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More Efficient Cryptosystems From k-th Power Residues
At Eurocrypt 2013, Joye and Libert proposed a method for constructing public key cryptosystems (PKCs) and lossy trapdoor functions (LTDFs) from (2)-power residue symbols. Their work can be viewed as non-trivial extensions of the well-known PKC scheme due to Goldwasser and Micali, and the LTDF scheme due to Freeman et al., respectively. In this paper, we will demonstrate that this kind of work c...
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Let p be a prime number and let k ≥ 2 be an integer such that k divides p − 1. Norton proved that the least k-th power non-residue modp is at most 3.9p log p unless k = 2 and p ≡ 3 (mod 4), in which case the bound is 4.7p log p. With a combinatorial idea and a little help from computing power, we improve the upper bounds to 0.9p log p and 1.1p log p, respectively.
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In [2] it was shown that A(&, 4) = co for fe^ 1048909 and it was conjectured that A(&, 4) = =° for all k. In this paper we establish this conjecture with the following Theorem. A(&, 4) = ». Proof. It suffices to prove the theorem for values of k which are prime. The proof makes use of the following proposition which is a special case of a result of Kummer [l] (see also [3]). Proposition. Let k ...
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Abstract. Let Rk(n) be the number of representations of an integer n as the sum of a prime and a k-th power for k ≥ 2. Furthermore, set Ek(X) = |{n ≤ X, n ∈ Ik, n not a sum of a prime and a k-th power}|. In the present paper we use sieve techniques to obtain a strong upper bound on Rk(n) for n ≤ X with no exceptions, and we improve upon the results of A. Zaccagnini to prove Ek(X) ≪k X 1−181 log...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1973
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-23-1-89-106