The Sidon cryptosystem [22] is a new multivariate encryption scheme based on the theory of Sidon spaces which was presented at PKC 2021. As is usual for this kind of schemes, its security relies on the hardness of solving particular instances of the MQ problem and of the MinRank problem. A nice feature of the scheme is that it enjoys a homomorphic property due the bilinearity of its public polynomials. Unfortunately, we will show that the Sidon cryptosystem can be broken by a polynomial time key-recovery attack. This attack relies on the existence of solutions to the underlying MinRank instance which lie in a subfield and which are inherent to the structure of the secret Sidon space. We prove that such solutions can be found in polynomial time. Our attack consists in recovering an equivalent key for the cryptosystem by exploiting these particular solutions, and this task can be performed very efficiently.