In the following picture $\kappa$ is the key length and $n$ is the block size.

I understand that $\{0,1\}^\kappa$ means all possible combination of keys and $\{0,1\}^n$ means all possible combination of a plain-text block.

But I don't understand the overall equation, specially the meaning of the arrow $\to$. Can anyone explain it in brief? (note: I am very weak in mathematics)

Let $E: \{0,1\}^\kappa \times \{0,1\}^n \to \{0,1\}^n$ be a block cipher with a $\kappa$ bit key and an $n$-bit state.


1 Answer 1


That definition is a standard definition which defines encryption as a function $E$. That function takes two inputs, a $\kappa$ bit key and a $n$ bit message. Hence it is defined over the cartesian product - denoted as $\times$ - over these two sets, i.e. all bitstring of length $\kappa$ and $n$ respectively. It maps - denoted as $\rightarrow$ - to an $n$ bit output (the ciphertext).

  • $\begingroup$ O! yes... Functions! Cartesian Product! ...totally forgot what those are... its been over a decade since I last study on those! Thanks DrLecter. But "cryptography" is getting hard right now since maths are involved. $\endgroup$
    – Giliweed
    Sep 7, 2014 at 6:33
  • $\begingroup$ I was wondering to know, who gave this standard definition and when? perhaps I can avoid going through mathematics and read the original constructor's writings of this definition, which might save some time. And hoping it would be easier to understand. Because, I believe the constructor might have explained it in his writings with simple examples ---its a guess. $\endgroup$
    – Giliweed
    Sep 7, 2014 at 7:32
  • $\begingroup$ @Giliweed That definition is very natural to formally model (block) ciphers. You will find a description in any textbook. For instance, take a look at the respective chapter of the handbook of applied cryptography (which is freely available online). $\endgroup$
    – DrLecter
    Sep 7, 2014 at 8:00

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