Topic
Exponentiation
About: Exponentiation is a research topic. Over the lifetime, 1483 publications have been published within this topic receiving 28075 citations. The topic is also known as: to the power.
Papers published on a yearly basis
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Papers
TL;DR: A powerdomain construction is developed, which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains, and a restricted class of algebraic inductive partial orders is found which is closed under this construction.
Abstract: We develop a powerdomain construction, $\mathcal{P}[ \cdot ]$, which is analogous to the powerset construction and also fits in with the usual sum, product and exponentiation constructions on domains. The desire for such a construction arises when considering programming languages with nondeterministic features or parallel features treated in a nondeterministic way. We hope to achieve a natural, fully abstract semantics in which such equivalences as $(p\textit{ par } p) = (q\textit{ par }p)$ hold. The domain ($D \to $ Truthvalues) is not the right one, and instead we take the (finitely) generable subsets of D. When D is discrete they are ordered in an elementwise fashion. In the general case they are given the coarsest ordering consistent, in an appropriate sense, with the ordering given in the discrete case. We then find a restricted class of algebraic inductive partial orders which is closed under $\mathcal{P}[ \cdot ]$ as well as the sum, product and exponentiation constructions. This class permits the...
731 citations
TL;DR: This paper surveys the known methods for fast exponentiation, examining their relative strengths and weaknesses.
624 citations
Journal Article•
TL;DR: A new coordinate system and a new mixed coordinates strategy are proposed, which significantly improves on the number of basic operations needed for elliptic curve exponentiation.
Abstract: Elliptic curve cryptosystems, proposed by Koblitz ([12]) and Miller ([16]), can be constructed over a smaller field of definition than the ElGamal cryptosystems ([6]) or the RSA cryptosystems ([20]) This is why elliptic curve cryptosystems have begun to attract notice In this paper, we investigate efficient elliptic curve exponentiation We propose a new coordinate system and a new mixed coordinates strategy, which significantly improves on the number of basic operations needed for elliptic curve exponentiation
487 citations
4 Mar 2006
TL;DR: In this paper, it was shown that if a set of players hold shares of a value $a \in \mathbb{F}_p $ for some prime p (where the set of shares is written [a]p), it is possible to compute, in constant rounds and with unconditional security, sharings of the bits of a, i.e., compute sharings [a0]p,..., [al−−1]p such that l = ⌈ log2p ⌉, a0,..., al−1∈
Abstract: We show that if a set of players hold shares of a value $a \in \mathbb{F}_p $ for some prime p (where the set of shares is written [a]p), it is possible to compute, in constant rounds and with unconditional security, sharings of the bits of a, i.e., compute sharings [a0]p, ..., [al−−1]p such that l = ⌈ log2p ⌉, a0,...,al−1∈{0,1} and a = ∑i=0l−−1ai 2i. Our protocol is secure against active adversaries and works for any linear secret sharing scheme with a multiplication protocol. The complexity of our protocol is $\mathcal{O}(l {\rm log} l)$ invocations of the multiplication protocol for the underlying secret sharing scheme, carried out in $\mathcal{O}(1)$ rounds.
This result immediately implies solutions to other long-standing open problems such as constant-rounds and unconditionally secure protocols for deciding whether a shared number is zero, comparing shared numbers, raising a shared number to a shared exponent and reducing a shared number modulo a shared modulus.
448 citations
12 Aug 1999
TL;DR: Three new types of power analysis attacks against smartcard implementations of modular exponentiation algorithms are described, each of which requires an adversary to exponentiate many random messages with a known and a secret exponent.
Abstract: Three new types of power analysis attacks against smartcard implementations of modular exponentiation algorithms are described. The first attack requires an adversary to exponentiate many random messages with a known and a secret exponent. The second attack assumes that the adversary can make the smartcard exponentiate using exponents of his own choosing. The last attack assumes the adversary knows the modulus and the exponentiation algorithm being used in the hardware. Experiments show that these attacks are successful. Potential countermeasures are suggested.
447 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 40 |
| 2024 | 32 |
| 2023 | 49 |
| 2022 | 165 |
| 2021 | 41 |
| 2020 | 48 |