear resilient boolean functions in odd number of input variables n whose nonlinearity in many cases exceeds the current best known values moreover we design some balanced functions for odd n that also achieve the highest nonlinearity known

on the existence of 10 2 7 488 resilient functions using a heuristic search combined with some algebraic techniques several examples for 10 variable boolean functions with nonlinearity 488 algebraic degree 7 and resiliency degree 2 were constructed this construction affirmatively answers the open problem about the existence of such functions

on syzygies of veronese embedding of arbitrary projective varieties abstract for a very ample line bundle l on a projective scheme x let l l x p h 0 x l l l 1 be the embedding defined by the complete linear series l l in this paper we study the problem how the castelnuovo mumford regularity of l x effects on the defining equations of l l x and the syzygies among them we show that if l x p h 0 x l is m regular then x l l satisfies property n 2 l m 1 for m 2 l m 2 and x l l satisfies property n l for all l m 1

on the existence of 9 3 5 240 resilient functions using a heuristic search technique several examples for 9 variable boolean functions with nonlinearity 240 algebraic degree 5 and resiliency degree 3 were constructed this construction affirmatively answers the open problem about the existence of such functions

secondary constructions of highly nonlinear boolean functions and disjoint spectra plateaued functions in this paper we modify a generalized indirect sum construction to construct functions with high nonlinearity by utilizing the modified construction highly nonlinear functions in n m variables can be obtained from known bent functions in n variables and highly nonlinear functions in m variables it is possible to obtain new n 15 variable functions with nonlinearity 2 n 1 5 1 2 n 1 5 1 2 20x2 n 2 and new 12 variable 2 resilient functions with nonlinearity 2000 and algebraic degree 8 which achieve optimal algebraic immunity moreover the modified construction can also be used as an iterative construction of a quadruple of disjoint spectra plateaued functions in addition we present sufficient conditions for a quadruple of disjoint spectra plateaued functions to have no nonzero linear structure

a spectral characterization of correlation immune combining functions it is shown that a boolean combining function f x of n variables is mth order correlation immune if and only if its walsh transform f omega vanishes for all omega with hamming weight between 1 and m inclusive this result is used to extend slightly siegenthaler s ieee trans comput vol c 34 pp 81 85 jan 1985 characterization of the algebraic normal form of correlation immune combining functions

constructions of balanced odd variable rotation symmetric boolean functions with optimal algebraic immunity and high nonlinearity abstract rotation symmetric boolean functions have been used as components of different cryptosystems in this paper two classes of balanced rotation symmetric boolean functions having optimal algebraic immunity on odd number of variables are constructed we give a lower bound on the algebraic degree of the first class of functions and prove that the n variable functions in the second class has optimal algebraic degree if n 2 m 1 for m 2 moreover it is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric boolean functions with optimal algebraic immunity and have good behavior against fast algebraic attacks at least for small numbers of input variables

brill noether theory of curves on enriques surfaces i the positive cone and gonality we study the existence of linear series on curves lying on an enriques surface and general in their complete linear system using a method that works also below the bogomolov reider range we compute in all cases the gonality of such curves we also give a new result about the positive cone of line bundles on an enriques surface and we show how this relates to the gonality

patterson wiedemann type functions on 21 variables with nonlinearity greater than bent concatenation bound nonlinearity is one of the most challenging combinatorial property in the domain of boolean function research obtaining nonlinearity greater than the bent concatenation bound for odd number of variables continues to be one of the most sought after combinatorial research problems the pioneering result in this direction has been discovered by patterson and wiedemann in 1983 ieee it which considered boolean functions on 5 times 3 15 variables that are invariant under the actions of the cyclic group gf 2 5 ast cdot gf 2 3 ast as well as the group of frobenius automorphisms some of these boolean functions possess nonlinearity greater than the bent concatenation bound the next possible option for exploring such functions is on 7 times 3 21 variables however obtaining such functions remained elusive for more than three decades even after substantial efforts as evident in the literature in this paper we exploit combinatorial arguments together with heuristic search to demonstrate such functions for the first time

on the covering radius of first order generalized reed muller codes we generalize to any finite fq fields a theorem about covering radius of codes of strength 2 proved by helleseth and coworkers then using this result and partial covering radius we give bounds for the covering radius of first order generalized reed muller codes finally using magma we get some improvements for f3

