Editing Index calculus algorithm (section)

==The Index Calculus family==
Index Calculus inspired a large family of algorithms. In finite fields <math>\mathbb{F}_{q} </math> with <math>q=p^n</math> for some prime <math>p</math>, the state-of-art algorithms are 
the Number Field Sieve for Discrete Logarithms, <math display="inline"> L_{q}\left[1/3,\sqrt[3]{64/9}\,\right]</math>, when <math> p </math> is large compared to <math>q</math>,<ref name="barbulescu2013thesis">{{cite thesis
|type=PhD
|last=Barbulescu
|first=Razvan
|date=2013
|title=Algorithms for discrete logarithm in finite fields
|publisher=University of Lorraine
|url=https://hal.univ-lorraine.fr/tel-01750438
}}</ref> the [[function field sieve]], <math display="inline">L_q\left[1/3,\sqrt[3]{32/9}\,\right]</math>,<ref name="barbulescu2013thesis"/> and Joux,<ref name="joux2013indexcalcverysmallchar">{{cite conference
|last=Joux
|first=Antoine
|author-link=Antoine Joux
|title=A new index calculus algorithm with complexity <math>L(1/4 + o(1))</math> in very small characteristic
|url=https://link.springer.com/chapter/10.1007/978-3-662-43414-7_18
|editor-last1=Lange
|editor-first1=Tanja
|editor-link1=Tanja Lange
|editor-last2=Lauter
|editor-first2=Kristin
|editor-link2=Kristin Lauter
|editor-last3=Lisoněk
|editor-first3=Petr
|conference=Selected Areas in Cryptography&mdash;SAC 2013
|date=August 2013
|conference-url=https://link.springer.com/book/10.1007/978-3-662-43414-7
|volume=8282
|series=Lecture Notes in Computer Science
|publisher=Springer
|location=Burnaby, BC, Canada
|isbn=978-3-662-43414-7
|pages=355&ndash;379
|doi=10.1007/978-3-662-43414-7_18
|doi-access=free
|url-access=subscription
}}</ref> <math>L_{q}\left[1/4+\varepsilon,c\right] </math> for <math>c>0</math>, when <math>p</math> is small compared to <math>q </math> and the Number Field Sieve in High Degree, <math>L_q[1/3,c]</math> for <math>c>0</math> when <math>p </math> is middle-sided. Discrete logarithm in some families of elliptic curves can be solved in time <math>L_q\left[1/3,c\right]</math> for <math> c>0</math>, but the general case remains exponential.