In 1996, Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman introduced the NTRU cryptographic system. The latter is regarded as a post quantum cryptographic system due to its resistance to quantum attacks. In this work we review the encryption and decryption algorithms for the NTRU and also one cryptanalysis method based on the LLL (Lenstra, Lenstra, Lov´asz) algorithm for finding short vectors in a lattice. We implemented the cryptanalysis in Magma to observe in which cases the private key can be quickly recovered, i.e., in which cases the cryptographic system becomes vulnerable. Another type of attack, with the objective of recovering the message, but not the key, was also implemented.