Basis of an ideal I'm trying to read Gentry's paper. What is the "basis of an ideal"? I know what rings and ideals are but the only "basis" I know is the "basis of a vector space". What's the connection between the two?
 A: I can't claim any kind of expertise in cryptography, but usually "basis" in the context of ideals just means a set of generators for that ideal. The most well-known occurrence of this is the name "Hilbert's Basis Theorem", which states, in modern terminology, that if $R$ is a ring such that every ideal of $R$ is finitely generated, then every ideal of the polynomial ring $R[x]$ is also finitely generated. At the time, it was commonplace to refer to any set of generators as a basis.
Note that this is different from the use of "basis" in linear algebra over a field: in that context, "basis" implies linear independence over the field of scalars, but in more general rings, there is not necessarily even any "field of scalars" to speak of.
In the paper, Gentry speaks of an algorithm "IdealGen" that takes as input a ring $R$ and a basis $\mathbf{B}_{I}$ of an ideal $I\subset R$ and outputs "public and secret bases $\mathbf{B}_{J}^{\text{pk}}$ and $\mathbf{B}_{J}^{\text{sk}}$ of some (variable) ideal $J$". The name of this algorithm seems like pretty strong evidence that Gentry is just using "basis" to mean generating set.
