Editing Discrete logarithm (section)

== Properties ==
Powers obey the usual algebraic identity <math>b^{k+l}=b^k\cdot b^l</math>.<ref name=":0" /> In other words, the [[function (mathematics)|function]]
:<math>f \colon \mathbf{Z} \to G</math>
defined by <math>f(k)=b^k</math> is a [[group homomorphism]] from the group of integers <math>\mathbf{Z}</math> under addition [[Surjection|onto]] the [[subgroup]] <math>H</math> of <math>G</math> [[generating set of a group|generated]] by <math>b</math>. For all <math>a</math> in <math>H</math>, <math>\log_b a</math> exists. [[Converse (logic)|Converse]]ly, <math>\log_b a</math> does not exist for <math>a</math> that are not in <math>H</math>.

If <math>H</math> is [[infinite group|infinite]], then <math>\log_b a</math> is also unique, and the discrete logarithm amounts to a [[group isomorphism]]

:<math>\log_b \colon H \to \mathbf{Z}.</math>

On the other hand, if <math>H</math> is [[finite group|finite]] of [[order of a group|order]] <math>n</math>, then <math>\log_b a</math> is 0 unique only up to [[modular arithmetic|congruence modulo]] <math>n</math>, and the discrete logarithm amounts to a group isomorphism
:<math>\log_b\colon H \to \mathbf{Z}_n,</math>
where <math>\mathbf{Z}_n</math> denotes the additive group of integers modulo <math>n</math>.

The familiar base change formula for ordinary logarithms remains valid: If <math>c</math> is another generator of <math>H</math>, then

:<math>\log_c a = \log_c b \cdot \log_b a.</math>