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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 nonadja-cent representations. Nonadjacent representations are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. In(More)
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(More)
Let w ≥ 2 be an integer and let Dw be the set of integers which 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 = P a i 2 i , with a i ∈ Dw, such that of any w consecutive a i 's at most one is nonzero. Such representations are called width-w nonadjacent forms (w-NAFs).(More)
So-called nonadjacent representations are commonly used in elliptic curve cryptography to facilitate computing a scalar multiple of a point on an elliptic curve. A nonadjacent representation having few non-zero coefficients would further speed up the computations. However, any attempt to use these techniques must also consider the impact on the security of(More)
Redundant number systems (e.g., signed binary representations) have been utilized to efficiently implement algebraic operations required by public-key cryptosystems, especially those based on elliptic curves. Several families of integer representations have been proposed that have a minimal number of nonzero digits (so-called minimal weight(More)
An online algorithm is presented that produces an optimal radix-2 representation of an input integer n using digits from the set D ,u = {a ∈ Z : ≤ a ≤ u}, where ≤ 0 and u ≥ 1. The algorithm works by scanning the digits of the binary representation of n from left-to-right (i.e., from most-significant to least-significant). The output representation is(More)
We present a simple algorithm for computing the arithmetic weight of an integer with respect to a given radix r ≥ 2. The arithmetic weight of n is the minimum number of nonzero digits in any signed radix-r representation of n. This algorithm leads to a new family of minimal weight signed radix-r representations which can be constructed using a left-to-right(More)