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

systematic methods of constructing bent functions and 2 rotation symmetric bent functions in this paper we first present two systematic constructions of bent functions by modifying the truth tables of rothaus s bent function and maiorana mcfarland s bent function respectively the number of the newly constructed bent functions by modifying the truth table of rothaus s bent function is also determined the methods of constructing self dual bent functions are then given after the dual functions of these bent functions being determined finally as an application two constructions of 2 rotation symmetric bent functions are presented in this paper

generalized lexicographic multiobjective combinatorial optimization application to cryptography this paper formalizes a family of prioritized multicriteria optimization problems and assesses the corresponding up to date known suboptimal solutions the resulting framework is then employed to characterize and search for boolean functions which are valuable for a robust symmetric mainly block cipher design the proposed optimality definitions generalize the lexicographic method by establishing an ordered sequence of multiobjective combinatorial optimization problems which in turn gathers the relative relevance of the criteria so that the optimal solutions can be obtained from a sequential application of the pareto efficiency the relationship among the different formulable problems is characterized in terms of both their respective solutions sets and computing costs since in practice only a limited set of functions can be evaluated i e are known the best known pareto efficient functions are also defined finally this framework is employed to obtain new functions having known pareto max

modifications of patterson wiedemann functions for cryptographic applications three basic properties of boolean functions to be useful for cryptographic purposes are balancedness high algebraic degree and high nonlinearity in addition strict avalanche criteria and propagation characteristics are required for design of s boxes we introduce methods to modify the patterson wiedemann 19983 1990 and bent functions to achieve the above cryptographic properties in the process we are able to answer some open questions about boolean functions

stochastic darboux transformations for quasi birth and death processes and urn models abstract we consider stochastic ul and lu block factorizations of the one step transition probability matrix for a discrete time quasi birth and death process namely a stochastic block tridiagonal matrix the simpler case of random walks with only nearest neighbors transitions gives a unique lu factorization and a one parameter family of factorizations in the ul case the block structure considered here yields many more possible factorizations resulting in a much enlarged class of potential applications by reversing the order of the factors also known as a darboux transformation we get new families of quasi birth and death processes where it is possible to identify the matrix valued spectral measures in terms of a geronimus ul or a christoffel lu transformation of the original one we apply our results to one example going with matrix valued jacobi polynomials arising in group representation theory we also give urn models for some particular cases

a note on the invariant distribution of a quasi birth and death process the aim of this paper is to give an explicit formula of the invariant distribution of a quasi birth and death process in terms of the block entries of the transition probability matrix using a matrix valued orthogonal polynomials approach we will show that the invariant distribution can be computed using the squared norms of the corresponding matrix valued orthogonal polynomials no matter if they are or not diagonal matrices we will give an example where the squared norms are not diagonal matrices but nevertheless we can compute its invariant distribution

affine equivalence of cubic homogeneous rotation symmetric functions homogeneous rotation symmetric boolean functions have been extensively studied in recent years because of their applications in cryptography little is known about the basic question of when two such functions are affine equivalent the simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009 this paper studies the much more complicated cubic case for such functions a new concept of patterns is introduced by means of which the structure of the smallest group g n whose action on the set of all such cubic functions in n variables gives the affine equivalence classes for these functions under permutation of the variables is determined we conjecture that the equivalence classes are the same if all nonsingular affine transformations not just permutations are allowed our method gives much more information about the equivalence classes for example in this paper we give a complete description of the equivalence classes when n is a prime or a power of 3

search for boolean functions with excellent profiles in the rotation symmetric class for the first time boolean functions on 9 variables having nonlinearity 241 are discovered that remained as an open question in literature for almost three decades such functions are found by heuristic search in the space of rotation symmetric boolean functions rsbfs this shows that there exist boolean functions on n odd variables having nonlinearity 2n 1 2n 1 2 if and only if n 7 using similar search technique balanced boolean functions on 9 10 and 11 variables are attained having autocorrelation spectra with maximum absolute value 2lceiln 2rceil on odd number of variables earlier such functions were known for 15 21 variables there was no evidence of such functions at all on even number of variables in certain cases our functions can be affinely transformed to obtain first order resiliency or first order propagation characteristics moreover 10 variable functions having first order resiliency and nonlinearity 492 are presented that had been posed as an open question at crypto 2000 the functions reported in this paper are discovered using a suitably modified steepest descent based iterative heuristic search in the rsbf class along with proper affine transformations it seems elusive to get a construction technique to match such functions

