a framework for non interactive instance dependent commitment schemes nic zero knowledge protocols are often studied through specific problems like graph isomorphism in many cases this approach prevents an important level of abstraction and leads to limited results whereas in fact the constructions apply to a wide variety of problems we propose to address this issue with a formal framework of non interactive instance dependent commitment schemes nic we define nic in the perfect statistical and computational settings and formally characterize problems admitting nic in all of these settings we also prove other useful lemmas such as closure properties consequently results that previously applied only to specific problems are now strengthened by our framework to apply to classes of problems by providing formal yet intuitive tools our framework facilitates the construction of zero knowledge protocols for a wide variety of problems in various settings without the need to refer to a specific problem our results are unconditional

cubic structures equivariant euler characteristics and lattices of modular forms we use the theory of cubic structures to give a fixed point riemann roch formula for the equivariant euler characteristics of coherent sheaves on projective at schemes over z with a tame action of a finite abelian group this formula supports a conjecture concerning the extent to which such equivariant euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus our results are applied to study the module structure of modular forms having fourier coeficients in a ring of algebraic integers as well as the action of diamond hecke operators on the mordell weil groups and tate shafarevich groups of jacobians of modular curves

nl printable sets and nondeterministic kolmogorov complexity p printable sets were defined by hartmanis and yesha and have been investigated by several researchers the analogous notion of l printable sets was defined by fortnow et al both p printability and l printability were shown to be related to notions of resource bounded kolmogorov complexity nondeterministic logspace nl printability was defined by jenner and kirsig but some basic questions regarding this notion were left open in this paper we answer a question of jenner and kirsig by providing a machine based characterization of the nl printable sets in order to relate nl printability to resource bounded kolmogorov complexity the paper introduces nondeterministic space bounded kolmogorov complexity we present some of the basic properties of this notion of kolmogorov complexity using similar techniques we investigate relationships among classes between nl and ul

the hidden subgroup problem and mktp abstract we show that the hidden subgroup problem for group families where products and inverses can be computed efficiently is in bpp mktp where mktp is the minimum kt problem using the techniques of allender et al 2018 1 we also show that the problem is in zpp mktp provided that there is a pac overestimator computable in zpp mktp for the logarithm of the order of the input group this last result implies that for permutation groups the dihedral group and many types of matrix groups the problem is in zpp mktp lastly we also show that two decision versions of the problem admit statistical zero knowledge proofs these results help classify the relative difficulty of the hidden subgroup problem

on twisted forms and relative algebraic k theory this paper introduces a new approach to the study of certain aspects of galois module theory by combining ideas arising from the study of the galois structure of torsors of finite group schemes with techniques coming from relative algebraic k theory

the master integrable two dimensional system with a quartic second integral in this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities this system is quite general and involves 21 parameters we also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters the results are applied to problems of particle and rigid body dynamics new integrable cases are obtained as special versions of the new system these cases include motions in a plane lobachevsky plane sphere and surfaces of variable curvature they also include generalizations of the classical cases of kovalevskaya chaplygin and goriatchev with the addition of certain singular terms to the potential

zero knowledge and circuit minimization we show that every problem in the complexity class szk statistical zero knowledge is efficiently reducible to the minimum circuit size problem mcsp in particular graph isomorphism lies in rp mcsp

power from random strings we show that sets consisting of strings of high kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions these sets are provably not complete under the usual many one reductions let r rm c r rm kt r rm ks r rm kt be the sets of strings x having complexity at least x 2 according to the usual kolmogorov complexity measure mbox rm c levin s time bounded kolmogorov complexity mbox rm kt l levin inform and control 61 1984 pp 15 37 a space bounded kolmogorov measure mbox rm ks and a new time bounded kolmogorov complexity measure mbox rm kt respectively our main results are as follows begin remunerate item r rm ks and r rm kt are complete for rm pspace and mbox rm exp respectively under mbox rm p poly truth table reductions similar results hold for other classes with rm pspace robust turing complete sets item mbox rm exp mbox rm np r rm kt item rm pspace mbox rm zpp r rm ks subseteq mbox rm p r rm c item the discrete log factoring and several lattice problems are solvable in mbox rm bpp r rm kt end remunerate our hardness result for rm pspace gives rise to fairly natural problems that are complete for rm pspace under mbox leq rm p rm t reductions but not under mbox leq rm log rm m reductions our techniques also allow us to show that all computably enumerable sets are reducible to r rm c via mbox rm p poly truth table reductions this provides the first efficient reduction of the halting problem to r rm c

