Discrete-Logarithm Case Tuple Exponentiation

The basic functions with a bundling property in the discrete-logarithm case are simply products of several exponentiations. This is called tuple exponentiation or, if the number of components is known to be jti,/x-tuple exponentiation. Pair exponentiation seems to have been first used like this in [B0CT88], larger tuples in [ChHP92]. 

Definition 8.1. Let H be an Abelian group of order q (not yet necessarily prime) and IN. 

The exponents only need to be defined modulo q because of Fermat s little theorem hence the same notation can be used for tuples x = (xj. 2.  

A more usual name is vector exponentiation however, there is not always a vector space, and it is useful to have ju in the name. 

Some simple mathematical properties of tuple exponentiation follow. 

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In abstract constructions, the domains Gg are written additively and the codomains Hg multiplicatively. This notation corresponds to the discrete-logarithm case, where hg is tuple exponentiation. Note that homomorphisms automatically have a bundling property if the domain is sufficiently larger than the codomain, as in Lemma 8.17. 

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