In the context of spoofing attacks in speaker recognition systems, we observed that the waveform probability mass function (PMF) of genuine speech differs significantly from the PMF of speech resulting from the attacks. This is true for synthesized or converted speech as well as replayed speech. We also noticed that this observation seems to have a significant impact on spoofing detection performance. In this article, we propose an algorithm, denoted genuinization, capable of reducing the waveform distribution gap between authentic speech and spoofing speech. Our genuinization algorithm is evaluated on ASVspoof 2019 challenge datasets, using the baseline system provided by the challenge organization. We first assess the influence of genuinization on spoofing performance. Using genuinization for the spoofing attacks degrades spoofing detection performance by up to a factor of 10. Next, we integrate the genuinization algorithm in the spoofing countermeasures and we observe a huge spoofing detection improvement in different cases. The results of our experiments show clearly that waveform distribution plays an important role and must be taken into account by anti-spoofing systems.