# How do Indistinguishability(IND) and Non-Malleability(NM) differ?

Reading through Katz & Lindell's Intro to Modern Crypto, I came across these notions which were introduced in quick succession without much differentiation between them.

How do Indistinguishability(IND) and Non-Malleability(NM) differ? Which is the stronger notion of the two, in which (if any) scenario are they equivalent?

For this answer I'll use the paper linked in the comments: "Relations Among Notions of Security for Public-Key Encryption Schemes" by Bellare, Desai, Pointcheval and Rogaway (PDF) which also contains this very nice overview graph on page 4 of the PDF: How do Indistinguishability(IND) and Non-Malleability(NM) differ?

Generally in the IND setup you try to infer some information (at least 1 bit) about a given plaintext using your given capabilities (encryption and decryption oracles at the various stages).

In the NM setting on the other hand you are given a ciphertext $y$ of a plaintext $x$ and now want to come up with some "meaningful" relation $R$ such that given $y$ you can find a vector $\bf y$ which when component-wise decrypted yields a vector of plaintexts $\bf x$ such that $R(x, {\bf x})$ holds.

In case you are asking yourself how the distinction between CCA1 and CCA2 is implemented for NM, in CCA1 you only get access to a decryption oracle before seeing $y$.

Which is the stronger notion of the two [...]?

As can be seen in the above graph, for a given attacker capability X, NM-X generally implies IND-X but the converse doesn't always hold.

[In] which (if any) scenario are they equivalent?

As can be seen in the above graph, NM-X and IND-Y are equivalent only if X=Y is the CCA2 attacker model.