# VE475 Introduction to Cryptography Assignment 10 solved

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Ex. 1 — Group structure on an elliptic curve
Prove proposition 8.7.
Ex. 2 — Number of points on an elliptic curve
Let E be the elliptic curve defined by y
2 = x
3 + 3x + 7 over F11 and P the point (8, 9) on E.
1. Compute [2]P, [5]P, and [10]P.
2. How many points are on E.
3. List all the points from E.
Ex. 3 — ECDSA
In chapter 6 we studied the Digital Signature Algorithm. Search and explain how it can be transposed to
elliptic curves. What are the benefits?
Ex. 4 — BB84
Describe and explain how the BB84 quantum key distribution protocol works.
Ex. 5 — Quantum key distribution
Alice and Bob are given two communication channels: a classical and a quantum one. The quantum
channel is isolated from any environment interaction, i.e. the environment does not alter the photons.
1. Describe a simple key agreement protocol where Alice and Bob take advantage of the two channels.
2. Assuming Eve can listen to the information on the classical channel while she can observe and
resend photons on the quantum channel, prove that Alice and Bob can detect Eve’s interaction.
Ex. 6 — Simple questions
1. Given four n × n matrices U1,U2, V1, V2, prove that (U1 ⊗ V1) · (U2 ⊗ V2) = (U1U2) ⊗ (V1V2).
2. Sow that the operator ⊗ is bilinear.