Explore the discrete logarithm problem on elliptic curves in this Eurocrypt 2012 conference talk. Delve into the index calculus methods and their application to elliptic curve cryptography. Learn about the transfer of the Elliptic Curve Discrete Logarithm Problem (ECDLP) via cover maps, including the GHS construction and decomposition attacks. Examine Nagao's approach for decompositions and its analysis, followed by a discussion on modified index calculus. Investigate a special case involving quadratic extensions in odd characteristics and the sieving technique. Discover the combined attack and its application to the sextic extension case. Conclude with a concrete attack demonstration on a 150-bit curve and explore scaling data for implementation.