Algorithmic Cryptography

Walter Unger WS 2013/2014

Sascha Geulen November 7, 2013

Exercise

Algorithmic Cryptography

Sheet 3

Exercise 3.1 (4 points)

(a) Letn be a positive integer anda∈Zn with gcd(n+ 1, a) =a.

Prove: a⁻¹ ≡ ⁿ⁺¹_a modn.

(b) Compute the multiplicative inverse of 153 mod 7802.

Exercise 3.2 (4 points)

Let a, bbe odd positive integers withb > a.

Prove: gcd (a, b) = gcd a,^b−a₂ .

Exercise 3.3 (4 points)

Let c = E_e,n^RSA(m) be the ciphertext belonging to the plaintext m if an RSA system is used. Assume that the public-key e ≤ 10. Furthermore, assume there is an oracle that gives for the unknown plaintext m and input r >0the value c_r =E_e,n^RSA(m+r).

Prove: The plaintext can be decrypted efficiently.

Exercise 3.4 (4 points)

Let n = pq be an RSA modulus, m be a plaintext, and r be the order of m, i.e., m^r≡1 mod n. Furthermore, let r be even and m^r/2 6≡ −1 modn.

Prove: gcd(m^r/2 −1, n)∈ {p, q}.

Deadline: Thursday, November 14, 2013, 10:15 a.m.,

in the lecture or in the letterbox in front of i1.

Walter Unger WS 2013/2014

Sascha Geulen November 7, 2013

Exercise

Algorithmic Cryptography

Sheet 3

Please fill in your name and your student number and mark the exercises that you can present.

Then staple this page in front of your solution sheet.

Presentation of your exercise solution

I want to present the exercises marked in the following table in the tutorial. Each mark gives two points if you are present in the tutorial and your solution is good (at least two points).

Name Student Number E 3.1 E 3.2 E 3.3 E 3.4