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[MrFreeDragon]

-snip-
My question is : why 13 = 8 + 4 +1 ? is 4 + 4 + 4 + 1 give the same result or must there be a certain rule ?
Not sure and i can't retreive the post but it seem i read something with the private key in it's binary form ?
-snip-

Your example with 13 will give the same result. You can also calculate 13 as 1 + 1 + 1 + 1 + ... + 1 + 1 (13 times), but here you should perform 12 additions.

As for the private key in binary form - I made a tool to play with binary numbers: https://bitcointalk.org/index.php?topic=5187401
That tool is very nice for learning purposes. And also good to create real wallets if you use physical coin for your random entropy source.

Every binary number could be represented as c255*2^255 + c254*2^254 + c253*2^253 + ... + c3*2^3 + c2^2 + c1*2 + c0, where coefficients c0, c1, c2, ... c254, c255 represent bit values (1 or 0). It is actually your binary number.
So in order to calculate public keys for various large binary numbers you can easily make the pre-calculations of 256 public keys for numbers 1, 2, 2^2, 2^3, 2^4, 2^5, ... 2^254, 2^255 and later just make up to 255 additions between pre-calculated public keys.

[1714079281]

1714079281

[archyone]

Hello everyone, good! thanks to the help of MrFreeDragon and BrewMaster I now have a good basis for adding 2 points on an elliptical curve.

As a reminder :

If you want to double point P in order to receive R = P + P, you should make the following:
c = 3 * P.x * Px * invert (2 * P.y)% modulo
R.x = (c * c - 2 * P.x)% modulo
R.y = (c * (P.x - R.x) - P.y)% modulo
modulo = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
and
If you want to add 2 points P and Q (2 different non-zero Points) there is another formula for R = P + Q:
dx = (Q.x - P.x)% modulo
dy = (Q.y - P.y)% modulo
c = dy * invert (dx)% modulo
R.x = (c * c - P.x - Q.x)% modulo
R.y = (c * (P.x - R.x) - P.y)% modulo

I have now no problem with this but as you can imagine, I still have 1 problem ^^, certainly due to my approximate understanding of English I am unable to find a formula for point to point substraction. Is this also possible?

Thanks in advance (again^^)

[BrewMaster]

I have now no problem with this but as you can imagine, I still have 1 problem ^^, certainly due to my approximate understanding of English I am unable to find a formula for point to point substraction. Is this also possible?

there is no special formula for point subtraction as far as i know. instead the P-Q is simply defined as P+(-Q) (same as addition) and -Q or negative of a point is defined as negating its y coordinate. or in other words -Q(x,y) = Q(x,-y) and since we don't use negative numbers in modular arithmetic -y becomes P-y where P is curve's prime.

[archyone]

there is no special formula for point subtraction as far as i know. instead the P-Q is simply defined as P+(-Q) (same as addition) and -Q or negative of a point is defined as negating its y coordinate. or in other words -Q(x,y) = Q(x,-y) and since we don't use negative numbers in modular arithmetic -y becomes P-y where P is curve's prime.

oOO !! wonderfully explained, first test -> total success .. I understood the first time when research for several weeks had not led to much.
Many thanks to you BrewMaster (again ^^)
It is this notation (-y becomes P-y where P is curve's prime) that I have not seen anywhere that I am lacking.

[bytcoin]

-snip-
I have a question.
If the private key is 3 ... do I need to change the formula this way?
R = P+P+P
c = 4*P.x*Px*invert(3*P.y) % modulo
R.x = (c*c - 3*P.x) % modulo

No, formula is ALWAYS the same. That was a formula for double the point:
c = 3*P.x*Px*invert(2*P.y) % modulo
R.x = (c*c - 2*P.x) % modulo
R.y = (c*(P.x - R.x) - P.y) % modulo

If you want to add 2 points P and Q (2 different non-zero Points) there is another formula for R = P + Q:
dx = (Q.x - P.x) % modulo
dy = (Q.y - P.y) % modulo
c = dy * invert(dx) % modulo
R.x = (c*c - P.x - Q.x) % modulo
R.y = (c*(P.x - R.x) - P.y) % modulo

If the formula is always the same, All public keys will also always be the same! "Where am I wrong?

(dx, dy, c, R.x, R.y, Q.x Q.y, P.x, P.y)Can someone explain to me what are this?

I think Rx and Ry are the coordinates of the public key, right?

[BrewMaster]

If the formula is always the same, All public keys will also always be the same! "Where am I wrong?

they aren't the same because this formula is called different number of times depending on the private key. in simple terms imagine if the formula was this: x+1 and you always called it with the same x (like you do with the generator point). if you use it with x=5 and call it 3 times (private key equal to 3) you get 8 and if you call it 6 times (private key equal to 6) you get 11 and so on.

[archyone]

If the formula is always the same, All public keys will also always be the same! "Where am I wrong?

(dx, dy, c, R.x, R.y, Q.x Q.y, P.x, P.y)Can someone explain to me what are this?

I think Rx and Ry are the coordinates of the public key, right?

