We describe a new variant of the well known Baby-Step Giant-Step algorithm in the case of some discrete logarithms with a special structure. More precisely, we focus on discrete logarithms equal to products in groups of unknown order. As an example of application, we show that this new algorithm enables to cryptanalyse a variant of the GPS scheme proposed by Girault and Lefranc at CHES 2004 conference in which the private key is equal to the product of two sub-private keys of low Hamming weight. We also describe a second attack based on a known variant of the Baby-Step Giant-Step algorithm using the low Hamming weight of the sub-private keys.