Algorithmic Cryptography

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Walter Unger WS 2017/2018

Janosch Fuchs December 21, 2017

Exercise

Algorithmic Cryptography

Sheet 9

• 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 9.1 (4 points)

Noticing the successes of Knud Knudson’s ideas for election protocols, he was invited to give the keynote talk at the respected SUPER (Symposium on Unknown Protocols for Election and Randomness) conference in Turitg, Svizra. One of the talks at the conference aroused Knud Knudson’s interest, an election protocol for yes/no votes where each voter can give a positive or negative vote or he can abstain. After the election only the result (yes or no) is revealed, but the number of positive and negative votes and abstentions stays secret. Unfortunately, Knud Knudson cannot remember the details of the protocol and, since he is not a computer scientist, he is not able to implement it. On the other hand, he has heard on the conference that a group of students from Aoke has studied a similar protocol in a lecture about cryptography. Can you design the protocol for Knud Knudson?

After the conference, Knud Knudson prepares his equipment for the next expedition to the North Pole. His colleagues gave Knud Knudson for an analysis of the Ocean current a prime number p, n generators g₁, . . . , g_n of Z^∗p, and n numbers y₁, . . . , y_n ∈ Z^∗p with yi =g_i^xⁱ modp for somexi not known by Knud Knudson. Knud Knudson knows that if there is a pair(y_i, y_j), i6=j for that the samexwas used, then the numbers will conflict with each other and cannot be used for the analysis.

Give an algorithm, that computes all pairs (y_i, y_j), i 6= j such that y_i = g^x_i modp and yj =g_j^x modpfor the same x.

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Exercise 9.3 (4 points) After Knud Knudson’s arrival at the North Pole, the Inuit told him that there is the Treasure of the Unknown Harbor Seal near the Geographic North Pole. In order to find it, Knud Knudson has to discover the coordinates of the Stones of the Average, each given by a point (x_i, y_i), i ∈ {1, . . . , n}. The treasure is located at the average point of them, i.e., at ¹_n·Pn

i=1x_i,_n¹ ·Pn i=1y_i

.

Unfortunately, Knud Knudson’s mental calculation skills do not allow him to calculate the average point using only his brain. So he uses his new Peach yPhone, which has a connection to a server that can compute complicated calculations. As a result of a misapprehension, the vendor sent Knud Knudson a smartphone that can save only one point at the same time. So Knud Knudson must use the server to do the computation.

But he does not trust the server operator. So he does not want that the server can see any point or the result. Hence, the computation has to be done using encrypted data.

Can you design a protocol that guarantees Knud Knudson’s privacy requirements? The Inuit gave Knud Knudson a machine that can compute the discrete logarithm efficiently, i.e., given a prime number p, a generator g ofZ^∗p, and ay∈Z^∗p, it can compute a x∈Z^∗p

with y≡g^x modp.

Construct an election protocol for the following problem:

LetK1,K2, andK3 be three candidates. The candidate with the most votes wins (absolute majority). The result of the vote must not be revealed, but it must be possible for the winner that he can convince the voters of being elected. Do not use a trusted center.

Merry Christmas and a Happy New Year!

Deadline:Thursday, January 11, 2018, 10:15 a.m.,

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