Analysis of Generalized Grover's Quantum Search Algorithms Using Recursion Equations

The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [PRA 60, 2742 (1999)] is generalized. It is applied to the large class of Grover's type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution is expressed using first order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, P_max. Both T and P_max are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.Comment: 8 pages, no figures. To appear in Phys. Rev.

Ideal hierarchical secret sharing schemes

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.Peer ReviewedPostprint (author's final draft

Secret Sharing Schemes on Access Structures With Intersection Number Equal To One

The characterization of ideal access structures and the search for bounds on the optimal information rate are two important problems in secret sharing

The contribution of maternal HIV seroconversion during late pregnancy and breastfeeding to mother-to-child transmission of HIV

The prevention of mother-to-child transmission (PMTCT) of HIV has been focused mainly on women who are HIV positive at their first antenatal visit, but there is uncertainty regarding the contribution to overall transmission from mothers who seroconvert after their first antenatal visit and before weaning. A mathematical model was developed to simulate changes in mother-to-child transmission of HIV over time, in South Africa. The model allows for changes in infant feeding practices as infants age, temporal changes in the provision of antiretroviral prophylaxis and counseling on infant feeding, as well as temporal changes in maternal HIV prevalence and incidence. The proportion of mother-to-child transmission (MTCT) from mothers who seroconverted after their first antenatal visit was 26% [95% confidence interval (CI): 22% to 30%] in 2008, or 15,000 of 57,000 infections. It is estimated that by 2014, total MTCT will reduce to 39,000 per annum, and transmission from mothers seroconverting after their first antenatal visit will reduce to 13,000 per annum, accounting for 34% (95% CI: 29% to 39%) of MTCT. If maternal HIV incidence during late pregnancy and breastfeeding were reduced by 50% after 2010, and HIV screening were repeated in late pregnancy and at 6-week immunization visits after 2010, the average annual number of MTCT cases over the 2010-2015 period would reduce by 28% (95% CI: 25% to 31%), from 39,000 to 28,000 per annum. Maternal seroconversion during late pregnancy and breastfeeding contributes significantly to the pediatric HIV burden and needs greater attention in the planning of prevention of MTCT programs.

On-line secret sharing

In a perfect secret sharing scheme the dealer distributes shares to participants so that qualified subsets can recover the secret, while unqualified subsets have no information on the secret. In an on-line secret sharing scheme the dealer assigns shares in the order the participants show up, knowing only those qualified subsets whose all members she has seen. We often assume that the overall access structure (the set of minimal qualified subsets) is known and only the order of the participants is unknown. On-line secret sharing is a useful primitive when the set of participants grows in time, and redistributing the secret when a new participant shows up is too expensive. In this paper we start the investigation of unconditionally secure on-line secret sharing schemes. The complexity of a secret sharing scheme is the size of the largest share a single participant can receive over the size of the secret. The infimum of this amount in the on-line or off-line setting is the on-line or off-line complexity of the access structure, respectively. For paths on at most five vertices and cycles on at most six vertices the on-line and offline complexities are equal, while for other paths and cycles these values differ. We show that the gap between these values can be arbitrarily large even for graph based access structures. We present a general on-line secret sharing scheme that we call first-fit. Its complexity is the maximal degree of the access structure. We show, however, that this on-line scheme is never optimal: the on-line complexity is always strictly less than the maximal degree. On the other hand, we give examples where the first-fit scheme is almost optimal, namely, the on-line complexity can be arbitrarily close to the maximal degree. The performance ratio is the ratio of the on-line and off-line complexities of the same access structure. We show that for graphs the performance ratio is smaller than the number of vertices, and for an infinite family of graphs the performance ratio is at least constant times the square root of the number of vertices. © 2011 Springer Science+Business Media, LLC