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x - Faculty of Computer Science - Technische Universität Dresden

x - Faculty of Computer Science - Technische Universität Dresden

secret area text with

secret area text with signature and test result x, s(x), t(x, s(x)) RSA as digital signature system “decryption” (s(x)) t =((h(x) s ) t � h(x) mod n collision-resistant hash function h - globally known - t, n key to test the signature, publicly known text with signature x, s(x) key generation: p, q prime number n := p�q t with gcd(t,(p -1)(q -1)) = 1 s � t -1 mod (p -1)(q -1) “encryption” s, n (h(x)) s mod n random number key to sign, kept secret text x 222

Faster calculation of the secret operation mod p, q separately: y d � w once and for all: every time: proof: d p := c -1 mod p-1 � (y d p) c � y mod p d q := c -1 mod q-1 � (y d q) c � y mod q set w := CRA ( y d p , y d q ) � w c � How much faster ? complexity exponentiation: � l 3 (y d p) c � y mod p (y d c q) � y mod q � w c � y mod n l complexity 2 exponentiations of half the length: � 2 � = 3 l 3 2 4 complexity CRA: 2 multiplications � 2 � l 2 1 addition � l So: � Factor 4 irrelevant 223

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