Dear students, welcome to the first lecture of privacy preserving cryptocurrencies.

So far we have only discussed the content which means the structure of the lecture and

how we can start with the real stuff.

We begin this lecture with the basic preliminaries and these consist mainly of two parts.

The first part are the mathematical foundations and here we will talk about groups, groups

of a specific structure such as cyclic groups and the group C and star.

In the second part we will discuss basic crypto and here most of the stuff you have already

seen in introductory class.

Nevertheless please take your time and learn these foundations well because they will be

used in all of the subsequent lectures.

So I look forward to seeing you in the question and answer session and I hope that these videos

help you to get into this stuff.

Thank you.

The purpose of the first lectures is to establish a common ground such that we are sure we are

on the same basis and we share the same knowledge.

So this is the first lecture so we will start with introducing the first basic math and

this includes the description of what is a group, what are the group operations, what

are subgroups and also we will take a look at specific groups such as cyclic groups and

the group C and star.

In the second part we will also take a look at basic crypto.

So here we will remind us what a reduction is, provable security and also the underlying

assumptions such as the discrete logarithm assumption for example.

We will also take a look at cryptographic schemes such as encryption schemes, signature

schemes and also commitment schemes.

So most of these schemes you should have heard before from the introductory class but in

case you missed this class feel free to watch the videos now that everything is online.

Nevertheless we try to keep this class even though it is an advanced class in such a way

such that you can essentially take it without taking the prior class of course it is quite

or it is more challenging to follow.

So let's begin with the actual content.

So we will start by defining and introducing the notion of groups.

So overall as you know the construction or the goal of this class is the construction

of efficient privacy preserving protocols in the context of cryptocurrencies.

Some of these techniques will be used to enhance bitcoin and others will essentially be used

to create a cryptocurrency from scratch that has or that have additional privacy preserving

properties.

For all of these construction we require certain proof techniques and also certain cryptographic

schemes and they all rely on certain mathematical structures.

Which means in some cases you could in principle take an arbitrary signature scheme and combine

it with a certain zero knowledge proof for example but in most cases such a generic combination

is not efficient.

And therefore we need to take a look at specific signature scheme that works well with a specific

zero knowledge proof for example.

And even though we haven't discussed what the zero knowledge proof is at this point

you can imagine that this is some proof technique that we will use.

In summary all of these construction rely on certain mathematical primitives and structures

and this is what we need to define now.

We begin with the definition of group.

A group is a very basic and nice structure that essentially consists of a set of elements

and within this set of elements you can define an operation.