Blockchain tutorial 11: Elliptic Curve key pair generation Video Lecture | Blockchain Tutorial: A Fundamental Guide - Cryptocurrency

in this video I will explain how the
elliptic curve is used to
keep air blockchain implementation
a bit bored or etherium uses ellipticals
to generate private and public key pairs
elliptic curve cryptography was attended
by Neil cobblers and victor Miller in
1985 but 256 ECC public key provides the
comparable security to a 3072 ayers our
public key the primary advantage of
using elliptic curve based pro
photographer is reduced key size and
hand speed elliptic curves has nothing
to do with ellipses ellipses are formed
by quadratic curves elliptic curves are
always cubed the standards for efficient
cryptography group is an international
consortium to develop commercial
standards for efficient and
interoperable cryptography based on
elliptic curve cryptography the secd
website is www.pevs.com UNS were
interested in this one stands for
efficiency cryptography part 1 stands
for efficient cryptography part 2 part 1
describe how the elliptic curve
cryptography works
part two describes the recommended
electrical domain parameters the SEC G
has published a document with a
recommended set of elliptic curve domain
parameters refer to the letters P a B G
and n H this data set is collectively
referred to as the elliptic curve domain
parameters that is this document part
two here are the recommended elliptic
curve domain parameters these parameters
have been given nicknames to enable them
to be easily identified for example SEC
P 256 K 1 in this table you will find a
set of elliptic curve domain parameters
the elliptic curves used smaller key
sizes with respect to errors a providing
comparable security if you look at SEC P
256 K 1 the elliptic curve uses 256 bit
key lengths its counterpart are s a user
3072 bit key lengths Bitcoin and
etherium both uses the same SEC P 256 K
1 elliptic curve domain parameters if
you look at part 2 of the recommended
elliptic curve domain parameters
document the section recommended
elliptic curve domain parameters SEC P
256 K 1 page 9
other recommended parameters SEC P 256 K
1 which is used by Bitcoin and ethereal
here are other parameters P a B G and H
SEC P 256 K 1 uses the following
elliptic curve equation I squared is X
cubed plus 8x plus B in the following
slides we will go through each parameter
P a B G N and H parameter a has the
value of 0 and parameter B has the value
7
thus the equation will be Y squared is X
cubed plus 7 this is how the elliptic
curve looks like please ignore this line
and I assume out
infinity
a finance field is a field with a finite
number of elements defined by parameter
P which is a prime number thus the final
field range lies between 0 and P minus 1
this means that module P should be used
in the equation the elliptic curve
equation is y squared is X cubed plus KX
plus b elliptic curve equation with the
model operation will be y squared is X
cube plus ax plus B minus P SEC P 256k 1
the finds parameter P having this value
this is in Texas decimal form
Base Point G also known as generator or
primitive element is a predetermined
point on the Left curve that everyone
uses to compute other points on the
curve often the base point G is
displayed in two ways
in the compress form with a prefix 0 2
so this is the base point G with the
prefix 0 2 over here if you remove the
prefix 0 2 this value is the
x-coordinate to get a y-coordinate you
use this equation
mataji has also an uncompressed form
with the prefix 0 for this is
uncompressed form with the prefix 0 for
if the prefix is removed the first half
from here to here the value is the
x-coordinate this value from here to
here is the x-value the last part from
here to here is the y-coordinate in my
discrete logarithm video part 9 I have
explained what the cyclic group is when
you apply a certain number of operations
to Base Point G the cycle starts all
over again in the same order when the
next cycle starts the first time it is
indicated by parameter n which is called
the order of Base Point G and have this
hexadecimal value the parameter n
determines what the maximum value is
that can be turned into a private key
and it's 256 bit number in the range 1
and n minus 1 is a valid private key the
parameter H is called the cofactor and
has the constant value 1 because it has
value 1 it does not play a role in the
key generation and I therefore will not
elaborate on the purpose of this
parameter so to recap I have gone
through all the SEC P 256 K 1 domain
parameters P a B G N and H here is P in
hexadecimal form and this is in decimal
form if you want to calculate it a has
this hexadecimal value which is 0 B has
this hexadecimal value which is certain
decimal the base point G in compress
form with 0 2 which means that this is
the x value to calculate the Y value you
have to use the equation which I've
mentioned in the slides earlier
and this is G in uncompressed form the
first part is the x value and second
part is y value and this is the order n
value and this is the H value there are
two operations often called dot
operations which can be applied to a
base point on the elliptic curve point
addition and point doubling before I
explain what point addition and point
doubling is I first has explained that
the elliptic curve with this equation
has the following properties if a line
intersects two points P and Q it
intersects a third point on the curve
indicated by the label - R if a line is
tangent to the curve it intersects
another point on the curve all vertical
lines intersects the curve at infinity I
will now demonstrate what I just told
you if a line intersects two points P
and Q it is the six a third point on the
curve wave of minus R a line goes
through P and Q and intersects the curve
on this point look at minus R this minds
are different move point P over here
look at minus R it always intersects the
curve
point was if a line is tangent to the
curve it intersects another point on the
curve if P and Q are on top of each
other it is tangent the line intersects
the curve if I move P over here and Q
over here the tangent line intersects
the curve yes - are against the last
points all vertical lines intersects the
curve at infinity let's say P is over
here and Q is over here then you have a
vertical line but if you look at the
curve if I go to infinity it is almost a
straight line point addition is adding
two points P and Q on an elliptic curve
whereby p and two are not the same
points first I demonstrate the geometry
approach draw a straight line between P
and Q the line will intersect the
elliptic curve at exactly one more point
labeled - are the reflection of the
point minus R with respect to the x axis
if the point R which is the result of
addition of course P and Q we draw a
straight line between P and - it will
intersect the curve at minus R and when
you mirror this point on the x-axis you