almost perfect nonlinear power functions on gf 2 sup n the welch case we summarize the state of the classification of almost perfect nonlinear apn power functions x sup d on gf 2 sup n and contribute two new cases to prove these cases we derive new permutation polynomials the first case supports a well known conjecture of welch stating that for odd n 2m 1 the power function x sup 2m 3 is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

further constructions of resilient boolean functions with very high nonlinearity one well known method of generating key stream sequences for stream ciphers is to combine the outputs of several linear feedback shift registers lfsr using a combining boolean function here we concentrate on the design of good combining boolean functions we provide resilient boolean functions with currently best known nonlinearity these functions were not known earlier and the issues related to their existence were posed as open questions in the literature some of the functions we construct here achieve the provable upper bound on nonlinearity for resilient boolean functions our technique interlinks mathematical results with classical computer search

systematic constructions of rotation symmetric bent functions 2 rotation symmetric bent functions and bent idempotent functions rotation symmetric bent functions and their generation two rotation symmetric bent functions are two classes of cryptographically significant boolean functions however few constructions have been presented in the literature which either have restriction on integer n or have algebraic degree no more than 4 in this paper for any even integer n ge 4 three classes of bent functions are presented respectively most notably the proposed n variable rotation symmetric bent functions and two rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to n 2 besides we obtain bent idempotent functions with the maximal algebraic degree n 2

constructions of almost optimal resilient boolean functions on large even number of variables in this paper a technique on constructing nonlinear resilient boolean functions is described by using several sets of disjoint spectra functions on a small number of variables an almost optimal resilient function on a large even number of variables can be constructed it is shown that given any m one can construct infinitely many re variable n even m resilient functions with nonlinearity 2n 1 2n 2 a large class of highly nonlinear resilient functions which were not known are obtained then one method to optimize the degree of the constructed functions is proposed last an improved version of the main construction is given

9 variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class we give a new lower bound to the covering radius of the first order reed muller code rm 1 n where n 9 11 13 equivalently we present the n variable boolean functions for n 9 11 13 with maximum nonlinearity found till now in 2006 9 variable boolean functions having nonlinearity 241 which is strictly greater than the bent concatenation bound of 240 have been discovered in the class of rotation symmetric boolean functions rsbfs by kavut maitra and yucel to improve this nonlinearity result we have firstly defined some subsets of the n variable boolean functions as the generalized classes of k rsbfs and k dsbfs k dihedral symmetric boolean functions where k is a positive integer dividing n secondly utilizing a steepest descent like iterative heuristic search algorithm we have found 9 variable boolean functions with nonlinearity 242 within the classes of both 3 rsbfs and 3 dsbfs thirdly motivated by the fact that rsbfs are invariant under a special permutation of the input vector we have classified all possible permutations up to the linear equivalence of boolean functions that are invariant under those permutations

equivalence of 2 rotation symmetric quartic boolean functions abstract a boolean function in n variables is 2 rotation symmetric if it is invariant under even powers of x 1 x n x 2 x n x 1 but not under the first power ordinary rotation symmetry we call such a function a 2 function a 2 function is called monomial rotation symmetric mrs if it is generated by applying powers of 2 to a single monomial if the quartic mrs 2 function in 2n variables has a monomial x1xqxrxs then we use the notation 2 1 q r s 2n for the function this paper gives a detailed theory of equivalence of quartic mrs 2 functions in 2n variables such a theory was provided for the cubic mrs 2 functions in two 2015 papers of cusick and johns as in the earlier papers the two main topics in the theory are describing the affine equivalence classes of the functions under certain groups of permutations and giving details of the linear recursions that the hamming weights of any sequence of functions 2 1 q r s 2n with q n s s 1 can be shown to satisfy the discussion for both of these topics uses new ideas because the quartic theory naturally divides into two cases

