Version: 0.1

Status: Ready.

By: Albert van der Sel

Remarks: It's a very simple note on encryption.

A crypto expert, could easily write a 800+ page text on this vast collection of subjects.

And..., then that would then cover only the basic theory.

Don't forget that new articles popup daily, covering new territory.

Also, a large part of Quantum Computing deals on crypto- and related theories.

I guess that I only want to say that the subject is "huge".

Here, just a few points will be touched upon, which might give a general impression

if you were really new to the subject.

CONTENTS:

1. A few words on Symmetric encryption (shared key) and Asymmetric encryption (Public key/Private key).

2. A few words on Hashfunctions, Hashes and Digests.

3. A few words on Certificates.

4. A few words on Elliptic curve encryption.

in other sciences too, like in Quantum Mechanics to illustrate Quantum effects and observations.

Suppose that Bob and Alice want to send "encrypted" information to each other.

That information could be in the form of files, messages, email, streamed data etc...

There usually is a stack of protocols on both sides, which makes it work.

However, the mechanisms that are directly involved in encryption/decryption, can be divided in

in Symmetric encryption/decryption and Asymmetric encryption/decryption.

If Bob and Alice want to exchange encrypted messages (meaning any type of data, like files, mail, http etc..),

then they may use

It's important to understand that both use the same key. This key must stay secret from other people and entities ofcourse.

This is often seen as a weaker point in the protocol: that is: how to safely get the key to both parties?

Maybe they need some sort of "secure" channel first, to transfer the key, for example from Alice to Bob.

All encryptions/decriptions go by using the same one key.

So indeed, Bob and Alice both use the same one key, which often is called the "shared key".

In this case, a user has a Public key and a Private key. It's a "key pair". So, that holds for Alice and Bob too.

The "private key" must be kept strictly private to the owner. The public key may be known to other user(s).

Such a Public key, and Private key (for a certain user) are related in such way, that an encryption with one key,

can be decrypted with the other.

So, if we take a look at Alice: she safeguards her private key, and never discloses that to another person.

However, she may send her public key to Bob. Likewise, Bob may send his public key to Alice.

Now, the "keypoint" here is:

-That Alice may use Bob's public key to encrypt messages. Bob can decrypt them, using his private key.

-Bob may use Alice's public key to encrypt messages. Alice can decrypt them, using her's private key.

-Any other user who intercepts such message, cannot do anyting with the encrypted data.

The Jip and Janneke figure below illustrates the difference between Public key/Private key, and Shared key encryptions.

Fig 2: Just an illustration of Public key/Private key, and Shared key encryptions.

Source: my own Jip and Janneke figure.

Public key/Private key cryptography is generally considered to be "better" than Shared key cryptography.

However, the first method is more involved and has a higher overhead.

For large data transfers, Shared key performs much better. However, how both the sender and receiver

gets the same one shared key, was always a point of concern. Obviosly, if the intended sender just

sends the (readable) Shared key to the intended receiver, then "someone" on the network

may intercept that key, and then this "someone", has the option to spy on the messages.

Therefore a "2 Stage" approach is often used.

Using Public key/Private key cryptography, the "shared key" is simply a message, which is

thus enveloped by the first type of cryptography. Once both parties posess the same shared key,

then the encryption of data fully switches to the symmetric type of encryption.

There are quite a few variants (sorts) of both "Shared key" and "Public/Private key" implementations.

Although the general methods described above is true, the variants differ in bitsize, contruction of key(s) etc..

AES-128, AES-192, AES-256, RC4, RC5, RC6, DES.

RSA (like RSA-256), DSA, Elliptic curve techniques.

The Bitcoin system uses "Elliptic Curve Digital Signature Algorithm" or ECDSA, based on

Elliptic curve cryptography.

What we have seen above, is typically used to encrypt/decrypt data. Something that is called

a "hash", or "digest", is an other sort of implementation.

and maps to a certain output.