a new integrable toda related three particle system a study of a three particle system that possesses in addition to the energy function two nonlinear constants of motion is presented herein the system is obtained by modification of the lagrangian of the toda lattice the explicit expression of the two integrals is obtained and it is proven that this new three particle lagrangian is integrable finally the possible generalization to the case of n particles is discussed

lax formalism for a family of integrable toda related n particle systems this article can be divided in two parts first a lax representation is obtained for a family of integrable three particle toda related systems the second part contains the generalization to the case of n particles

integrable three particle systems hidden symmetries and deformations of the calogero moser system the properties of some three particle systems are studied using as an approach the theory of generalized or hidden symmetries it is proven that the existence of a family that possesses in addition to the energy function two nonlinear constants of motion in involution this family of integrable systems is obtained by considering deformations of the lagrangian of the calogero moser system finally the generalization of the results to the general case of n particles is discussed

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

invariants de classes exemples de non annulation en dimension superieure the so called class invariant homomorphism measures the galois module structure of torsors under a finite flat group scheme g which lie in the image of a coboundary map associated to an isogeny between neron models of abelian varieties with kernel g when the varieties are elliptic curves with semi stable reduction and the order of g is coprime to 6 it is known that the homomorphism vanishes on torsion points in this paper using weil restrictions of elliptic curves we give the construction for any prime number p 2 of an abelian variety a of dimension p endowed with an isogeny with kernel p whose coboundary map is surjective in the case when a has rank zero and the p part of the picard group of the base is non trivial we obtain examples where does not vanish on torsion points

integrability of toda lattice by generalized variational symmetry approach a systematic method of finding nontrivial generalized variational symmetries are presented herein and the corresponding first integrals for nonlinear dynamical system with two degrees of freedom are derived then a method is applied to a toda lattice given by the potential v ex y c1e2y c2ey c3e 2x c4 e x where c1 c2 c3 and c4 are parameters the symmetries and constants of motion are presented explicitly

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

graph isomorphism is in spp we show that graph isomorphism is in the complexity class spp and hence it is in p in fact in modkp for each k 2 these inclusions for graph isomorphism were not known prior to membership in spp we derive this result as a corollary of a more general result we show that a generic problem find group has an fpspp algorithm this general result has other consequences for example it follows that the hidden subgroup problem for permutation groups studied in the context of quantum algorithms has an fpspp algorithm also some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism e g coset intersection are in spp and thus in modkp for each k 2

a note on dimensions of polynomial size circuits in this paper we use resource bounded dimension theory to investigate polynomial size circuits we show that for every i 0 p poly has ith order scaled p3 strong dimension 0 we also show that p polyi o has p3 dimension 1 2 and p3 strong dimension 1 our results improve previous measure results of lutz almost everywhere high nonuniform complexity j comput syst sci 44 2 1992 220 258 and dimension results of hitchcock and vinodchandran dimension entropy rates and compression in proc 19th ieee conf computational complexity 2004 pp 174 183 j comput syst sci to appear additionally we establish a supergale dilation theorem which extends the martingale dilation technique introduced implicitly by ambos spies terwijn and zheng resource bounded randomness and weakly complete problems theoret comput sci 172 1 2 1997 195 207 and made explicit by juedes and lutz weak completeness in e and e2 theoret comput sci 143 1 1995 149 158

solvable black box group problems are low for pp in this paper we show that several algorithmic problems for abelian black box groups and solvable black box groups are low for pp in particular we prove that membership testing group intersection order verification and group isomorphism for abelian groups are in spp and group factorization coset intersection and double coset membership for abelian groups are in lwpp similarly we show that membership testing order verification and group isomorphism for solvable groups are in zppspp we also show that group intersection group factorization coset intersection and double coset membership for solvable groups are low for pp

nonlinear equations with superposition formulas and the exceptional group g2 i complex and real forms of g2 and their maximal subalgebras in order to study nonlinear ordinary differential equations with superposition principles related to the exceptional simple lie group g2 the complex and real forms of its lie algebra are examined and their maximal subalgebras are summarized in particular the parabolic subalgebras of the noncompact real form gnc2 r are determined explicit matrix realizations of the fundamental representation d 1 0 are used and studied in connection with invariant subspaces in a seven dimensional complex or real vector space the results are collected in three tables of specific interest for the study of nonlinear differential equations which will be developed in paper ii of this series