[/quote]

ok, I will try to explain as simply as possible because it is true that I myself struggled to understand the system.
So it is a question here of adding 2 points (not the same). In this example we use :

first point: (all values are in décimal for a better comprehension)

X coordinate: 21262057306151627953595685090280431278183829487175876377991189246716355947009 (it is Qx)
Y coordinate: 41749993296225487051377864631615517161996906063147759678534462689479575333124 (it is Qy)
The Private key for this point is 0000000000000000000000000000000000000000000000000000000000000008

second point to add:

X coordinate: 89565891926547004231252920425935692360644145829622209833684329913297188986597 (it is Px)
Y coordinate: 12158399299693830322967808612713398636155367887041628176798871954788371653930 (it is Py)
The Private key for this point is 0000000000000000000000000000000000000000000000000000000000000002

So to add the 1st point to the second we use the formula : (here modulo is 115792089237316195423570985008687907853269984665640564039457584007908834671663 )

dx = (Q.x - P.x) % modulo
dy = (Q.y - P.y) % modulo
c = dy * invert(dx) % modulo
R.x = (c*c - P.x - Q.x) % modulo
R.y = (c*(P.x - R.x) - P.y) % modulo

in our example we have

dx = (21262057306151627953595685090280431278183829487175876377991189246716355947009 - 89565891926547004231252920425935692360644145829622209833684329913297188986597) % modulo
dx = 47488254616920819145913749673032646770809668323194230583764443341328001632075

dy = (41749993296225487051377864631615517161996906063147759678534462689479575333124 - 12158399299693830322967808612713398636155367887041628176798871954788371653930) % modulo
dy = 29591593996531656728410056018902118525841538176106131501735590734691203679194

invert of dx = 70279122268919195963430815486314537773961171454828771794853116552210630553734

c = dy * invert(dx) % modulo
c = 16132032934385503768504319366562120314980927452732756733183380715276156205226

So the new point (8 + 2)

R.x = (c*c - P.x - Q.x) % modulo
R.x --> X coordinate of (8+2) = 72488970228380509287422715226575535698893157273063074627791787432852706183111

R.y = (c*(P.x - R.x) - P.y) % modulo
R.y --> Y coordinate of (8+2) = 62070622898698443831883535403436258712770888294397026493185421712108624767191

If we check these coordinates, we find that it corresponds to the private key: 000000000000000000000000000000000000000000000000000000000000000a (10)

There you go, I hope I was as clear as possible and apologies for my broken English ^^

[ricardosuperx]

I learned to double the point, but I cannot calculate the public keys of the private keys in sequence

Duplication of points;
Compressed public key;
In decimal;
Equation:
c = (3px^2 + a) / 2py
rx = c^2 – 2px

Prime Modulo: 115792089237316195423570985008687907853269984665640564039457584007908834671663

Base Point: (55066263022277343669578718895168534326250603453777594175500187360389116729240, 32670510020758816978083085130507043184471273380659243275938904335757337482424)

Order: 115792089237316195423570985008687907852837564279074904382605163141518161494337

Private key:  0000000000000000000000000000000000000000000000000000000000000002


c= 3px^2 + a) / 2py

c=(3*55066263022277343669578718895168534326250603453777594175500187360389116729240^2)/2*32670510020758816978083085130507043184471273380659243275938904335757337482424

c=(3*60300556597753154781239923047219078515410877540607532238537983597388018023497)/2*32670510020758816978083085130507043184471273380659243275938904335757337482424

c=65109580555943268920148784132969327692962647956182032676156366784255219398828/2*32670510020758816978083085130507043184471273380659243275938904335757337482424

c=65109580555943268920148784132969327692962647956182032676156366784255219398828/65341020041517633956166170261014086368942546761318486551877808671514674964848

c=91914383230618135761690975197207778399550061809281766160147273830617914855857

rx=(91914383230618135761690975197207778399550061809281766160147273830617914855857^2)-2*55066263022277343669578718895168534326250603453777594175500187360389116729240

rx=83906328733785496146839373207584853159875368071536834145227120626166587773414-2*55066263022277343669578718895168534326250603453777594175500187360389116729240

rx=83906328733785496146839373207584853159875368071536834145227120626166587773414-110132526044554687339157437790337068652501206907555188351000374720778233458480

rx=89565891926547004231252920425935692360644145829622209833684329913297188986597

rx Compressed in hex = 02c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5

Now i would like to understand how to do for private key 3
Can someone do a tutorial like I did IN DECIMAL OF PRIVATE KEY 3

[ricardosuperx]

If the formula is always the same, All public keys will also always be the same! "Where am I wrong?

(dx, dy, c, R.x, R.y, Q.x Q.y, P.x, P.y)Can someone explain to me what are this?

I think Rx and Ry are the coordinates of the public key, right?

ok, I will try to explain as simply as possible because it is true that I myself struggled to understand the system.
So it is a question here of adding 2 points (not the same). In this example we use :

first point: (all values are in décimal for a better comprehension)

X coordinate: 21262057306151627953595685090280431278183829487175876377991189246716355947009 (it is Qx)
Y coordinate: 41749993296225487051377864631615517161996906063147759678534462689479575333124 (it is Qy)
The Private key for this point is 0000000000000000000000000000000000000000000000000000000000000008

second point to add:

X coordinate: 89565891926547004231252920425935692360644145829622209833684329913297188986597 (it is Px)
Y coordinate: 12158399299693830322967808612713398636155367887041628176798871954788371653930 (it is Py)
The Private key for this point is 0000000000000000000000000000000000000000000000000000000000000002

So to add the 1st point to the second we use the formula : (here modulo is 115792089237316195423570985008687907853269984665640564039457584007908834671663 )

dx = (Q.x - P.x) % modulo
dy = (Q.y - P.y) % modulo
c = dy * invert(dx) % modulo
R.x = (c*c - P.x - Q.x) % modulo
R.y = (c*(P.x - R.x) - P.y) % modulo

in our example we have

dx = (21262057306151627953595685090280431278183829487175876377991189246716355947009 - 89565891926547004231252920425935692360644145829622209833684329913297188986597) % modulo
dx = 47488254616920819145913749673032646770809668323194230583764443341328001632075

dy = (41749993296225487051377864631615517161996906063147759678534462689479575333124 - 12158399299693830322967808612713398636155367887041628176798871954788371653930) % modulo
dy = 29591593996531656728410056018902118525841538176106131501735590734691203679194

invert of dx = 70279122268919195963430815486314537773961171454828771794853116552210630553734

c = dy * invert(dx) % modulo
c = 16132032934385503768504319366562120314980927452732756733183380715276156205226