In math, this is ofcourse an extremely well-known concept, like for example f(x)=2x+7, or f(x)=3x

So, for example with "f(x)=3x

find the corresponding (output) value "3x

A special class of functions in cryptography exists too. These are the hashfunctions.

They can take data of variable length, and produce output of a fixed size.

This output is often called "the hash", or "the hash value", or "the digest".

Almost always, different inputs will generate different outputs. Usually, if in an exceptional case,

two different inputs result in the same output, it's called a "collision".

However, it should be understood that "in crypto appliances" these should be extremely low or non-existent.

-If you let operate the hashfunction on some input, it produces certain output.

If you again let operate the hashfunction on that same input again, then it must produce the same output again.

In crypto language, it is often said "

then it always results in the same hash (or digest).

-Generally, it is understood that a hashfunction is "one way". It's easy to get an output,

but say you have a given output, it should be practically impossible to find the associated input.

Suppose we have a simple example of a "password hash", created by the

Then for example, Harry could use the readable password "MyStrongPWD", which might be mapped as:

Η(MyStrongPWD) → 045A77BBC190FFBB7CC7745

Also, suppose Mirja uses the readable password "IwantSummer", which might be mapped as:

Η(IwantSummer) → 095A76EBC140FCBA7CA3212

Note that the length of the hashes are equal.

There exists quite a few cryptographic hash functions. A few, which once were considered

to be "strong", were MD5 and SHA-0. However, around the period 2003-2009, some weaknesses

were found, and MD5 is currently not advised anymore.

Some weaknesses were found in SHA-1 too. Currently the SHA-2 and SHA-3 families, are still

considered to be very strong.

It should be understood, that if a genuine "brute force" is applied, many so-called

secure stuff, can be broken. That is, if enormous loads of CPU power is applied,

and for example runs for a year (*on average*), before a hash is cracked, you might still

consider it to be rather secure.

Indeed, it should be extremely difficult to derive the original text from a hash.

Note: some hashfunctions use variable properties in order to establish the hash,

like the system date/time, or cpu params etc..

These would not be "deterministic" (e.g. the same input must give the same output over and over),

so, in general, these are not suited for many functions in cryptography,

but ofcourse they can be useful in other applications.

You should see this feature, as a (sort of) special case of asymmetric encription.

A Digital Signature, at creation time at the sender, uses the Private key from Asymmetric encryption,

and with verification at the recipient, the Public key of the sender is used.

The main purpose is to provide trust to a recipient, that a message is indeed from an intended sending party.

If the intended sending party, indeed already have a Private key, and a Public key to it's disposal,

then a

"to sign" the message. The recipient then, uses the Public key of the sender to verify the signature.

keys are used in encryption/decription.

However, in a public network, there still is an issue. How does Alice knows for sure that the public key

that she received from Bob, really came from Bob?

This get's more urgent, if Bob is actually a Webshop, or financial institute, and Alice would be a customer.

In the following, let's pretend that Bob and Alice indeed have such roles.

Ofcourse, instead of a Webshop, you can substitute it by any Website that participates in transactions,

or otherwise needs secure (encrypted) datatransfer.

There are multiple types of certificates. We have "digital certificates", which is a generic term

used at all sorts of purposes. And we have SSL/TLS certificates.

Most of the time, people actually mean the SSL/TLS certificates, when talking about "certificates".

Those are used in Internet traffic, to prove the identity of a Website, and to make encrypted datatransfer possible.

Certificates can not only be used to enable secure Internet (SSL), but also for driver signing, code signing etc..

The (SSL) certificate is a file. It thus serves two main objectives:

- The certificate is issued by a Trusted Authority, who checked the Webshop. It contains

some identifiers which uniquely identify the Webshop (like the Domain, Organization, email etc..).

