## Description

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.