So the new point (8 + 2)

R.x = (c*c - P.x - Q.x) % modulo
R.x --> X coordinate of (8+2) = 72488970228380509287422715226575535698893157273063074627791787432852706183111

R.y = (c*(P.x - R.x) - P.y) % modulo
R.y --> Y coordinate of (8+2) = 62070622898698443831883535403436258712770888294397026493185421712108624767191

If we check these coordinates, we find that it corresponds to the private key: 000000000000000000000000000000000000000000000000000000000000000a (10)

There you go, I hope I was as clear as possible and apologies for my broken English ^^

[/quote]


That's what I wanted! Could you make private key 3?

[archyone]

That's what I wanted! Could you make private key 3?

We must not forget that there are two formulas
The one you used to find the point corresponding to the private key: 2 (or rather 0000000000000000000000000000000000000000000000000000000000000002 to be more precise) it is Duplication of points.
You did it in your example with the base point--> 1 + 1 =2  (but that could be another point)
To go now with 3 we need to do --> 2 + 1 = 3 (we have now 2 différents points and we can't doubling them)

For that you need to use the second formula:

modulo = 115792089237316195423570985008687907853269984665640564039457584007908834671663
Px = 89565891926547004231252920425935692360644145829622209833684329913297188986597 (x coordinate point 2)
Py = 12158399299693830322967808612713398636155367887041628176798871954788371653930 (y coordinate point 2)
Qx = 55066263022277343669578718895168534326250603453777594175500187360389116729240 (x coordinate point 1) not because it is the base point, just because it is the point n°1
Qy = 32670510020758816978083085130507043184471273380659243275938904335757337482424 (y coordinate point 1)

dx = (Qx - Px) % modulo             --> 34499628904269660561674201530767158034393542375844615658184142552908072257357
dy = (Qy - Py) % modulo             --> 95279978516251208768455708490894263304954079172022948940317551626939868843169
c = dy * invert(dx) % modulo       --> 23578750110654438173404407907450265080473019639451825850605815020978465167024
Rx = (c*c - Px - Qx) % modulo     --> 112711660439710606056748659173929673102114977341539408544630613555209775888121 (x coordinate of point (2+1 =3)
Ry = (c*(Px - Rx) - Py) % modulo --> 25583027980570883691656905877401976406448868254816295069919888960541586679410   (y coordinate of point (2+1 =3)

Can't explain better

[BASE16]

You are mixing BASE16 with BASE10 there  
Your private key = 0x0000000000000000000000000000000000000000000000000000000000000002
It's called BITcoin not BYTEcoin !! or even INTcoin hahaha

Since this is a tutorial you can not really leave out the part of the binary private key so here goes.  

You start with a 256 Bits BASE2 Binary Private Key. (Aka. the BIT coin flips)
A bit is either 1 or 0 no in between.

Code:

1011010000000101011011011111011001101001000111111000110111000111001011100101011000110000001011011101101011010011010001011101011001011111111010101101001111101010110110010010100110010110000010011010100000100110111000100011010001001110101101100011101010100100

Then you can use the following table which is about point doubling:

Code:

