Technical history of discrete logarithms in small characteristic finite fields

Due to its use in cryptographic protocols such as the Diffie--Hellman

key exchange, the discrete logarithm problem attracted a considerable

amount of attention in the past 40 years. In this talk, we summarize

the key technical ideas and their evolution for the case of discrete

logarithms in small characteristic finite fields. This road leads from

the original belief that this problem was hard enough for

cryptographic purpose to the current state of the art where the

algorithms are so efficient and practical that the problem can no

longer be considered for cryptographic use.