on arithmetic class invariants let f be a number field with ring of integers of and let s be a finite set of places of f assume that s contains the set s of archimedean places of f and write sf for the set of finite places contained in s let os or o when there is no danger of confusion denote the ring of sf integers of f write f for an algebraic closure of f let y be any scheme over spec o suppose thatg is a finite flat commutative group scheme over y of exponent n and let g denote the cartier dual of g let x y be ag torsor and write 0 g y for the trivialg torsor thenox is anog comodule and so it is also anogd module see 12 as an ogd module ox is locally free of rank one and it therefore gives a line bundle m over g set l m m 1 0 then the map h y g pic g l

on counting rings of integers as galois modules let k be a number field and g a finite abelian group we study the asymptotic behaviour of the number of tamely ramified g extensions of k with ring of integers of fixed realisable class as a galois module

a note on elliptic curves and galois module structure in global function fields in this paper we study the galois module structure of certain kummer orders obtained by dividing torsion points on an elliptic curve defined over a global function field we prove that such kummer orders are globally free as galois modules this is the analogue over function fields of a conjecture first stated by m j taylor for cm elliptic curves defined over number fields

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

an integrable three particle system a study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized dynamical or hidden symmetries two lagrangians are considered both obtained by modifying the toda lagrangian first a two particle system is studied and then the results are generalized to a three particle system it is shown that in both cases the lagrangians possess nonlinear constants of motion in involution and thus they are integrable

swan modules and realisable classes for kummer extensions of prime degree abstract letl 2 be a prime number let o fdenote a ring of algebraic integers of a number fieldfand letgbe the cyclic group of orderl consider the ring of integers o las a locally free o k g module wherel kis a tamely ramified galois extension of number fields with galois group isomorphic tog the classes of rings of integers obtained in such a way for a fixed number fieldkcontaining thelth roots of unity form the subgroup of realisable classesrin the locally free class groupcl o k g on the other hand one may also look at the swan subgrouptofcl o k g formed by the classes of locally free o k g ideals s wheresin o kis relatively prime toland denotes the sum of the elements ofg we show thatt l 1 2 r d wheredis the kernel group ofcl o k g we also determine necessary and sufficient conditions for when a realisable class is a swan class last we show thatr dis nontrivial fork q l whenl 3 by providing a nontrivial lower bound for the size of the swan subgroupt

galois structure of de rham cohomology of tame covers of schemes a class on in the grothendieck group ko z g of all finitely generated projective z g modules the class group cl z g of z g is defined to be the quotient of ko z g by the subgroup generated by the class of z g let on stab be the image of on in cl z g in 34 taylor proved fr6hlich s conjecture that on stab is equal to another invariant wn k in cl z g which cassou nogues had defined by means of the root numbers of symplectic representations of the galois group g the root number of a representation v of g is the constant which appears in the functional equation of the artin lfunction of v the connection between galois structure invariants and artin l functions has been a basic theme in research on galois structure for further discussion see 12 and 4

integrals of motion for toda systems with unequal masses we present new integrals of motion for the toda lattice chain of particles in one dimension with exponential interaction for two special cases of boundary conditions the free end lattice with three non equal mass particles and the fixed end lattice for two particles in both cases we use two distinct approaches in order to identify the integrable cases direct search of the integral of motion and group theoretical methods our results are in agreement with the predictions of painleve analysis

polylog depth highness and lowness for e abstract we study the relations between the notions of highness lowness and logical depth in the setting of complexity theory we introduce a new notion of polylog depth based on time bounded kolmogorov complexity we show polylog depth satisfies all basic logical depth properties namely sets in p are not polylog deep sets with time bounded kolmogorov complexity greater than polylog are not polylog deep and only polylog deep sets can polynomially turing compute a polylog deep set we prove that if np does not have p measure zero then np contains polylog deep sets we show that every high set for e contains a polylog deep set in its polynomial turing degree and that there exist low e exp polylog deep sets keywords algorithmic information theory kolmogorov complexity bennett logical depth

