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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 (8) Continuation squares and roots mod p, prime: Euler criterion : (i.e. fast algorithm to test whether square) Proof using little Theorem of Fermat: x p -1 � 1 mod p co-domain ok : � {±1}, because 2 � 1 x square : x p�1 2 x p p �1 x p �1 � x x nonsquare : The solutions of 2 x �1 are the 2 squares. So no nonsquare satisfies the equation. x � p�1 2 p�1 x p�1 2 � ( y � �1 Therefore: 2 . 2 ) mod p�1 2 � p ( x y p�1 p�1 p�1 ) 2 �1 175

Calculating with and without p,q (9) squares and roots mod p � 3 mod 4 • extracting roots is easy: given x � QRp mod p is root proof : 1. p � 3 mod 4 � � N 2. Euler, x � QR p In addition: w � QR p (power of x � QR p) � extracting roots iteratively is possible • -1 p w: � w 2 x p�1 4 �x p�1 2 � p �1 4 � -1 � QR p � of the roots � w: -w � QR p (otherwise –1 = (-w) • w -1 � QR p ) x p�1 �1 2 � x � p�1 2 � x �1� p-1 4r+2 2r+1 2 � (-1) 2 = (-1) = (-1) = -1 p = 4r+3 x 176

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