Discrete logs and Fourier Coefficients

We present a probabilistic polynomial time reduction from the dis- crete logarithm problem in the multiplicative group IFq , where q de- notes a prime number, to the problem of calculating the Fourier coef- cien ts of a Hecke eigenform of level q. This is done using the recursion formula for the coecien ts of an eigenform with associated Dirichlet character , using the character to change the discrete log problem from IFq to the group q 1 of complex q 1-th roots of unity. It is worth mentioning that Bas Edixhoven has outlined an algo- rithm (Edix) that would calculate the p-th Fourier coecien t of a given modular form in polynomial time. Another result related with the reduction presented in this note is in Dennis Charles' thesis (Char), where he proves that being able to compute the values of Ramanujan's -function is not more dicult that being able to factor RSA moduli, a dierence between his ap- proach and ours is that he considers the problem of calculating the n-th Fourier coecien t for arbitrary n, whereas we restrict ourselves to computing Fourier coecien ts for a prime number and the square of a prime number. Charles' result supports a claim by Edixhoven, saying that in order to compute the n-th Fourier coecien t of an eigenform, one must be able to factor n.