adams operations and integral hermitian galois representations we give an algebraic interpretation of the action of the j th adams operator j on hermitian galois structure classes for tame covers of schemes of dimension 1 this generalizes work of erez and of erez taylor concerning rings of algebraic integers when j 2

integrability of the toda chain with free boundaries the hamilton function for the toda chain with free boundaries is studied for different interaction and mass parameters for certain systems the connection to simple lie algebras is shown and the integrability is proved by constructing the corresponding lax representations for the four particle system a painleve analysis is presented

on the polynomial depth of various sets of random strings we introduce a general framework for defining the depth of an infinite binary sequence with respect to a class of observers we show that our general framework captures all depth notions introduced in computability complexity theory so far we review most such notions show how they are particular cases of our general depth framework and review some classical results about the different depth notions we use our framework to define new notions of polynomial depth called monotone poly depth based on a polynomial version of monotone kolmogorov complexity we show that monotone poly depth satisfies all desirable properties of depth notions we give two natural examples of deep sets by showing that both the set of levin random strings and the set of kolmogorov random strings are monotone poly deep

how to construct integrable fokker planck and electromagnetic hamiltonians from ordinary integrable hamiltonians integrable hamiltonians with velocity dependent potentials including those of fokker planck type h 1 2 p2x p2y kxpx kypy are constructed from integrable hamiltonians of type h 1 2 p2x p2y v x y using certain canonical and noncanonical transformations some of the hamiltonians obtained this way are integrable only for zero energy candidates for the potential which is of interest for fokker planck models are constructed in several cases

the class group pairing and p descent on elliptic curves we give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field then we relate it to the descent relative to a suitable cyclic isogeny this allows us to connect the resulting selmer group with the logarithmic class group of the base these constructions are explicit and suitable for computer experimentation from a conceptual point of view the questions that arise here are analogues of visibility questions in the sense of cremona and mazur

on the galois structure of equivariant line bundles on curves let it be a finite field and let x be a smooth protective curve over k with structure sheaf o let g be a finite group and write cl 0 g for the reduced grothendieck group of the category of g vector bundles in this paper we describe explicitly the subgroup of cl 0 g which is generated by the classes arising from g stable invertible sheaves on tame galois covers of x which have galois group g introduction let k be a finite field of characteristic p9 and let g be a finite abelian group suppose that y x is a tamely ramified galois covering of smooth projective curves over k with galois group g we shall refer to such coverings as tame g covers of x in this paper we study the structure of g stable line bundles on such curves y in order to be more precise we let cl ox g 9 respectively cl k g 9 denote the reduced grothendieck group of x g vector bundles respectively of finitely generated fc g modules which are cohomologically trivial for g suppose that a is a g stable line bundle on y then 4 is an ox g vector bundle and it gives rise to a class f a g c ox g we shall say that an element of c1 0x g is realizable if it may be obtained in this manner for some choice of y and a we shall combine techniques of c m and bu to obtain an explicit description of the subgroup 9 x g of cl ox g which is generated by the subset of realizable classes in conjunction with a riemann roch theorem these results are sufficient to describe explicitly those elements of c k g which arise as the image of the vector bundles a under the refined euler poincare characteristic

relative galois module structure and steinitz classes of dihedral extensions of degree 8 abstract let k be a number field and ok its ring of integers let be the dihedral group of order 8 let m be a maximal ok order in k containing ok and c l m its class group we denote by r m the set of realizable classes that is the set of classes c c l m such that there exists a galois extension n k at most tamely ramified with galois group isomorphic to and the class of m ok on equal to c where on is the ring of integers of n in this article we prove that r m is a subgroup of c l m provided that k and the fourth cyclotomic field of q are linearly disjoint and the class number of k is odd to this end we will solve an embedding problem connected with steinitz classes of galois extensions in addition for any k with odd class number we show that the set of steinitz classes of tame dihedral extension of k is the full class group of k

tame realisable classes over hopf orders abstract letobe the ring of algebraic integers in a number fieldkand letgbe a finite abelian group mcculloh has characterised the classes in the locally free classgroup cl og which are realisable by rings of integers of tame normal extensions ofkwith galois groupg we extend this to certain extensions which in general need be neither tame nor normal by replacingogwith a hopf order a and introducing the strongly a tame orders as an analogue of tame rings of integers we also describe the classes realised by principal homogeneous spaces over the dual of a

