Editing Discrete logarithm (section)

== Definition ==
Let <math>G</math> be any group. Denote its [[group operation]] by multiplication and its [[identity element]] by <math>1</math>. Let <math>b</math> be any element of <math>G</math>. For any positive integer <math>k</math>, the expression <math>b^k</math> denotes the product of <math>b</math> with itself <math>k</math> times:<ref name=":0">{{Cite book |author-last1=Lam |url=https://link.springer.com/book/10.1007/978-3-0348-8295-8 |title=Cryptography and Computational Number Theory |author-last2=Shparlinski |author-last3=Wang |author-last4=Xing |editor-first1=Kwok-Yan |editor-first2=Igor |editor-first3=Huaxiong |editor-first4=Chaoping |editor-last1=Lam |editor-last2=Shparlinski |editor-last3=Wang |editor-last4=Xing |publisher=[[Birkhäuser Basel]] |isbn=978-3-7643-6510-3 |edition=1 |series=Progress in Computer Science and Applied Logic |date=2001 |pages=54–56 |language=en |doi=10.1007/978-3-0348-8295-8 |eissn=2297-0584 |issn=2297-0576}}</ref>
:<math>b^k = \underbrace{b \cdot b \cdot \ldots \cdot b}_{k \; \text{factors}}.</math>
Similarly, let <math>b^{-k}</math> denote the product of <math>b^{-1}</math> with itself <math>k</math> times. For <math>k=0</math>, the <math>k</math><sup>th</sup> power is the identity: <math>b^0=1</math>.

Let <math>a</math> also be an element of <math>G</math>. An integer <math>k</math> that solves the equation <math>b^k=a</math> is termed a '''discrete logarithm''' (or simply '''logarithm''', in this context) of <math>a</math> to the base <math>b</math>. One writes <math>k=\log_b a</math>.