high meets low construction of strictly almost optimal resilient boolean functions via fragmentary walsh spectra this paper considers the construction of resilient boolean functions on an odd number of variables with strictly almost optimal sao nonlinearity through introducing the fragmentary walsh transform a construction technique called high meets low is proposed the detailed design procedures of a 39 variable 3 resilient boolean function with sao nonlinearity 2 38 2 19 2 16 2 14 are given it is shown that the nonlinearity of an n variable t resilient boolean function can reach 2 n 1 2 n 1 2 5 cdot 2 n 11 2 or 2 n 1 2 n 1 2 2 n 7 2 which are the largest known values for the corresponding n and t values finally by constructing a 29 variable balanced boolean function with sao nonlinearity 2 28 2 14 2 10 2 9 we show an alternative method to realize the high meets low construction technique

on the picard group of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a non singular projective surface s with h s d s h2 s cs 0 and 2ks 0 the unramitied double cover defined by ks is a k3 surface r a non singular projective surface with hi r d r 0 kr 0 illusie has shown i that the group of divisors modulo numerical equivalence is isomorphic to the enriques lattice u e 8 1 where u and e s 1 denote respectively the unique even unimodular lattices of index of inertia 1 1 and 0 8 the purpose of this note is to use this isomorphism to study the picard group of s we prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for s and r for example we prove the following results

cryptographically significant boolean functions with five valued walsh spectra we describe methods to construct balanced resp 1 resilient functions of odd number of variables n achieving the bent concatenation nonlinearity and having algebraic degree n1 resp n 2 the technique is to algebraically modify the concatenation of two properly chosen n 1 variable maiorana mcfarland bent functions the constructed functions can be used with certain recursive operators to provide higher order resilient functions with maximum possible degree and high nonlinearity such functions are well suited for stream cipher applications interestingly all the constructed functions have a five valued walsh spectr

the karlin mcgregor formula for a variant of a discrete version of walsh s spider we consider a variant of a discrete space version of walsh s spider see walsh 1978 temps locaux asterisque vol 52 53 paris soc math de france as well as evans and sowers 2003 ann probab 31 486 527 and its references this process can be seen as an instance of a quasi birth and death process a class of random walks for which the classical theory of karlin and mcgregor can be nicely adapted as in dette reuther studden and zygmunt 2006 siam j matrix anal appl 29 117 42 grunbaum 2007 probability geometry and integrable systems ed pinsky and birnir vol 55 berkeley ca msri publication pp 241 60 see also arxiv math pr 0703375 grunbaum 2007 dagstuhl seminar proc 07461 on numerical methods in structured markov chains ed bini grunbaum 2008 proceedings of iwota and grunbaum and de la iglesia 2008 siam j matrix anal appl 30 741 63 we give here a weight matrix that makes the corresponding matrix valued orthogonal polynomials orthogonal to each other we also determine the polynomials themselves and thus obtain all the ingredients to apply a matrix valued version of the karlin mcgregor formula

on some cosets of the first order reed muller code with high minimum weight we study a family of particular cosets of the first order reed muller code r 1 m those generated by special codewords the idempotents thus we obtain new maximal weight distributions of cosets of r 1 7 and 84 distinct almost maximal weight distributions of cosets of r 1 9 that is with minimum weight 240 this leads to crypotographic applications in the context of stream ciphers

on the norm and covering radius of the first order reed muller codes let spl rho 1 m and n 1 m be the covering radius and norm of the first order reed muller code r 1 m respectively it is known that spl rho 1 2k 1 spl les lower bound 2 sup 2k 2 sup 2k 1 2 and n 1 2k 1 spl les 2 lower bound 2 sup 2k 2 sup 2k 1 2 k 0 we prove that spl rho 1 2k 1 spl les 2 lower bound 2 sup 2k 1 2 sup 2k 3 2 and n 1 2k 1 spl les 4 lower bound 2 sup 2k 1 2 sup 2k 3 2 k 0 we also discuss the connections of the two new bounds with other coding theoretic problems