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

nontrivial galois module structure of cyclotomic fields we say a game galois field extension l k with galois group g has trivial galois module structure if the rings of integers have the property that ol is a free ok g module the work of greither replogle rubin and srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame galois field extension of degree l so that l k has nontrivial galois module structure however the proof does not directly yield specific primes l for a given algebraic number field k for k any cyclotomic field we find an explicit l so that there is a tame degree l extension l k with nontrivial galois module structure

relative galois module structure of octahedral extensions abstract let k be a number field o k its ring of integers and cl k its class group let be the symmetric octahedral group s 4 let m be a maximal o k order in the semisimple algebra k containing o k cl m its locally free class group and cl m the kernel of the morphism cl m cl k induced by the augmentation m o k let n k be a galois extension with galois group isomorphic to and o n the ring of integers of n when n k is tame i e at most tamely ramified extension of scalars allows us to assign to o n the class of m o k o n denoted m o k o n in cl m we define the set r m of realizable classes to be the set of classes c cl m such that there exists a galois extension n k which is tame with galois group isomorphic to and for which m o k o n c in the present article we prove that r m is the subgroup cl m of cl m provided that the class number of k is odd

limits on the computational power of random strings how powerful is the set of random strings what can one say about a set a that is efficiently reducible to r the set of kolmogorov random strings we present the first upper bound on the class of computable sets in p r and np r the two most widely studied notions of kolmogorov complexity are the plain complexity c x and prefix complexity k x this gives rise to two common ways to define the set of random strings r r c and r k of course each different choice of universal turing machine u in the definition of c and k yields another variant r c u or r k u previous work on the power of r for any of these variants has shown bpp a a t t pr pspace p r nexp np r since these inclusions hold irrespective of low level details of how r is defined and since bpp pspace and nexp are all in d 1 0 the class of decidable languages we have e g nexp d 1 0 unp r k u our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to r k u we show bpp d 1 0 u a a t t pr k u pspace nexp d 1 0 unp r k u expspace hence in particular pspace is sandwiched between the class of sets polynomial time turing and truth table reducible to r as a side product we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions

on the galois module structure of cyclic kummer extensions abstract let p be an odd prime number let k k be a cyclic totally ramified kummer extension of degree p n with the galois group g and assume k has a kummer generator satisfying some conditions let o and o be the rings of integers in k and k respectively in case k is a p adic number field let o l be the ring of integers in the subextension of k k of degree p l over k we obtain conditions that all subrings o l of o o n 0 l n are free over associated orders a l of o l and we prove that a a n is stable under the action of automorphisms of g in case k is an algebraic number field for tamely ramified abelian extensions k k mcculloh gave the characterization of the set r o g of realizable galois module classes cl o and proved that r o e contains cl o e s c where o e is a certain quotient ring of o g and s c is the stickelberger ideal for cyclic wildly ramified kummer extensions k k we obtain the relation between two sets r o e and cl o e s c which is the analogous result to his result

a zero one law for rp and derandomization of am if np is not small we show that if rp does not have p measure zero then zpp exp as corollaries we obtain a zero one law for rp in exp and that both probabilistic classes zpp and rp have the same measure in exp we also prove that if np does not have p measure zero then np am

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

equivariant epsilon constants discriminants and etale cohomology let l k be a finite galois extension of number fields we formulate and study a conjectural equality between an element of the relative algebraic k group k 0 mathbb z mathrm gal l k mathbb r which is constructed from the equivariant artin epsilon constant of l k and a sum of structural invariants associated to l k the precise conjecture is motivated by the requirement that a special case of the equivariant refinement of the tamagawa number conjecture of bloch and kato as formulated by flach and the second named author should be compatible with the functional equation of the associated l function we show that more concretely our conjecture also suggests a completely systematic refinement of the central approach and results of classical galois module theory in particular the evidence for our conjecture that we present here already strongly refines many of the main results of galois module theory

computational complexity studies of synchronous boolean finite dynamical systems on directed graphs abstract a finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule this paper studies a subclass of finite dynamical systems the synchronous boolean finite dynamical system synchronous bfds for short where the states are boolean and the state update takes place in discrete time and at the same on all objects the paper is concerned with three problems convergence path intersection and cycle length of the synchronous bfds in which the state update functions or the local state transition functions are chosen from a predetermined finite basis of boolean functions b the paper results characterize their computational complexity