Fachbereich Mathematik Mohamed Barakat
Wintersemester 2010/11 Simon Hampe
Cryptography
Homework assignment 10
Due date: Wednesday 19/01 at 13:45
Exercise 1. Let e=n= 21825283495649be a public key of the Rabin cryptosys- tem. Decrypt c∈Z/nZ encoded as
111000101110111110000111010101010101111000
in binary digits.
Exercise 2. Complete the proof of Theorem 6.1.8 by proving:
Letp and q be distinct odd primes and t∈N with
ord(Z/(p−1)Z,+)(t) = ord(Z/(q−1)Z,+)(t) = 2k. Then
ord(Z/(p−1)Z,+)(xt)6= ord(Z/(q−1)Z,+)(t) for half of all pairs(x, y)∈Z/(p−1)Z×Z/(q−1)Z.
Exercise 3. Prove the first statement of Remark 6.4.1 and the last statement of Remark 6.4.2:
(1) The Blum-Goldwasser cryptosystem does not satisfy the security model IND-CCA2.
(2) If the assumptions QR and SQROOT are equally strong, then the Blum- Goldwasser does not satisfy the security model ASYMMETRY-CCA2.
Exercise 4. Let(n, e1) and (n, e2) be two public RSA-keys,x < n a plaintext, and ci = E(n,e
i)(x) the corresponding ciphertexts. Show that if e1 and e2 are coprime1 then xcan be easily computed using c1, c2, and the two public keys.
1German: Teilerfremd
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