weight recursions for any rotation symmetric boolean functions let f n x 1 x 2 ldots x n denote the algebraic normal form polynomial form of a rotation symmetric boolean function of degree d in n geq d variables and let wt f n denote the hamming weight of this function let 1 a 2 ldots a d n denote the function f n of degree d in n variables generated by the monomial x 1 x a 2 ldots x a d such a function f n is called monomial rotation symmetric mrs it was proved in a 2012 paper that for any mrs f n with d 3 the sequence of weights w k wt f k k 3 4 ldots satisfies a homogeneous linear recursion with integer coefficients in this paper it is proved that such recursions exist for any rotation symmetric function f n such a function is generated by some sum of t monomials of various degrees a mathematica program is available on arxiv org which explicitly computes the homogeneous linear recursion for the weights given any rotation symmetric f n the reader who is only interested in finding some recursions can use the program and not be concerned with the details of the rather complicated proofs in this paper

a classification of the cosets of the reed muller code r 1 6 the weight distribution of a coset of a reed muller code m 1 m is invariant under a large transformation group consisting of all affine rearrangements of a vector space with dimension m we discuss a general algorithm that produces an ordered list of orbit representatives for this group action as a byproduct the procedure finds the order of the symmetry group of a coset with m 6 we can implement the algorithm on a computer and find that there are 150357 equivalence classes these classes produce 2082 distinct weight distributions their symmetry groups have 122 different orders

on 2 4 complete intersection threefolds that contain an enriques surface we study nodal complete intersection threefolds of type 2 4 in mathbb p 5 which contain an enriques surface in its fano embedding we completely determine calabi yau birational models of a generic such threefold these models have hodge numbers h 11 2 h 12 32 we also describe calabi yau varieties with hodge numbers h 11 h 12 equal to 2 26 23 5 and 31 1 the last two pairs of hodge numbers are to the best of our knowledge new

construction of rotation symmetric s boxes with high nonlinearity and improved dpa resistivity in this paper we provide an n times n bijective rotation symmetric s box rssb construction with improved resistance to differential power analysis dpa using rotation symmetric boolean functions rsbfs the rssb class is generated from an instance of a proposed rssb construction and then iteratively applying a simulated annealing algorithm in the respective neighborhood of the rssb followed by a hill climbing algorithm to obtain a good tradeoff of cryptographic properties the constructed 8 times 8 rssbs have a nonlinearity of 102 and transparency order value 7 709 whereas the rijndael s box has a higher transparency order of 7 86 the evaluation of security metric called guessing entropy on the constructed rssbs shows that a side channel adversary requires more effort to exploit information leakage from the simulated power traces in comparison to rijndael s box the correlation based dpa on rssbs which when incorporated in aes 128 shows requirement of significantly more power traces when implemented on xilinx virtex 5 fpga device on sasebo gii development board while the distributed memory and block memory implementations of the rijndael s box required 500 and 2 000 power traces to extract the last round key our proposed rssbs required 2 000 and 12 000 power traces respectively

results on rotation symmetric s boxes we give an efficient exhaustive search strategy to enumerate 6x6 bijective rotation symmetric s boxes rssbs having nonlinearity 24 which is found to be the maximum nonlinearity within the class of 6x6 bijective rssbs it is shown that there are 3072 rssbs achieving the cryptographic properties of the inverse function over gf 2 6 i e nonlinearity 24 differential uniformity 4 and algebraic degree 5 such that among them there are only four which are not affine equivalent among these four rssbs we find a non affine transformation under which the cryptographic properties of the inverse function are invariant then we define the generalized classes of k rssbs as the polynomials of gf 2 n with coefficients in gf 2 k where k divides n moreover motivated by the fact that rssbs are symmetric under a special permutation we classify all possible permutations up to the linear equivalence of s boxes that are symmetric under them

monomial and quadratic bent functions over the finite fields of odd characteristic considered are p ary bent functions having the form f x tr sub n spl sigma sub i 0 sup s a sub i x sup di a new class of ternary monomial regular bent function with the dillon exponent is discovered the existence of dillon bent functions in the general case is an open problem of deciding whether a certain kloosterman sum can take on the value 1 also described is the general gold like form of a bent function that covers all the previously known monomial quadratic cases the weak regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given finally some criteria for an arbitrary quadratic function to be bent are proven