256 '1' - represents 1*G         or 1 * the generator point
255 '2' - represents 1*G + 1*G   or 2 * the generator point 
254 '4' - represents 2*G + 2*G   or 4 * the generator point 
253 '8' - represents 4*G + 4*G   or 8 * the generator point and the point doubling goes on 256 times... 
252 '16'
251 '32'
250 '64'
249 '128'
248 '256'
247 '512'
246 '1024'
245 '2048'
244 '4096'
243 '8192'
242 '16384'
241 '32768'
240 '65536'
239 '131072'
238 '262144'
237 '524288'
236 '1048576'
235 '2097152'
234 '4194304'
233 '8388608'
232 '16777216'
231 '33554432'
230 '67108864'
229 '134217728'
228 '268435456'
227 '536870912'
226 '1073741824'
225 '2147483648'
224 '4294967296'
223 '8589934592'
222 '17179869184'
221 '34359738368'
220 '68719476736'
219 '137438953472'
218 '274877906944'
217 '549755813888'
216 '1099511627776'
215 '2199023255552'
214 '4398046511104'
213 '8796093022208'
212 '17592186044416'
211 '35184372088832'
210 '70368744177664'
209 '140737488355328'
208 '281474976710656'
207 '562949953421312'
206 '1125899906842624'
205 '2251799813685248'
204 '4503599627370496'
203 '9007199254740992'
202 '18014398509481984'
201 '36028797018963968'
200 '72057594037927936'
199 '144115188075855872'
198 '288230376151711744'
197 '576460752303423488'
196 '1152921504606846976'
195 '2305843009213693952'
194 '4611686018427387904'
193 '9223372036854775808'
192 '18446744073709551616'
191 '36893488147419103232'
190 '73786976294838206464'
189 '147573952589676412928'
188 '295147905179352825856'
187 '590295810358705651712'
186 '1180591620717411303424'
185 '2361183241434822606848'
184 '4722366482869645213696'
183 '9444732965739290427392'
182 '18889465931478580854784'
181 '37778931862957161709568'
180 '75557863725914323419136'
179 '151115727451828646838272'
178 '302231454903657293676544'
177 '604462909807314587353088'
176 '1208925819614629174706176'
175 '2417851639229258349412352'
174 '4835703278458516698824704'
173 '9671406556917033397649408'
172 '19342813113834066795298816'
171 '38685626227668133590597632'
170 '77371252455336267181195264'
169 '154742504910672534362390528'
168 '309485009821345068724781056'
167 '618970019642690137449562112'
166 '1237940039285380274899124224'
165 '2475880078570760549798248448'
164 '4951760157141521099596496896'
163 '9903520314283042199192993792'
162 '19807040628566084398385987584'
161 '39614081257132168796771975168'
160 '79228162514264337593543950336'
159 '158456325028528675187087900672'
158 '316912650057057350374175801344'
157 '633825300114114700748351602688'
156 '1267650600228229401496703205376'
155 '2535301200456458802993406410752'
154 '5070602400912917605986812821504'
153 '10141204801825835211973625643008'
152 '20282409603651670423947251286016'
151 '40564819207303340847894502572032'
150 '81129638414606681695789005144064'
149 '162259276829213363391578010288128'
148 '324518553658426726783156020576256'
147 '649037107316853453566312041152512'
146 '1298074214633706907132624082305024'
145 '2596148429267413814265248164610048'
144 '5192296858534827628530496329220096'
143 '10384593717069655257060992658440192'
142 '20769187434139310514121985316880384'
141 '41538374868278621028243970633760768'
140 '83076749736557242056487941267521536'
139 '166153499473114484112975882535043072'
138 '332306998946228968225951765070086144'
137 '664613997892457936451903530140172288'
136 '1329227995784915872903807060280344576'
135 '2658455991569831745807614120560689152'
134 '5316911983139663491615228241121378304'
133 '10633823966279326983230456482242756608'
132 '21267647932558653966460912964485513216'
131 '42535295865117307932921825928971026432'
130 '85070591730234615865843651857942052864'
129 '170141183460469231731687303715884105728'
128 '340282366920938463463374607431768211456' 
127 '680564733841876926926749214863536422912'
126 '1361129467683753853853498429727072845824'
125 '2722258935367507707706996859454145691648'
124 '5444517870735015415413993718908291383296'
123 '10889035741470030830827987437816582766592'
122 '21778071482940061661655974875633165533184'
121 '43556142965880123323311949751266331066368'
120 '87112285931760246646623899502532662132736'
119 '174224571863520493293247799005065324265472'
118 '348449143727040986586495598010130648530944'
117 '696898287454081973172991196020261297061888'
116 '1393796574908163946345982392040522594123776'
115 '2787593149816327892691964784081045188247552'
114 '5575186299632655785383929568162090376495104'
113 '11150372599265311570767859136324180752990208'
112 '22300745198530623141535718272648361505980416'
111 '44601490397061246283071436545296723011960832'
110 '89202980794122492566142873090593446023921664'
109 '178405961588244985132285746181186892047843328'
108 '356811923176489970264571492362373784095686656'
107 '713623846352979940529142984724747568191373312'
106 '1427247692705959881058285969449495136382746624'
105 '2854495385411919762116571938898990272765493248'
104 '5708990770823839524233143877797980545530986496'
103 '11417981541647679048466287755595961091061972992'
102 '22835963083295358096932575511191922182123945984'
101 '45671926166590716193865151022383844364247891968'
100 '91343852333181432387730302044767688728495783936'
99 '182687704666362864775460604089535377456991567872'
98 '365375409332725729550921208179070754913983135744'
97 '730750818665451459101842416358141509827966271488'
96 '1461501637330902918203684832716283019655932542976'
95 '2923003274661805836407369665432566039311865085952'
94 '5846006549323611672814739330865132078623730171904'
93 '11692013098647223345629478661730264157247460343808'
92 '23384026197294446691258957323460528314494920687616'
91 '46768052394588893382517914646921056628989841375232'
90 '93536104789177786765035829293842113257979682750464'
89 '187072209578355573530071658587684226515959365500928'
88 '374144419156711147060143317175368453031918731001856'
87 '748288838313422294120286634350736906063837462003712'
86 '1496577676626844588240573268701473812127674924007424'
85 '2993155353253689176481146537402947624255349848014848'