will get the point
R
please note that the elliptic curve is
symmetric around the x-axis when you add
P and Q you get the final point R and
this point is coordinate is what we want
to know point addition does not mean
edition of the risk or eye coordinates
of P and Q it is just a good name given
for this approach point addition has
nothing to do with adding the coordinate
of point P and Q together point addition
is just a name it just means draw a
straight line between P and Q when it
intersects the curve at this point minus
R that will mirror this point on the
x-axis to get this point at this point R
is the coordinate we want to know here
is the mathematical approach to
calculate this coordinate with these
three equations left that is the slope
of this line
and why are the coordinates of point P
and 22 is the base point G and when you
use these three equations you calculate
the if a y-coordinate of points are
point W of point P on an elliptic curve
it is the same as moving point two to
the same location as 40 which means P
have the same quadrant as Point Q let's
demonstrate a geometry approach draw a
tangent line to the elliptic curve at
Point P the line intersects the elliptic
curve at the point labeled - are the
reflection of point months are with
respect to x-axis give the point R which
is the result of doubling of point P we
have points P and point - if we move
point P to the same location as point
two what we now have is a tangent line
at Point P which intersects the curve at
this point labeled - are the reflection
of point R with respect to the x-axis
this point R point R is the result of
point doubling we want to know this
coordinate
but w/e does not mean multiplying the X
or y coordinate of point P it's just a
name different for this approach there's
also a mathematical approach to
calculate this coordinate or you need
these three equations lambda is a slope
of this line x and y are the coordinates
of point P this point now you have
enough information to calculate the x
and y coordinate of point R let's create
two functions called PCEHR and easy
double this is point addition and this
is point doubling the ECS function
contains these equations and the easy
double function contains these equations
to calculate the x and y coordinate of
point are more information about these
mathematical equations can be found at
his web address in the description below
you can find this Web press before I
start explaining how the public key is
generated you need some additional
information the following procedure
describes how to generate a bit called
public key for other blocking
implementation it may differ when the
rope bitcoin public key is generated
using the ECA that easy double functions
it looked like this a large hexadecimal
number the actual Bitcoin address looks
like this additional conversion steps
need to be applied on the raw Bitcoin
public key to get the actual Bitcoin
address which will be explained in
another video in the following slides I
will only focusing on how to generate a
raw bit code public key using the easy s
and easy double functions I will be
using the spider script to demonstrate
how to generate the public key the
original Python script is created by GMC
Angelo who also created an excellent
YouTube movie called Bitcoin 101
elliptic curve cryptography the link for
both vital script and YouTube movie can
be found in description below I have
slightly modified the original Python
script
in the piper script the SEC P 256 K 1
domain parameters are defined the PABT
and n parameters these two basic
parameters are the same as defined in
the SEC G the main premises document
here are the SEC P 256 K 1 domain
parameters P a B G and n as you can see
these two main parameters are the same
the P in the same as this one the a and
B 0 and the 7 in a Python script the
uncompressed form is being used this is
the x value and here is the Y value just
seems over here and the parameter n
which you can see over here it's the
same as this one
the G parameter also called the
generator point here's the x coordinate
and the y coordinate in hexadecimal form
I've converted the hexadecimal values
into decimal values which you can see
over here to give you an idea how large
this number is the x coordinate is
approximately the same as five point
five times 10 to the power of
seventy-six and the y coordinate is
approximately the same as three point
two times 10 to the power of 76 so where
is this point located in the little
ticker so here's the elliptic curve and
as you can see this is 10 to the power 8
11 you get idea
so I'm now 10 to the power of 28 but now
you have an idea where this generator
point may be located on the elliptic
curve
the Pfizer script DC ad and the easy
double functions are defined these
functions are the same as the functions
we have been talked about earlier in the
slides let's see how the public key is
generated first you need to generate a
private key this is my generated private
key over here so these are all
definitions here is where it starts EGC
multiplier uses the generator port and
the private key here's the method ECC
multiplied this is generator point and
this is the private key the first step
is to check if the private key is fell
it as you can see this is my private key
and this is the end parameter you can
see the private key is much smaller than
the n parameter so that private key is
felid my price is not the same as 0 next
it converts my private key into a binary
value is my private key and this my
private key converted into a binary
value the generated point is assigned to
variable Q this for loop goes through
each bit of my private key the first
step is to use the generator point as
input for the EC double we generate a
new coordinate which is the R point next
is check if by binary failure is one
which is not because it's now 0 then it
reads the next bit value that is one it
applies the easy double function again
then it checks if the best value is 1
which it is it applies the easy add
function with the previous calculated
point and the generated point and the
next bit value of my private key is
being read that is this one and it
continues the loop when it reaches the
end of my private key it has final
accorded and this final coordinate is
assigned to this variable public key
this script first prints are the private
key then it prints out the
fellow of my public key the wife value
of my public key and here it brings out
the X and y value of the public key
concatenated so what it does is it
prints out this value the x value the y
value and the x and y value concatenated
let's test this here's my private key
here's the x value of my public key the
y value of my public key and the public
key where the x and y value are
concatenated to a is here to a and 0 3 3
is over here the x and y coordinates are
concatenated I've now demonstrated how
you generate the public and private key
pair using elliptic curve