In modern cryptologic theory, the design of cryptographic protocols is often based on the assumed difficulty of number theoretic problems. Especially, in order to prove the security of a cryptographic protocol based upon the assumed difficulty of a problem, a very important role is played by the legitimacy (as concerns the security of the cryptographic protocol) of that cryptographic assumption. Recently, Kurosawa, Ogata, and Tsujii proposed a new cryptographic assumption, the Chosen Discrete Logarithm Assumption (CDLA), and showed that any language in NP has a four-move, zero-knowledge interactive proof (ZKIP) under the CDLA. In this paper, we define the modified CDLA and consider its legitimacy. Our principal result (that the modified CDLA is not a legitimate cryptographic assumption) follows from a theoretical analysis of expected polynomial-time algorithms and the concrete construction of an algorithm based on the Artin conjecture.