84 '5986310706507378352962293074805895248510699696029696'
83 '11972621413014756705924586149611790497021399392059392'
82 '23945242826029513411849172299223580994042798784118784'
81 '47890485652059026823698344598447161988085597568237568'
80 '95780971304118053647396689196894323976171195136475136'
79 '191561942608236107294793378393788647952342390272950272'
78 '383123885216472214589586756787577295904684780545900544'
77 '766247770432944429179173513575154591809369561091801088'
76 '1532495540865888858358347027150309183618739122183602176'
75 '3064991081731777716716694054300618367237478244367204352'
74 '6129982163463555433433388108601236734474956488734408704'
73 '12259964326927110866866776217202473468949912977468817408'
72 '24519928653854221733733552434404946937899825954937634816'
71 '49039857307708443467467104868809893875799651909875269632'
70 '98079714615416886934934209737619787751599303819750539264'
69 '196159429230833773869868419475239575503198607639501078528'
68 '392318858461667547739736838950479151006397215279002157056'
67 '784637716923335095479473677900958302012794430558004314112'
66 '1569275433846670190958947355801916604025588861116008628224'
65 '3138550867693340381917894711603833208051177722232017256448'
64 '6277101735386680763835789423207666416102355444464034512896'
63 '12554203470773361527671578846415332832204710888928069025792'
62 '25108406941546723055343157692830665664409421777856138051584'
61 '50216813883093446110686315385661331328818843555712276103168'
60 '100433627766186892221372630771322662657637687111424552206336'
59 '200867255532373784442745261542645325315275374222849104412672'
58 '401734511064747568885490523085290650630550748445698208825344'
57 '803469022129495137770981046170581301261101496891396417650688'
56 '1606938044258990275541962092341162602522202993782792835301376'
55 '3213876088517980551083924184682325205044405987565585670602752'
54 '6427752177035961102167848369364650410088811975131171341205504'
53 '12855504354071922204335696738729300820177623950262342682411008'
52 '25711008708143844408671393477458601640355247900524685364822016'
51 '51422017416287688817342786954917203280710495801049370729644032'
50 '102844034832575377634685573909834406561420991602098741459288064'
49 '205688069665150755269371147819668813122841983204197482918576128'
48 '411376139330301510538742295639337626245683966408394965837152256'
47 '822752278660603021077484591278675252491367932816789931674304512'
46 '1645504557321206042154969182557350504982735865633579863348609024'
45 '3291009114642412084309938365114701009965471731267159726697218048'
44 '6582018229284824168619876730229402019930943462534319453394436096'
43 '13164036458569648337239753460458804039861886925068638906788872192'
42 '26328072917139296674479506920917608079723773850137277813577744384'
41 '52656145834278593348959013841835216159447547700274555627155488768'
40 '105312291668557186697918027683670432318895095400549111254310977536'
39 '210624583337114373395836055367340864637790190801098222508621955072'
38 '421249166674228746791672110734681729275580381602196445017243910144'
37 '842498333348457493583344221469363458551160763204392890034487820288'
36 '1684996666696914987166688442938726917102321526408785780068975640576'
35 '3369993333393829974333376885877453834204643052817571560137951281152'
34 '6739986666787659948666753771754907668409286105635143120275902562304'
33 '13479973333575319897333507543509815336818572211270286240551805124608'
32 '26959946667150639794667015087019630673637144422540572481103610249216'
31 '53919893334301279589334030174039261347274288845081144962207220498432'
30 '107839786668602559178668060348078522694548577690162289924414440996864'
29 '215679573337205118357336120696157045389097155380324579848828881993728'
28 '431359146674410236714672241392314090778194310760649159697657763987456'
27 '862718293348820473429344482784628181556388621521298319395315527974912'
26 '1725436586697640946858688965569256363112777243042596638790631055949824'
25 '3450873173395281893717377931138512726225554486085193277581262111899648'
24 '6901746346790563787434755862277025452451108972170386555162524223799296'
23 '13803492693581127574869511724554050904902217944340773110325048447598592'
22 '27606985387162255149739023449108101809804435888681546220650096895197184'
21 '55213970774324510299478046898216203619608871777363092441300193790394368'
20 '110427941548649020598956093796432407239217743554726184882600387580788736'
19 '220855883097298041197912187592864814478435487109452369765200775161577472'
18 '441711766194596082395824375185729628956870974218904739530401550323154944'
17 '883423532389192164791648750371459257913741948437809479060803100646309888'
16 '1766847064778384329583297500742918515827483896875618958121606201292619776'
15 '3533694129556768659166595001485837031654967793751237916243212402585239552'
14 '7067388259113537318333190002971674063309935587502475832486424805170479104'
13 '14134776518227074636666380005943348126619871175004951664972849610340958208'
12 '28269553036454149273332760011886696253239742350009903329945699220681916416'
11 '56539106072908298546665520023773392506479484700019806659891398441363832832'
10 '113078212145816597093331040047546785012958969400039613319782796882727665664'
9 '226156424291633194186662080095093570025917938800079226639565593765455331328'
8 '452312848583266388373324160190187140051835877600158453279131187530910662656'
7 '904625697166532776746648320380374280103671755200316906558262375061821325312'
6 '1809251394333065553493296640760748560207343510400633813116524750123642650624'
5 '3618502788666131106986593281521497120414687020801267626233049500247285301248'
4 '7237005577332262213973186563042994240829374041602535252466099000494570602496'
3 '14474011154664524427946373126085988481658748083205070504932198000989141204992'
2 '28948022309329048855892746252171976963317496166410141009864396001978282409984'
1 '57896044618658097711785492504343953926634992332820282019728792003956564819968'

