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I understand all of the previous steps, but I don't understand what is happening to "S2" in that the number "775" showed up.

Where does the 775 come from? I can't seem to get it, and I've tried modding it, but received the wrong answer.

The answer to this is to add the modulo to the a when a is negative. As such, this worked out fine after I added the modulo to the negative number.

Solved!

El Gamal Problem Signature Problem

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    $\begingroup$ $(300 - 61\cdot 425) \equiv 775\ \pmod{880}$ $\endgroup$ – Maeher Dec 10 '18 at 8:01
  • $\begingroup$ To see it, just look at both side of $(300−61⋅425)* 553 ≡775 * 533 $. $\endgroup$ – kelalaka Dec 10 '18 at 8:02
  • $\begingroup$ Thanks, but that's not the answer I get on my calculator. Should I solve within the parentheses first and follow order of operations, then mod by 880? $\endgroup$ – J. Doe Hue Dec 10 '18 at 8:04
  • $\begingroup$ I see how you got the answer now. However, I get a negative, and my calculator doesn't process negative numbers for the MOD operation. How else can I solve this problem using the negative in the parenthesis? $\endgroup$ – J. Doe Hue Dec 10 '18 at 8:09
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    $\begingroup$ When you get negative add the modulus. $\endgroup$ – kelalaka Dec 10 '18 at 8:19
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The question basically is how $300 - 61 \cdot 425 \bmod 880$ can turn into the value $775$, while a calculator returns a different value, likely $-105$. The difference is that sometimes a remainder is taken to have a value in the range $(-N, N)$ rather than $[0, N)$.

$300 - 61 \cdot 425 = -25625$. So you can say that we're trying to find a value $x \cdot 880 + y = -25625$. Now is can be $-29 \cdot 880 - 105 = -25625$ to find a value within $(-N, N)$, similar to finding $29 \cdot 880 + 105 = 25625$. $-105$ is then commonly called "the remainder"; it's what you would get if you would perform simple tail division yourself.

However, for cryptographic operations we generally try and perform operations in a group of order $N$ instead, with $N$ possible values in the range $[0, N)$. so in that case we can take $-30 \cdot 880 + 775 = -25625$, and the result will be $775$, exactly $880$ more than $-105$ in other words. This is commonly called taking the modulus, although this value can also be called a remainder.

So to find the right value within your calculator you can simply add $N = 880$ if the result is negative. A little trick in programming languages is to perform ((X % N) + N) % N, so that an if statement can be avoided. Note though that the + N may cause an overflow for large values of $N$.

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