Algorithmic Cryptography

Walter Unger WS 2017/2018

Janosch Fuchs November 02, 2017

Exercise

Algorithmic Cryptography

Sheet 3

• Write the name, group number and enrollment number of each group member on every sheet that you hand in.

• To achieve the permission for the exam you must earn50% of the sum of all points and present one of your solutions at least once.

• You can earn 50% bonus points by presenting your solution. At the beginning of every exercise session, you can mark the exercises that you want to present.

• If a student is not able to present a correct solution although he/she marked the exercise as presentable, he/she will lose all of his/her points on the exercise sheet.

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 09, 2017, 10:15 a.m.,

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