2

So I am a computer science student and have some problems reading cryptographic mathematics notations, so I need help from mathematicians. I was reading a 2018 IEEE journal about identity-based data integrity checking, and was met with this:

...Assume the user's identity is l-bit, which is described as

$ID = (ID_1, ID_2, ID_3, ..., ID_l) \in \{0,1\}^l$

...

I searched about sets with powers, sets of elements with powers, but garnered no useful results. I remembered about this in automata course, which more or less translates that to this:

On a set of l amount of ID, each ID is constructed with l digits of 0s and 1s.

But I'm still skeptical about that, am I wrong?

Thank you!

rareguy
  • 33

1 Answers1

0

It is the number of copies of the set. For example, $$ A^3= A \times A \times A= \{(a_1,a_2,a_3) \colon a_1,a_2,a_3 \in A\} $$ The notation was chosen to resemble ordinary powers: $x \cdot x \cdot x= x^3$. As another example, $$ \{0,1\}^3= \{(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)\} $$ The 'exponent' is the number of copies of the set that you have. You can even used cardinal powers for infinite (or 'more than infinite') copies.

mathematics2x2life
  • 13,322
  • So ID_x is either 0 or 1, and ID is a set of ID_x which comes from one of the elements of {0,1}^l, right? – rareguy Feb 04 '20 at 03:15
  • @rareguy I do not know what ID is in this context, but given the context, I would assume that the elements of ID$^l$ are of the form $(a_1,\ldots,a_l)$, where each $a_i$ is either 0 or 1. – mathematics2x2life Feb 04 '20 at 05:12
  • 1
    @mathematics2x2life Hi, I'm also interested in this question. Regarding your answer, is the following interpretation correct.

    1. If $l=2$ we have $ID=(ID_1,ID_2)\in {0,1}^2$. And we can write ${0,1}^2={(0,0), (0,1), (1,0), (1,1)}$, so we end up with $$ ID=(ID_1,ID_2)\in {(0,0), (0,1), (1,0), (1,1)} \quad ? $$

    2. Does this now mean $ID_1$ and $ID_2$ can be $0$ OR $1$? So we end up with the following possibilities: \begin{align} (ID_1,ID_2)&=(0,0), , \ (ID_1,ID_2)&=(0,1)\ (ID_1,ID_2)&=(1,0)\ (ID_1,ID_2)&=(1,1) \end{align}

    Thanks!

     – JDoeDoe Feb 05 '20 at 20:15
  • 1
    @JDoeDoe Yes it is – mathematics2x2life Feb 05 '20 at 21:59
  • @mathematics2x2life Thanks! I asked because I didn't follow your first comment (from Feb 4). With "the elements of ID$^l$ are of the form $(a_1,\dots,a_l)$", Does the notation $(a_1,\dots,a_l)$ corresponds to $(ID_1, \dots, ID_l)$ in the original question? So in my case $a_1=ID_1$ and $a_2=ID_2$? And also, why do you use the superscript $l$ in ID$^l$? – JDoeDoe Feb 08 '20 at 20:09
  • @JDoeDoe Probably because you read element versus set, which have in this case have, unfortunately, the same name, i.e ID$^l$ the set of $l$-tuples and ID$_1$, the first component of some $l$-tuple. For the latter part, I wanted to try to avoid this double name and also wanted to avoid using two symbols, ID, when I could just as easily use $a_i$ and then denote what it was later. – mathematics2x2life Feb 08 '20 at 20:57