Let w ≥ 2 be an integer and let D w be the set of integers that includes zero and the odd integers with absolute value less than 2 w−1. Every integer n can be represented as a finite sum of the form n = a i 2 i , with a i ∈ D w , such that of any w consecutive a i 's at most one is nonzero. Such representations are called width-w nonadjacent forms (w-NAFs). When w = 2 these representations use the digits {0, ±1} and coincide with the well-known nonadjacent forms. Width-w nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the w-NAF. We show that w-NAFs have a minimal number of nonzero digits and we also give a new characterization of the w-NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on w-NAFs and show that any base 2 representation of an integer, with digits in D w , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

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Abstract.Chloro ( The coordination in MONO is nearly linear, with a P-Au-CI angle of 177.0 (2) ° and Au-P and Au-C1 bond distances of 2.242(4) and 2.279 (5)A, respectively. In the BIS complex the C1 is ionic. The cation [(Cy3P)2Au] + is at a center of symmetry, so the two phosphines are staggered with exactly linear P--Au-P coordination and the P-Au distance is 2.321 (2) A.

Abstract. It is known that every positive integer n can be represented as a finite sum of the form n = P ai2 i , where ai ∈ {0, 1, −1} for all i, and no two consecutive ai's are non-zero. Such sums are called nonadjacent representations. Nonadjacent representations are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. In this paper, we investigate if other digit sets of the form {0, 1, x}, where x is an integer, provide each positive integer with a nonadjacent representation. If a digit set has this property we call it a nonadjacent digit set (NADS). We present an algorithm to determine if {0, 1, x} is a NADS; and if it is, we present an algorithm to efficiently determine the nonadjacent representation of any positive integer. We also present some necessary and sufficient conditions for {0, 1, x} to be a NADS. These conditions are used to exhibit infinite families of integers x such that {0, 1, x} is a NADS, as well as infinite families of x such that {0, 1, x} is not a NADS.

Internet geolocation technology aims to determine the physical (geographic) location of Internet users and devices. It is currently proposed or in use for a wide variety of purposes, including targeted marketing, restricting digital content sales to authorized jurisdictions, and security applications such as reducing credit card fraud. This raises questions about the veracity of claims of accurate and reliable geolocation. We provide a survey of Internet geolocation technologies with an emphasis on adversarial contexts; that is, we consider how this technology performs against a knowledgeable adversary whose goal is to evade geolocation. We do so by examining first the limitations of existing techniques, and then, from this base, determining how best to evade existing geolocation techniques. We also consider two further geolocation techniques which may be of use even against adversarial targets: (1) the extraction of client IP addresses using functionality introduced in the 1.5 Java API, and (2) the collection of round-trip times using HTTP refreshes. These techniques illustrate that the seemingly straightforward technique of evading geolocation by relaying traffic through a proxy server (or network of proxy servers) is not as straightforward as many end-users might expect. We give a demonstration of this for users of the popular Tor anonymizing network.