Iterative conditional branchings appear in various sensitive algorithms, like the modular exponentiation in the RSA cryptosystem or the scalar multiplication in elliptic-curve cryptography. The Montgomery ladder is a common example of such algorithm with desirable security properties against some side-channel and fault-injection attacks. In this paper, we abstract away and generalize these security properties by formalizing, using systems of equations, what we call semi-interleaved and fully-interleaved ladders. This fruitful approach allows us to design novel fault-injection attacks, able to obtain some/all bits of the secret against different ladders, breaking the common Montgomery ladder. We also demonstrate the generality of our approach by applying our ladder equations to the modular exponentiation and the scalar multiplication, both in the semi- and fully-interleaved cases, to propose novel and more secure algorithms.