Fachbereich Mathematik Mohamed Barakat
Wintersemester 2010/11 Simon Hampe
Cryptography
Homework assignment 9
Due date: Wednesday 12/01 at 13:45
Exercise 1. Define Jn:=Qn∪˙ Qen={a∈(Z/nZ)∗ | a
n
= 1}. Prove:
(1) Let n be a square-free odd number. Then Jn is a subgroup of (Z/nZ)∗ of index 2.
(2) If n is a Blum numbers, then [Jn:Qn] = 2 and Jn =Qn∪ −˙ Qn.
Exercise 2. Determine all Blum numbers ≤ 100. Determine for the smallest four Blum numbers:
(1) the cardinality of Qn,Qen, andQn, (2) all elements of Qn,
(3) all orbits of the Rabinfunction (on Qn)
(4) all possible output sequences of the Blum-Blum-Shub generator.
Exercise 3. Using the algorithm introduced in the proof of Theorem 5.2.18:
(1) Compute the square root of 5 modulo 19.
(2) Compute the square root of 2 modulo 17.
(3) It is known that
2 p
= (−1)p
2
−1
8 . Use this information to determine a class of primes for which the probabilistic algorithm (introduced in the proof of Theorem 5.2.18) can be turned into a deterministic one.
Now we consider square root modulon, wheren is a composite number.
(4) Let n= 713. Show how to use1852 ≡1 modn to factor n. (You can guess that n is a Blum number.)
Exercise 4. Let n = 77 be the public key of the Blum-Goldwasser cryptosys- tem1. Decipher the ciphertext 1000111111.
1Elements in(Z/nZ)∗ are encoded by their smallest positive representative as binary numbers (with the lowest bit to the left).
1