The first number in the list is the bit number, and the second number is the multiplier, the number of times you will add the generator point to itself later on.

It's important to know that we start with bit 256, this is the last bit of the binary private key so you have to read the private key backwards and start with the last bit first.

Adding all the numbers is simple:
If the bit value is true or a '1' then you will write down the multiplier value and if it is false or a '0' you will write down nothing and move to the next bit.

You do this for all the bits and when you are done, you simply add all these multiplier values together, and this will give you the number of times you have to add the generator point to itself to get to the public key point.

Code:

1011010000000101011011011111011001101001000111111000110111000111001011100101011000110000001011011101101011010011010001011101011001011111111010101101001111101010110110010010100110010110000010011010100000100110111000100011010001001110101101100011101010100100

Let's calculate the private key in the example above and remember we start with the last bit first:

Code:

bit 256 = 0 -
bit 255 = 0 -
bit 254 = 1 - 4
bit 253 = 0 -
bit 252 = 0 -
bit 251 = 1 - 32
bit 250 = 0 -
bit 249 = 1 - 128
bit 248 = 0 -
bit 247 = 1 - 512
bit 246 = 0 -
bit 245 = 1 - 2048
bit 244 = 1 - 4096
bit 243 = 1 - 8192
bit 242 = 0 -
bit 241 = 0 -
bit 240 = 0 -
bit 239 = 1 - 131072
bit 238 = 1 - 262144
bit 237 = 0 -
bit 236 = 1 - 1048576
bit 235 = 1 - 2097152
bit 234 = 0 -
bit 233 = 1 - 8388608
bit 232 = 0 -
bit 231 = 1 - 33554432
bit 230 = 1 - 67108864
bit 229 = 1 - 134217728
bit 228 = 0 -
bit 227 = 0 -
bit 226 = 1 - 1073741824
bit 225 = 0 -
bit 224 = 0 -
bit 223 = 0 -
bit 222 = 1 - 17179869184
bit 221 = 0 -
bit 220 = 1 - 68719476736
bit 219 = 1 - 137438953472
bit 218 = 0 -
bit 217 = 0 -
bit 216 = 0 -
bit 215 = 1 - 2199023255552
bit 214 = 0 -
bit 213 = 0 -
bit 212 = 0 -
bit 211 = 1 - 35184372088832
bit 210 = 1 - 70368744177664
bit 209 = 1 - 140737488355328
bit 208 = 0 -
bit 207 = 1 - 562949953421312
bit 206 = 1 - 1125899906842624
bit 205 = 0 -
bit 204 = 0 -
bit 203 = 1 - 9007199254740992
bit 202 = 0 -
bit 201 = 0 -
bit 200 = 0 -
bit 199 = 0 -
bit 198 = 0 -
bit 197 = 1 - 576460752303423488
bit 196 = 0 -
bit 195 = 1 - 2305843009213693952
bit 194 = 0 -
bit 193 = 1 - 9223372036854775808
bit 192 = 1 - 18446744073709551616
bit 191 = 0 -
bit 190 = 0 -
bit 189 = 1 - 147573952589676412928
bit 188 = 0 -
bit 187 = 0 -
bit 186 = 0 -
bit 185 = 0 -
bit 184 = 0 -
bit 183 = 1 - 9444732965739290427392
bit 182 = 1 - 18889465931478580854784
bit 181 = 0 -
bit 180 = 1 - 75557863725914323419136
bit 179 = 0 -
bit 178 = 0 -
bit 177 = 1 - 604462909807314587353088
bit 176 = 1 - 1208925819614629174706176
bit 175 = 0 -
bit 174 = 0 -
bit 173 = 1 - 9671406556917033397649408
bit 172 = 0 -
bit 171 = 1 - 38685626227668133590597632
bit 170 = 0 -
bit 169 = 0 -
bit 168 = 1 - 309485009821345068724781056
bit 167 = 0 -
bit 166 = 0 -
bit 165 = 1 - 2475880078570760549798248448
bit 164 = 1 - 4951760157141521099596496896
bit 163 = 0 -
bit 162 = 1 - 19807040628566084398385987584
bit 161 = 1 - 39614081257132168796771975168
bit 160 = 0 -
bit 159 = 1 - 158456325028528675187087900672
bit 158 = 0 -
bit 157 = 1 - 633825300114114700748351602688
bit 156 = 0 -
bit 155 = 1 - 2535301200456458802993406410752
bit 154 = 1 - 5070602400912917605986812821504
bit 153 = 1 - 10141204801825835211973625643008
bit 152 = 1 - 20282409603651670423947251286016
bit 151 = 1 - 40564819207303340847894502572032
bit 150 = 0 -
bit 149 = 0 -
bit 148 = 1 - 324518553658426726783156020576256
bit 147 = 0 -
bit 146 = 1 - 1298074214633706907132624082305024
bit 145 = 1 - 2596148429267413814265248164610048
bit 144 = 0 -
bit 143 = 1 - 10384593717069655257060992658440192
bit 142 = 0 -
bit 141 = 1 - 41538374868278621028243970633760768
bit 140 = 0 -
bit 139 = 1 - 166153499473114484112975882535043072
bit 138 = 1 - 332306998946228968225951765070086144
bit 137 = 1 - 664613997892457936451903530140172288
bit 136 = 1 - 1329227995784915872903807060280344576
bit 135 = 1 - 2658455991569831745807614120560689152
bit 134 = 1 - 5316911983139663491615228241121378304
bit 133 = 1 - 10633823966279326983230456482242756608
bit 132 = 1 - 21267647932558653966460912964485513216
bit 131 = 0 -
bit 130 = 1 - 85070591730234615865843651857942052864
bit 129 = 0 -
bit 128 = 0 -
bit 127 = 1 - 680564733841876926926749214863536422912
bit 126 = 1 - 1361129467683753853853498429727072845824
bit 125 = 0 -
bit 124 = 1 - 5444517870735015415413993718908291383296
bit 123 = 0 -
bit 122 = 1 - 21778071482940061661655974875633165533184
bit 121 = 1 - 43556142965880123323311949751266331066368
bit 120 = 1 - 87112285931760246646623899502532662132736
bit 119 = 0 -
bit 118 = 1 - 348449143727040986586495598010130648530944
bit 117 = 0 -
bit 116 = 0 -
bit 115 = 0 -
bit 114 = 1 - 5575186299632655785383929568162090376495104
bit 113 = 0 -
bit 112 = 1 - 22300745198530623141535718272648361505980416
