In the generalized Russian cards problem, Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of size a + b + c. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card she does not hold. Unlike many traditional problems in cryptography, however, they are not allowed to encode or hide the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. We propose a general four-step solution based on finite vector spaces, and call it the “colouring protocol”, as it involves colourings of lines. Our main results show that the colouring protocol may be used to solve the generalized Russian cards problem in cases where a is a power of a prime, c = O(a2) and b = O(c2). This improves substantially on the set of parameters for which solutions are known to exist; in particular, it had not been shown previously that the problem could be solved in cases where the eavesdropper has more cards than one of the communicating players.Ministerio de Economía y Competitividad FFI2011-15945-EEuropean Research Council ERC Starting Grant EPS 313360Junta de Andalucía P08-HUM-0415

In the generalized Russian cards problem, the three players Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of a + b + c cards. Players only know their own cards and what the deck of cards is. Alice and Bob are then required to communicate their hand of cards to each other by way of public messages. The communication is said to be safe if Cath does not learn the ownership of any specific card; in this paper we consider a strengthened notion of safety introduced by Swanson and Stinson which we call k-safety.An elegant solution by Atkinson views the cards as points in a finite projective plane. We propose a general solution in the spirit of Atkinson's, although based on finite vector spaces rather than projective planes, and call it the 'geometric protocol'. Given arbitrary c, k > 0, this protocol gives an informative and k-safe solution to the generalized Russian cards problem for infinitely many values of (a, b, c) with b = O(ac). This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees k-safety when Cath has more than one card.

Given an interpreted system, we investigate ways for two agents to communicate secrets by public announcements. For card deals, the problem to keep all of your cards a secret (i) can be distinguished from the problem to keep some of your cards a secret (ii). For (i): we characterize a novel class of protocols consisting of two announcements, for the case where two agents both hold n cards and the third agent a single card; the communicating agents announce the sum of their cards modulo 2n + 1. For (ii): we show that the problem to keep at least one of your cards a secret is equivalent to the problem to keep your local state (hand of cards) a secret; we provide a large class of card deals for which exchange of secrets is possible; and we give an example for which there is no protocol of less than three announcements.

Key words First order arithmetic, parameter free induction, exponentiation. MSC (2010) 03F30, 03H15We characterize the sets of all Π2 and all B(Σ1 ) (= Boolean combinations of Σ1 ) theorems of IΠ − 1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ − n + 1 is conservative over IΣ − n with respect to B(Σn + 1 ) sentences cannot be extended to Πn + 2 sentences.

In this paper we continue the study of the theories I n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that I n+1 (T) is n+2-axiomatizable. In particular, I n+1 (I n+1) gives an axiomatization of Th n 2 (I n+1) and is not finitely axiomatizable. This fact relates the fragment I n+1 (I n+1) + to induction rule for n +1-formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons' conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for n+2 and n+1 formulas (see [2]). In [7], this result is used to separate the fragments of Arithmetic introduced there: I n+1 (I n+1) and B * n+1 (I n+1). A basic result on n+1-induction rule is the following conservativeness theorem of C. Parsons (see [16] and 6.5): I n+1 is a n+2-conservative extension of