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I often encounter the term “security parameter” when I read crypto related stuff. My basic understanding is that it just denotes some bit-length however, I'm not so sure.

For example, when it says

… output strings of length 2K where K is the security parameter

how should I understand that ?

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Quoting the obvious (Wikipedia article about the term “security parameter”.)

In cryptography, the security parameter is a variable that measures the input size of the computational problem. Both the resource requirements of the cryptographic algorithm or protocol as well as the adversary's probability of breaking security are expressed in terms of the security parameter.

The security parameter is usually expressed in unary representation (for example, a security parameter of $n$ is expressed as a string of $n$ 1s) so that the time complexity of the cryptographic algorithm is polynomial in the size of the input. For example, in the RSA cryptosystem, the security parameter $k$ denotes the length in bits of the modulus $n$; the positive integer $n$ must therefore be a number in the set ${0, …, 2^k - 1}$.

Also note that the parties and adversary are assumed to run in time that is polynomial in the security parameter, and that security must hold except with probability that is negligible in the security parameter.

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