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I have come across the following sentence in my cryptography class regarding asymmetric cryptography:

Number $x$ is any number from $\mathbb{Z}_{30}$.

What does the symbol $\mathbb{Z}_{30}$ means?

Xander Henderson
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Martin
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    Are you sure that your book doesn't explain it anywhere? That your lecturer hasn't talked about it? – Arthur Nov 27 '19 at 17:33
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    Well, the simple "lies we tell to children" answer is one of the integers from $0,......, 29$ and do modular arithmetic on integers where if you go above thirty or below 0 you just circularly round back. (So for isntance $21+14 \equiv 5$ because $21+14=35\le 30$ so we round back down to zeor. And $6*5 \equiv 0$ and $7-11 \equiv 26$ [because $-4 \le 0$ so we round back up to $30$ and go down $4$.) But it's really more subtle than that (but just as easy). – fleablood Nov 27 '19 at 17:37
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    Where, precisely, did you see this comment? Is it lecture notes, or in a text? If it is in your lecture notes, you might want to ask your instructor or look at earlier notes. If it is from a text, please provide a more complete citation---which text? what page? etc. – Xander Henderson Nov 27 '19 at 19:16
  • See Wiki/Modular arithmetic/Integers modulo n – Bill Dubuque Nov 27 '19 at 20:05
  • Arthur and Xander Henderson, it's fair enough to assume the concept may have been introduced earlier. But it could be that the text assumes the student is familiar with the concept and notation. In any event, it is surely the set of integers modulo $30$ and we can, in one way or another, explain it. – fleablood Nov 27 '19 at 20:34

2 Answers2

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Probably "the integers modulo 30".

Eric Towers
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  • So basically all integers between 1 and the number specified? – Martin Nov 27 '19 at 17:39
  • It is more common to think of this as the range $[0,29]$. "In reality" these are sets, $0$ in $\Bbb{Z}_{30}$ for instance represents the set of integers ${\dots, -60, -30, 0, 30, 60, \dots}$ and the other number represent similar sets. When thinking about these as sets, perhaps the first surprise is that there is coherent arithmetic for them. – Eric Towers Nov 27 '19 at 17:41
  • Thanks. So just to verify, if I have to choose number from $\mathbb{Z}_{n}$, any number from 0 to n-1 will technically be correct? – Martin Nov 27 '19 at 17:50
  • " if I have to choose number from Zn, any number from 0 to n-1 will technically be correct? " More or less. The idea though is you can do addition subtraction, multiplication (but NOT division) on them simply be treating numbers and their remainders when divided by $n$ as the same thing. That is in $\mathbb Z_{17}$ then $3, 20, 37, -14,-31$ are all considered to be the "same" thing. – fleablood Nov 27 '19 at 17:55
  • @Martin Beware the "in reality" claims above. There are numerous constructions and neither is more "real". A strong argument can be made against revealing the internal structure of the elements. In fact one of the primary goals of the definition of an algebraic structure is to abstract away from such inessential internal structiure of the elements - a point which is explained at length in the prior linked answer. – Bill Dubuque Nov 27 '19 at 20:38
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$\mathbb Z_{30} = \{0,1,2,3,4,5,........,27,28,29\}$ [1]

With modular arithmetic. That is $a, b\in \mathbb Z_{30}$ then

$a+b = \begin{cases} a+b & \text{if }a+b <30\\a+b-30 &\text{if} a+ b \ge 30\end{cases}$

$a\cdot b = \begin{cases} a\cdot b & \text{if }a\cdot b <30\\a+b-30*k &\text{if } k*30 \le a\cdot b - k*30 < 30(k+1)\end{cases}$

$a-b = \begin{cases} a-b & \text{if }a-b \ge 0\\a-b+30 &\text{if} a- b < 0\end{cases}$

Division is not a basic operation.

This is often called "remainder" arithmetic because you are doing arithmetic because it is exactly normal arithmetic but where you are only concerned with the remainders of things after you divide by $30$.

That is: "$57$ is really $27$ when we take the remainder; and $62$ is really $2$ when we take the remainder; so $57*62$ is really $27*2 = 54$ and $54$ is really $24$ when we take the remainder so $57*62\equiv 54 \pmod{30}$".

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[1] This is actually ..... well, one way of thinking about it.... In abstract mathematics this explanation is .. an oversimplification. The following is a more common interpretations. It's a bit more subtle and theoretically quite different. But they are equivalent and in practice, tend to be the same:

$\mathbb Z_{30}$ is a way of splitting all the numbers into thirty sets. The thirty sets are $[0], [1],.......,[29]$ where each set $[k] = \{$ all integers that have $k$ as a remainder when you divide by $30\}$.

So $21$ is not just the number, twenty-one, it is actually the set of all numbers that have $21$ as a remainder... $\{....-39,-9,21,51,81,.....\}$.

And you can add and multiply these sets together.

$$\{......,-39,-9,21,51,81,.....\} + \{......,-46,-16,14, 44,74...\} = \{....,-55, -25,5,35,65,.....\}$$. Or in other words: $21 + 14 = 35= 30+5=5$.

And $$21\times 14 = \{......,-39,-9,21,51,81,.....\} \times \{......,-46,-16,14, 44,74...\}= \{...., -66,-36,-6,24,54,.....,264,294,324,......\}$$. Or in other words $21\times 14 = 294 = 9*30 + 24 = 24$.

But it's just easier to think of it as "remainder" arithmetic. And in practice it is exactly the same thing.

........

fleablood
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  • It may be true in the OP's construction of $,\Bbb Z_{30},,$ esp. in more applied contexts such as the OP's (crypotgraphy). – Bill Dubuque Nov 27 '19 at 20:00
  • What may be true? Oh, "This is actually not true" Point taken. I was partially tongue in cheek. – fleablood Nov 27 '19 at 20:15
  • See my comment on Eric's answer for elaboration. – Bill Dubuque Nov 27 '19 at 20:18
  • No, you are correct. Phrases like "in reality" and "it is actually" are to be taken with a grain of salt. I'm not sure the best way to teach a novice is.... I definitely learned modular arithmetic as "remainder" arithmetic and I still think of it that way..... but that was a huge concept block when I tried to learn them as equivalence class. In fact the entire idea of "equivalence classes" is one of the great abstraction barriers in math education in my mind. Which is a shame because by nature they should be ... evident. – fleablood Nov 27 '19 at 20:28
  • That's one (of many) reasons why we should avoid injecting any prejudice into the formative minds of our students. When you write that one view is the "actual" or "real" one then you risk doing so. – Bill Dubuque Nov 27 '19 at 20:34
  • True.............. On the other hand when you tell them there are two ways of thinking of concepts, they tend to ask "which one is the real way" and .... well, we still have to get the idea of abstract concepts being equivalent. – fleablood Nov 27 '19 at 20:36