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x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

Calculating with and

Calculating with and without p,q (2) Z n * : multiplicative group a � Z n * � gcd (a,n) = 1 • Inverting is fast (extended Euclidean Algorithm) Determine to a,n the values u,v with a • u + n • v = 1 Then: u � a -1 mod n example: 3 -1 mod 11 ? = -11 + 4 • 3 11 = 3 • 3 +2 = 1 • 3 - 1 • (11 - 3 •3) 3 = 1 • 2 +1 1 = 1 • 3 – 1 • 2 � 3 -1 � 4 mod 11 169

Calculating with and without p,q (3) Number of elements of Z n * The Euler �- Function is defined as �(n) := �{a � {0,...,n-1} � gcd (a,n)=1}�, whereby for any integer n � 0 holds: gcd (0,n)=�n�. It immediately follows from both definitions, that �Z n * � = �(n). For n = p�q, p,q prime and p�q we can easily calculate �(n): �(n) = (p-1) • (q-1) gcd ≠ 1 have the numbers 0, then p, 2p, …, (q-1)p and q, 2q, …, (p-1)q, and these 1+(q-1)+(p-1) = p+q-1 numbers are for p≠q all different. 170

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