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x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

to show : 1) Permutation

to show : 1) Permutation -1 � QR n To factor is difficult (2) 2) To calculate the inverse is easy using p,q 3) If there is a fast collision finding algorithm, then there is a fast algorithm to factor. x 2 � n -(2y) 2 cannot hold, since (2y) 2 � QR n. Therefore x 2 � n (2y) 2 � (x+2y)(x-2y) � n 0. x � 2y Because = 1 and = -1 it follows that n n x � n �2y Therefore gcd (x �2y,n) provides a non-trivial factor of n, i.e. p or q. 203 = one-to-one mapping with co-domain = domain

Solution of problem 1 (1) Tree of references generate (� sign) Sig r � r 0r 1 Sig r 0 r 00r 01 Sig r 00 R 00 Sig R 00 m 00 r 00 R 00 m 00 r 0 r 01 R 01 m 01 r � r 1 The attacker gets to know R i only after choosing m i. signature system1 . . . signature system 2 r j -1 Sig = F ( r j ) r j0r j1 n, ri -1 n, Sig = F ( r i ) R i Ri -1 n‘, Sig = F ( R i ) test m i reference R; rj F (Sig ) = rj ? n, rj0rj1 ri F (Sig ) = ri ? n, Ri Ri F (Sig ) = Ri ? n‘, mi 204 signature system 1 no active attack probabilistic signature system 2

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