bit 111 = 1 - 44601490397061246283071436545296723011960832
bit 110 = 0 -
bit 109 = 0 -
bit 108 = 1 - 356811923176489970264571492362373784095686656
bit 107 = 0 -
bit 106 = 1 - 1427247692705959881058285969449495136382746624
bit 105 = 1 - 2854495385411919762116571938898990272765493248
bit 104 = 0 -
bit 103 = 1 - 11417981541647679048466287755595961091061972992
bit 102 = 0 -
bit 101 = 1 - 45671926166590716193865151022383844364247891968
bit 100 = 1 - 91343852333181432387730302044767688728495783936
bit 99 = 0 -
bit 98 = 1 - 365375409332725729550921208179070754913983135744
bit 97 = 1 - 730750818665451459101842416358141509827966271488
bit 96 = 1 - 1461501637330902918203684832716283019655932542976
bit 95 = 0 -
bit 94 = 1 - 5846006549323611672814739330865132078623730171904
bit 93 = 1 - 11692013098647223345629478661730264157247460343808
bit 92 = 0 -
bit 91 = 1 - 46768052394588893382517914646921056628989841375232
bit 90 = 0 -
bit 89 = 0 -
bit 88 = 0 -
bit 87 = 0 -
bit 86 = 0 -
bit 85 = 0 -
bit 84 = 1 - 5986310706507378352962293074805895248510699696029696
bit 83 = 1 - 11972621413014756705924586149611790497021399392059392
bit 82 = 0 -
bit 81 = 0 -
bit 80 = 0 -
bit 79 = 1 - 191561942608236107294793378393788647952342390272950272
bit 78 = 1 - 383123885216472214589586756787577295904684780545900544
bit 77 = 0 -
bit 76 = 1 - 1532495540865888858358347027150309183618739122183602176
bit 75 = 0 -
bit 74 = 1 - 6129982163463555433433388108601236734474956488734408704
bit 73 = 0 -
bit 72 = 0 -
bit 71 = 1 - 49039857307708443467467104868809893875799651909875269632
bit 70 = 1 - 98079714615416886934934209737619787751599303819750539264
bit 69 = 1 - 196159429230833773869868419475239575503198607639501078528
bit 68 = 0 -
bit 67 = 1 - 784637716923335095479473677900958302012794430558004314112
bit 66 = 0 -
bit 65 = 0 -
bit 64 = 1 - 6277101735386680763835789423207666416102355444464034512896
bit 63 = 1 - 12554203470773361527671578846415332832204710888928069025792
bit 62 = 1 - 25108406941546723055343157692830665664409421777856138051584
bit 61 = 0 -
bit 60 = 0 -
bit 59 = 0 -
bit 58 = 1 - 401734511064747568885490523085290650630550748445698208825344
bit 57 = 1 - 803469022129495137770981046170581301261101496891396417650688
bit 56 = 1 - 1606938044258990275541962092341162602522202993782792835301376
bit 55 = 0 -
bit 54 = 1 - 6427752177035961102167848369364650410088811975131171341205504
bit 53 = 1 - 12855504354071922204335696738729300820177623950262342682411008
bit 52 = 0 -
bit 51 = 0 -
bit 50 = 0 -
bit 49 = 1 - 205688069665150755269371147819668813122841983204197482918576128
bit 48 = 1 - 411376139330301510538742295639337626245683966408394965837152256
bit 47 = 1 - 822752278660603021077484591278675252491367932816789931674304512
bit 46 = 1 - 1645504557321206042154969182557350504982735865633579863348609024
bit 45 = 1 - 3291009114642412084309938365114701009965471731267159726697218048
bit 44 = 1 - 6582018229284824168619876730229402019930943462534319453394436096
bit 43 = 0 -
bit 42 = 0 -
bit 41 = 0 -
bit 40 = 1 - 105312291668557186697918027683670432318895095400549111254310977536
bit 39 = 0 -
bit 38 = 0 -
bit 37 = 1 - 842498333348457493583344221469363458551160763204392890034487820288
bit 36 = 0 -
bit 35 = 1 - 3369993333393829974333376885877453834204643052817571560137951281152
bit 34 = 1 - 6739986666787659948666753771754907668409286105635143120275902562304
bit 33 = 0 -
bit 32 = 0 -
bit 31 = 1 - 53919893334301279589334030174039261347274288845081144962207220498432
bit 30 = 1 - 107839786668602559178668060348078522694548577690162289924414440996864
bit 29 = 0 -
bit 28 = 1 - 431359146674410236714672241392314090778194310760649159697657763987456
bit 27 = 1 - 862718293348820473429344482784628181556388621521298319395315527974912
bit 26 = 1 - 1725436586697640946858688965569256363112777243042596638790631055949824
bit 25 = 1 - 3450873173395281893717377931138512726225554486085193277581262111899648
bit 24 = 1 - 6901746346790563787434755862277025452451108972170386555162524223799296
bit 23 = 0 -
bit 22 = 1 - 27606985387162255149739023449108101809804435888681546220650096895197184
bit 21 = 1 - 55213970774324510299478046898216203619608871777363092441300193790394368
bit 20 = 0 -
bit 19 = 1 - 220855883097298041197912187592864814478435487109452369765200775161577472
bit 18 = 1 - 441711766194596082395824375185729628956870974218904739530401550323154944
bit 17 = 0 -
bit 16 = 1 - 1766847064778384329583297500742918515827483896875618958121606201292619776
bit 15 = 0 -
bit 14 = 1 - 7067388259113537318333190002971674063309935587502475832486424805170479104
bit 13 = 0 -
bit 12 = 0 -
bit 11 = 0 -
bit 10 = 0 -
bit 9 = 0 -
bit 8 = 0 -
bit 7 = 0 -
bit 6 = 1 - 1809251394333065553493296640760748560207343510400633813116524750123642650624
bit 5 = 0 -
bit 4 = 1 - 7237005577332262213973186563042994240829374041602535252466099000494570602496
bit 3 = 1 - 14474011154664524427946373126085988481658748083205070504932198000989141204992
bit 2 = 0 -
bit 1 = 1 - 57896044618658097711785492504343953926634992332820282019728792003956564819968

