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Modular Numbers and Cryptography

Addition and Subtraction

Properties of addition in modular arithmetic:

  1. If \(a + b = c\), then \(a \pmod{N} + b \pmod{N} \equiv c \pmod{N}\).
  2. If \(a \equiv b \pmod{N}\), then \(a + k \equiv b + k \pmod{N}\) for any integer \(k\).
  3. If \(a \equiv b \pmod{N}\), and \(c \equiv d \pmod{N}\), then \(a + c \equiv b + d \pmod{N}\).
  4. If \(a \equiv b \pmod{N}\), then \(-a \equiv -b \pmod{N}\).

Examples

1. Find the residue of \( (9+7) \pmod{5} \)

One method is to find each residue and add them together.

\begin{align} 9 &\pmod{5} + 7 \pmod{5} \\ \equiv 4 &\pmod{5} + 2 \pmod{5} \\ \equiv 6 &\pmod{5} \\ \equiv 1 &\pmod{5} \end{align}

However, based on the first property above, we can perform the operation quicker if we add first.

\begin{align} &(9+7) \pmod{5} \\ \equiv &16 \pmod{5} \\  \equiv &1 \pmod{5} \end{align}

2. Find the residue of \( (81 +38 +72 +64) \pmod{11} \)

\begin{align} &\equiv  255 \pmod{11} \\ &\equiv 2 \pmod{11} \end{align}

3. Perform the operation \((72 - 18) \pmod{13} \)

The same method can be applied to subtraction because it is simply an addition of a negative integer.

\begin{align} &\equiv  54\pmod{13} \\ &\equiv 2 \pmod{13} \end{align}

4. Perform the operation \((23 - 77 + 32) \pmod{9} \)

\begin{align} &\equiv  -22\pmod{9} \\ &\equiv 5 \pmod{9} \end{align}

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