Topic
Prime (order theory)
About: Prime (order theory) is a research topic. Over the lifetime, 15324 publications have been published within this topic receiving 157801 citations.
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TL;DR: In particular, for embedding degree k = 2q where q is prime, the authors showed that the ability to handle log(D)/log(r) ∼ (q - 3)/(q - 1) enables building elliptic curves with p ∼ q/(q- 1).
Abstract: Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree k ≤ 6. More general methods produce curves over Fp where the bit length of p is often twice as large as that of the order r of the subgroup with embedding degree k; the best published results achieve p = log(p)/log(r) ∼ 5/4. In this paper we make the first step towards surpassing these limitations by describing a method to construct elliptic curves of prime order and embedding degree k = 12. The new curves lead to very efficient implementation: non-pairing operations need no more than F p 4 arithmetic, and pairing values can be compressed to one third of their length in a way compatible with point reduction techniques. We also discuss the role of large CM discriminants D to minimize p; in particular, for embedding degree k = 2q where q is prime we show that the ability to handle log(D)/log(r) ∼ (q - 3)/(q - 1) enables building curves with p ∼ q/(q - 1).
469 citations
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TL;DR: The authors showed that masked-priming effects in the lexical decision task can be obtained for graphemically related pairs such as bontrast-constrast, but not for four-letter pairs for bamp-CAMP.
Abstract: Form-priming occurs when a prime that is graphemically similar to the target word facilitates processing of the target. In an activation model (such as Morton's logogen model), such an effect can be interpreted as a partial-activation effect. A prime that shares letters with the target must inevitably produce activation in the detectors for both the prime and the target. Alternatively, form-priming could be seen as a special case of repetition-priming, in which the prime actually accesses the entry for the target. It is shown that masked-priming effects in the lexical decision task can be obtained for graphemically related pairs such as bontrast-CONTRAST, but not for four-letter pairs such as bamp-CAMP. It is suggested that the priming effect is controlled by neighbourhood density, short words usually having many neighbours, long words having very few. This hypothesis is supported by the finding that form-priming does occur for four-letter words if the prime and target are drawn from low-density neighbour...
441 citations
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TL;DR: Lower bounds on the degree of a Chevalley group over a field of characteristic other than p have been obtained by Landazuri et al. as mentioned in this paper, who showed that for most types of groups and for most primes p it is not difficult to obtain reasonable lower bounds for the complex irreducible characters of G = G(q), using the existence of certain p-subgroups of G resembling extraspecial groups.
416 citations
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01 Dec 1985TL;DR: This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n), finding that in order to be safe from attacks using these algorithms, the value of n for which GF( 2n) is used in a cryptosystem has to be very large and carefully chosen.
Abstract: Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ? GF(q) is that integer k, 1 ? k ? q-1, for which u = gk. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2n) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2n) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2n) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.
384 citations