Consider the following expression:

`(a - b) mod N`

Which of the following is equivalent to the above expression?

`1) ((a mod N) + (-b mod N)) mod N 2) ((a mod N) - (b mod N)) mod N`

Also, how is (-b mod N) calculated, i.e., how is the mod of a negative number calculated?

Thanks.

**Answer**

Other answers have addressed the immediate question, so I’d like to address a philosophical one.

I think that the way you’re thinking of “mod” is a bit misleading. You seem to be thinking of “mod” as an operator: so that “13 mod 8” is another way to write the number “5”. This is the way that modulo operators often work in programming languages: in Python you can write “13 % 8” and get back the number 5.

Mathematically, though, I think it is better to think of “mod 8” as an adverb modifying “=”: when we say “5 = 13 (mod 8)” we are really saying “5 is equal to 13, *if* you think of equality as working modulo 8″. When you think of “mod” this way, it doesn’t really make sense to ask about the expression “((a mod N) + (-b mod N)) mod N”: it’s not even really an expression, under this interpretation.

I’m not trying to say that you are *wrong* for thinking of “mod” as an operation, because the operation of “taking a residue mod m” is a useful operation. However, I think it is also useful to keep the other meaning of “mod” in mind.

(After writing this answer I see that the question was posted more than a year ago. Well, maybe someone else will find this helpful.)

**Attribution***Source : Link , Question Author : J.P. , Answer Author : Gregory J. Puleo*