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x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

squares and roots

squares and roots Calculating with and without p,q (6) QR n := { x � Z n * | � y � Zn * : y 2 � x mod n } x : “quadratic residue” y : “root of x” -y is also a root (-1) 2 = 1 but attention: e.g. mod 8 1 2 � 1 3 2 � 1 4 QR n multiplicative group: 7 2 � 1 5 2 � 1 roots x 1, x 2 � QR n � x 1 • x 2 � QR n : (y 1y 2) 2 = y 1 2 y2 2 = x1x 2 x 1 -1 � QRn : (y 1 -1 ) 2 = (y1 2 ) -1 = x1 -1 173

Calculating with and without p,q (7) squares and roots mod p, prime: Z p field � as usual � 2 roots x � 0, p � 2 : 0 or 2 roots p �1 2 � |QR p| = (square function is 2 � 1) x 0 1 2 . . . p �1 �1 � . . . –2 –1 = p - 1 2 2 p x 2 0 1 4 . . . . . . 4 1 Jacobi symbol x 1 if x � QR p (for x � Z p * ) := p -1 else 174

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