What does the $\|$ operation mean in cryptographic notation? I am studying elliptic curves problems, which also includes study of related protocols such as ECIES. The problem is that I don't understand the notation $\|$. What does this operation mean? 

Some stuff is mentioned on Wikipedia's Integration Encryption Scheme article, but this algorithm is better explained in the book Guide to Elliptic Curves. This operation is mentioned in Alice's 3rd step:

$$ k_E \| k_M = \operatorname{KDF}(S \| S_1) $$
In terms of algorithms, it usually means concatenate - as in, join two binary (or otherwise defined) strings together (in this order). For wikipedia's cryptography articles that's nearly always the case.

When it is on the left side of an assignment, this means split the string from the right side into these component strings. This makes only sense if the length of most components is already fixed, of course – like here for the encryption and MAC keys.

In terms of programming languages such as C, however, it means logical or. Whilst it is unlikely the authors have confused them, it is something to be aware of.



Different notations may be used for string concatenation, including the single pipe sign $s_1 | s_2$ and the plus sign $s_1 + s_2$, although the double pipe sign as in the question seems prevalent.