Since the Trusted Authority is indeed worldwide reckognized as an "Authority", people like Alice

can trust the certificate and the organization which holds the certificate (the Webshop). - The certificate is also involved in the transfer the Public key of the Webshop, and a signature

which proves the validity of the Public key.

This can be useful in a

If, at a Webshop, a certificate is installed, that site can activate SSL which is a protocol

which sits just "above" TCPIP, and which is involved in encryption/decription.

When at a site SSL is used, the application protocol changes to HTTPS instead of HTTP,

where the "S" in HTTPS refers to "secure".

Most Operating systems, and browsers, are equipped with a list of Trusted Authorities.

The certificate, and the Public key/Private key infrastructure (PKI infrastructure), does not mean that

all Internet trafic is using Asymmetric encryption, as we have seen in chapter 1.

Bulk data encryption uses a shared (common) key, often also called the "session key".

Here is what often happens in Internet communication:

-Client connects to the Webshop.

-Webshop responds with sending it's certificate with it's signed Public key.

-Client creates a session key, encrypts it with the Webshop's Public key, and sends it to Webshop.

-Webshop decrypts it with it's private key.

-Both entities now have a shared session key, used in further encryption/decryption of data.

a form of "Asymmetric encryption".

But if you are interrested, then let's see what this remarkable stuff is about.

For example, the Bitcoin system uses "Elliptic Curve Digital Signature Algorithm" or ECDSA, based on

Elliptic curve cryptography.

Asymmetric encryption can be used for any sort of encryption/decription in general.

What we are going to discuss here is:

-The general principles of Elliptic Curve encryption.

-A specific case, namely how a Public key is derived from a Private key in the Bitcoin system.

So, let's first take a look at the general principles of Elliptic Curve encryption.

An "elleptic curve" has a general equation y

Note:

You know that functions (or relations) of one variable, that is, in the form of "y=f(x)" are often plot

in a two dimensional plane, having an x-axis and y-axis.

So, for example, a sketch of y=2x

ranging from -5 to +5, thus if x takes on the values -5,-4,-3,-2,-1,0,1,2,3,4,5, and for each of them,

calculate the corresponding y value. You then will see that y=2x

Ofcourse, here I took "x" to have integer values, but generally, this math is based on "real" numbers.

You could plot an elliptic curve too.

The Bitcoin system uses ECDSA, based on the elliptic curve y

You might wonder why this specific equation is used, and also you might wonder how a Public key

In ECDSA, once a Private Key is established, we go to work using the elliptic curve.

That is, we use the elliptic curve, to derive the Public key.

Fig 3: Just an illustration of a particular elliptic curve.

Source: my own Jip and Janneke figure.

Depending on the coefficients "a" and "b" in the relation above, the form of such curve

can vary greatly: just google on pictures of "elliptic curve", and you will see that variety.

Such a curve defines an abelian group, if the members in the group (thus the points on the curve),

adhere to a few basic rules. One of the most important rules is "closure", meaning that

an "operation" must exist, say denoted by "+", in such way, that for all members "P" and "Q",

must hold that P+Q is also member of the group (thus also a point on the curve).

It's tempting to view "+" as the addition operator, as is used in arithmetic.

Almost, but not exactly the same thing. However, in all crypto literature, or math articles

dealing on elliptic curves, the operator is indeed called "addition".

However,

looks a tiny bit different from simple regular addition, as is used in arithmetic.

Thus the only requirement is that "adding" points, no matter how many times, must produce

again a point "somewhere" on the curve.

How would one geometrically, or algebraically, further define the exact mechanics of such an "addition"

for any members on an elliptic curve?

- Viewed from a geometrical perspective, you only have one logical option:

and see where it intersects the elliptic curve

from a negative sign). You cannot opt for another solution. For example, any sort of curve between P and Q would not work,

since you want one result for P+Q, and not an unknown (variable) number of other solutions.

A line is logical, but also remember that

that P+Q actually works. That is: producing a third point on the curve (member of the curve).

