I'm reading an article which defines something called a hash function.

Let $n\in\mathbb{N}$ and let $H:\{0,1\}^*\to\{0,1\}^n:m\to h=H(m)$...

I know that $\{0,1\}^n$ is the cartesian product of the set with itself $n$ times, but I am not familiar with the meaning of the asterisk in this context for $\{0,1\}^*$

Is anyone familiar with the notation?

  • 9
    $\begingroup$ I think this is Kleene star. $\endgroup$
    – Wojowu
    Commented Feb 26, 2016 at 20:13
  • 1
    $\begingroup$ It should be emphasized that the answer given here (the Kleene star) applies to this particular case, as easily inferred from the context. There are dozens, if not hundreds, of different uses of a superscript asterisk in mathematics (and whatever text you're reading is not doing a very good job if the notation has not been clearly introduced and occasionally recalled). $\endgroup$
    – Gro-Tsen
    Commented Feb 26, 2016 at 22:29

1 Answer 1


In your case $$\{0,1\}^*=\bigcup_{n=0}^\infty\{0,1\}^n$$

In context of hash-functions, $\{0,1\}^*$ would be the set of strings with "letters" 0,1; including the empty string but no single infinite string.


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