Generic-Group Delay Functions Require Hidden-Order Groups

Despite the fundamental importance of delay functions, underlying both the classic notion of a time-lock puzzle and the more recent notion of a verifiable delay function, the only known delay function that offers both sufficient structure for realizing these two notions and a realistic level of practicality is the ``iterated squaring\u27\u27 construction of Rivest, Shamir and Wagner. This construction, however, is based on rather strong assumptions in groups of hidden orders, such as the RSA group (which requires a trusted setup) or the class group of an imaginary quadratic number field (which is still somewhat insufficiently explored from the cryptographic perspective). For more than two decades, the challenge of constructing delay functions in groups of known orders, admitting a variety of well-studied instantiations, has eluded the cryptography community. In this work we prove that there are no constructions of generic-group delay functions in cyclic groups of known orders: We show that for any delay function that does not exploit any particular property of the representation of the underlying group, there exists an attacker that completely breaks the function\u27s sequentiality when given the group\u27s order. As any time-lock puzzle and verifiable delay function give rise to a delay function, our result holds for these two notions we well, and explains the lack of success in resolving the above-mentioned long-standing challenge. Moreover, our result holds even if the underlying group is equipped with a $d$-linear map, for any constant $d \geq 2$ (and even for super-constant values of $d$ under certain conditions)

A Computationally Efficient Method for Large-Scale Concurrent Mapping and Localization

Decoupled stochastic mapping (DSM) is a computationally efficient approach to large-scale concurrent mapping and localization. DSM reduces the computational burden of conventional stochastic mapping by dividing the environment into multiple overlapping submap regions, each with its own stochastic map. Two new approximation techniques are utilized for transferring vehicle state information from one submap to another, yielding a constant-time algorithm whose memory requirements scale linearly with the size of the operating area. The performance of two different variations of the algorithm is demonstrated through simulations of environments with 110 and 1200 features. Experimental results are presented for an environment with 93 features using sonar data obtained in a 3 by 9 by 1 meter testing tank

Second order balance property on Christoffel words

International audienceIn this paper we study the balance matrix that gives the order of balance of any binary word. In addition, we define for Christoffel words a new matrix called second order balance matrix. This matrix gives more information on the balance property of a word that codes the number of occurrences of the letter 1 in successive blocks of the same length for the studied Christoffel word. By taking the maximum of the Second order balance matrix we define the second order of balance and we are able to order the Christoffel words according to these values. Our construction uses extensively the continued fraction associated with the slope of each Christoffel word, and we prove a recursive formula based on fine properties of the Stern-Brocot tree to construct second order matrices. Finally, we show that an infinite path on the Stern-Brocot tree, which minimizes the second order of balance is given by a path associated with the Fibonacci word