Viewed this way, we can say that a line through P and Q, exactly fits our "construct".

There is only one small catch. Using the principle above, we find the point "S" (see figure 3),

but ultimately we need to arrive at point "R". Here then, is a small difficulty to explain why

we must "flip" the "y-coordinate" of point S, to get the coordinates of R.

- Viewed from an algebraical perspective, you can say this:

See figure 3. You see just some randomly chosen points P and Q. I could have taken any other set,

with one exception only: when P and Q have the same "x" value, since then a line between P and Q

would go to "infinity", and never intersects the curve.

Suppose we need to express "P + Q = R" in coordinates (x,y). Here then, the aim is to find

the coordinates (x

But, we will introduce an intermediate step, namely finding the first true intersection "S" first, and

if that succeeds, we will immediately find the point "R", since the points R and S only differ in the sign

of the "y coordinate". Why we need to flip the y coordinate, will be explained later.

Please refer to figure 3 again. We are going to solve "P + Q = S" in terms of coordinates.

We then would have: (x

Note that the "slope" of the line through P and Q, or in better words, the "gradient" of the line through P and Q is:

grad = (y

Let's call this gradient "λ".

The general equation for the line between P and Q is:

y=y

We need to find the intersection of that line, with the elliptic curve. So, having two unknown variables (the coordinates),

and two equations, provides us mathematically with everything needed to find the intersection S,

and thus we can find the coordinates of R too.

Both equations together, that is:

| y=y

| y

will provide the solution for the untersection point "S". Believe me, it's just a bit of math, and nothing more.

Above is just an outline. Before we can make this more concrete, we need to return to

So, what is the purpose anyway? Knowing that adding points which are on an elliptic curve, must result in another point

on that curve, might be nice, but so what?

Instead of adding different points, like P and Q, we can also do "an experiment" like Q+Q.

If you look at figure 3 again, you can visualize what it means. Take a look at figure 3 again.

Now, let point P, along the curve, "crawl" towards point Q. At a certain moment, they coincide.

The line through P and Q (actually through Q and Q), is now exactly the tangent line along the curve, at the point Q.

The tangent line will intersect the curve somewhere at "T". This new point "T", is the result of "Q+Q",

except for the fact that we still need to flip the y-coordinate of T, to get the real result of "Q+Q",

but I ignore that for a moment.

You can repeat such addition over and over. You can actually try for example:

5 * Q = Q+Q+Q+Q+Q

or:

n * Q = Q+Q+...+Q (n times)

A specific implementation of "Elliptic encryption" is used by the Bitcoin system.

That system might serve as an example of such an implementation.

-Here, the "secp256k1" SEC (Standards for Efficient Cryptography) specification is used.

This specification lists all relevant parameters for a specific elliptic curve and encryption.

-It uses the elliptic curve "y

from the general equation "y

Fig 3: Just an illustration of the secp256k1 elliptic curve y

Source: Wikimedia commons (Public Domain).

-However, the x domain (the x-axis) is not the familiar set of (continues) real numbers, but a discrete set

of numbers for which the specification is set in "secp256k1".

Using this set of "x values", produces a discontinue scatter diagram (not shown in this note).

-A

Usually, this base point (or reference point) is denoted by "G", but other letters are found

in the literature too.

Ofcourse, since we have now a discrete set of x-values, we cannot actually speak of "a curve",

but "G" is indeed an element on the scatter diagram, quite similar as if we still would have

used a continues curve.

-Using the analogy of a continues curve, just like above, we are able to perform a calculation

like:

5 * G = G+G+G+G+G

or:

n * G = G+G+...+G (n times)

-Likewise, a Public key can be derived using such system, using the equation:

Note that the equation above, is not much different from an equation as "n * G = G+G+...+G (n times)".

It's a one-way system. It is commonly believed that it is not practically possible

to calculate the Private Key, from such Public Key.