computing affine equivalence classes of boolean functions by group isomorphism affine equivalence classification of boolean functions has significant applications in logic synthesis and cryptography previous studies for classification have been limited by the large set of boolean functions and the complex operations on the affine group although there are many research on affine equivalence classification for parts of boolean functions in recent years there are very few results for the entire set of boolean functions the best existing result has been achieved by harrison with 15768919 affine equivalence classes for 6 variable boolean functions this paper presents a concise formula for affine equivalence classification of the entire set of boolean functions as well as a formula for affine classification of boolean functions with distinct on set size respectively the method outlined in this paper greatly simplifies the affine group s action by constructing an isomorphism mapping from the affine group to a permutation group by this method we can compute the affine equivalence classes for up to 10 variables experiment results indicate that our scheme for calculating the affine equivalence classes for more than 6 variables is a significant advancement over previous published methods

constructions of quadratic and cubic rotation symmetric bent functions in this paper we consider constructions of rotation symmetric bent functions which are of the forms fc x i 1m 1 ci j 0n 1 xjxi j cm j 0m 1 xjxm j and ft x i 0n 1 xixt ixm i xixt i i 0m 1 xixm i where n 2m ci 0 1 the subscript u of xu in the previous expressions is taken as u modulo n for each case a necessary and sufficient condition is obtained to the best of our knowledge this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions

generalized maiorana mcfarland construction of resilient boolean functions with high nonlinearity and good algebraic properties a new framework concerning the construction of small order resilient boolean functions whose nonlinearity is strictly greater than 2 n 1 2 n 2 is given first a generalized maiorana mcfarland construction technique is described which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner it is shown that for any given m this technique can be used to construct a large class of n variable n both even and odd m resilient degree optimized boolean functions with currently best known nonlinearity this class may also provide functions with excellent algebraic properties measured through the resistance to fast algebraic attacks if the number of n 2 variable affine subfunctions used in the construction is relatively low due to a potentially low hardware implementation cost along with overall good cryptographic properties this class of functions is an attractive candidate for the use in certain stream cipher schemes

affine equivalence of quartic monomial rotation symmetric boolean functions in prime power dimension in this paper we analyze and exactly compute the number of affine equivalence classes under permutations for quartic monomial rotation symmetric functions in prime and prime power dimensions

construction of n variable n equiv 2 bmod 4 balanced boolean functions with maximum absolute value in autocorrelation spectra 2 frac n 2 in this paper we consider the maximum absolute value delta f in the autocorrelation spectrum not considering the zero point of a function f in an even number of variables n bent functions possess the highest nonlinearity with delta f 0 the long standing open question for two decades in this area is to obtain a theoretical construction of balanced functions with delta f so far there are only a few examples of such functions for n 10 14 but no general construction technique is known in this paper we mathematically construct an infinite class of balanced boolean functions on n variables having absolute indicator strictly lesser than delta n 2 n 2 2 n 6 4 nonlinearity strictly greater than rho n 2 n 1 2 n 2 2 n 2 3 5 cdot 2 n 2 4 and algebraic degree n 1 where n equiv 2 pmod 4 and n geq 46 while the bound n geq 46 is required for proving the generic result our construction starts from n 18 and we could obtain balanced functions with delta f and nonlinearity 2 n 1 2 n 2 for n 18 22 and 26

smooth varieties of almost minimal degree abstract in this article we study non linearly normal smooth projective varieties x p r of deg x codim x p r 2 we first give geometric characterizations for x theorem 1 1 indeed x is the image of an isomorphic projection of smooth varieties x p r 1 of minimal degree also if x is not the veronese surface then there exists a smooth rational normal scroll y p r which contains x as a divisor linearly equivalent to h 2 f where h is the hyperplane section of y and f is a fiber of the projection morphism y p 1 by using these characterizations 1 we determine all the possible types of y from the type of x theorem 1 2 and 2 we investigate the relation between the betti diagram of x and the type of y theorem 1 3 in particular we clarify the relation between the number of generators of the homogeneous ideal of x and the type of y as an application we construct non linearly normal examples where the converse to theorem 1 1 in d eisenbud m green k hulek s popescu restriction linear syzygies algebra and geometry compos math 141 2005 1460 1478 fails to hold remark 2