We now add up all these numbers, and this gives us a total of privateKeyBase10 = 81425905913881293233417886915456929825803636934140198496261481046664027716260 in BASE10
Or a total of privateKeyBase16 = b4056df6691f8dc72e56302ddad345d65fead3ead9299609a826e2344eb63aa4 in BASE16, which is the hexadecimal private key.



This is the private key number in a format that few people are familiar with but it can also be represented in various WIF Formats depending on the compression parameter:

WIF: 5KBZytdkzttdedEB7xLom6YnHRCVTNQPUrJiTDJKS7VJUbiX6Di  // uncompressed
WIF: L3Fea6uFCY2tm4u9fhGRPdVGsToibPUEDvdBxhHSMqdv8odQMTAZ // compressed

As most of us know the full uncompressed public key will start with prefix '04' and have both the X and Y coordinates.
The compressed public key only holds the X coordinate, and it can be odd, or even.
Public keys of the even type start with a prefix '2' while public keys of the odd type start with a prefix '03'
You can discover, or calculate the key type of the compressed public key by looking at the least significant bit of the Y coordinate of the public key.

It's important to note that this is not regular multiplication but epileptic curve multiplication.
More about the algorithm here: https://paulmillr.com/posts/noble-secp256k1-fast-ecc/

[MrFreeDragon]

-snip-
Now i would like to understand how to do for private key 3
Can someone do a tutorial like I did IN DECIMAL OF PRIVATE KEY 3

Formula for point doubling is:

def mul2(Pmul2, p = modulo):
    R = Point(0,0)
    c = 3*Pmul2.x*Pmul2.x*gmpy2.invert(2*Pmul2.y, p) % p
    R.x = (c*c-2*Pmul2.x) % p
    R.y = (c*(Pmul2.x - R.x)-Pmul2.y) % p
    return R

Formula for points addition is (used for different points only):

def add(Padd, Q, p = modulo):
    R = Point()
    dx = (Q.x - Padd.x) % p
    dy = (Q.y - Padd.y) % p
    c = dy * gmpy2.invert(dx, p) % p     
    R.x = (c*c - Padd.x - Q.x) % p
    R.y = (c*(Padd.x - R.x) - Padd.y) % p
    return R

gmpy2.invert here is the inverse calculation.

Now you can easily calculate public key for private number 2 - just double the basis point G. In order to calculate the private key for private number 3 you can apply addition for two points - point receive in first stage (doubling of G) and G (because 3 = 1*2 + 1, where 1 is the basis point G):

Code:

>>> PG.x, PG.y
(55066263022277343669578718895168534326250603453777594175500187360389116729240, 32670510020758816978083085130507043184471273380659243275938904335757337482424)
>>> P2=mul2(PG)
>>> int(P2.x), int(P2.y)
(89565891926547004231252920425935692360644145829622209833684329913297188986597, 12158399299693830322967808612713398636155367887041628176798871954788371653930)
>>> P3=add(P2,PG)
>>> int(P3.x), int(P3.y)
(112711660439710606056748659173929673102114977341539408544630613555209775888121, 25583027980570883691656905877401976406448868254816295069919888960541586679410)

[ricardosuperx]

That's what I wanted! Could you make private key 3?

We must not forget that there are two formulas
The one you used to find the point corresponding to the private key: 2 (or rather 0000000000000000000000000000000000000000000000000000000000000002 to be more precise) it is Duplication of points.
You did it in your example with the base point--> 1 + 1 =2  (but that could be another point)
To go now with 3 we need to do --> 2 + 1 = 3 (we have now 2 différents points and we can't doubling them)

For that you need to use the second formula:

modulo = 115792089237316195423570985008687907853269984665640564039457584007908834671663
Px = 89565891926547004231252920425935692360644145829622209833684329913297188986597 (x coordinate point 2)
Py = 12158399299693830322967808612713398636155367887041628176798871954788371653930 (y coordinate point 2)
Qx = 55066263022277343669578718895168534326250603453777594175500187360389116729240 (x coordinate point 1) not because it is the base point, just because it is the point n°1
Qy = 32670510020758816978083085130507043184471273380659243275938904335757337482424 (y coordinate point 1)

dx = (Qx - Px) % modulo             --> 34499628904269660561674201530767158034393542375844615658184142552908072257357
dy = (Qy - Py) % modulo             --> 95279978516251208768455708490894263304954079172022948940317551626939868843169
c = dy * invert(dx) % modulo       --> 23578750110654438173404407907450265080473019639451825850605815020978465167024
Rx = (c*c - Px - Qx) % modulo     --> 112711660439710606056748659173929673102114977341539408544630613555209775888121 (x coordinate of point (2+1 =3)
Ry = (c*(Px - Rx) - Py) % modulo --> 25583027980570883691656905877401976406448868254816295069919888960541586679410   (y coordinate of point (2+1 =3)

Can't explain better

Order: 115792089237316195423570985008687907852837564279074904382605163141518161494337

Base Point: (55066263022277343669578718895168534326250603453777594175500187360389116729240, 32670510020758816978083085130507043184471273380659243275938904335757337482424)

Modulo: 115792089237316195423570985008687907853269984665640564039457584007908834671663

Private key:  0000000000000000000000000000000000000000000000000000000000000003

Point addition:(2G+1G =3G)
Compressed public key;
In decimal;
Equation:

c = (qy – py) / (qx – px)
rx = c^2 – px – qx

px = 89565891926547004231252920425935692360644145829622209833684329913297188986597
py = 12158399299693830322967808612713398636155367887041628176798871954788371653930
qx = 55066263022277343669578718895168534326250603453777594175500187360389116729240
qy = 32670510020758816978083085130507043184471273380659243275938904335757337482424

c = 32670510020758816978083085130507043184471273380659243275938904335757337482424-
12158399299693830322967808612713398636155367887041628176798871954788371653930=
20512110721064986655115276517793644548315905493617615099140032380968965828494


c = 55066263022277343669578718895168534326250603453777594175500187360389116729240-
89565891926547004231252920425935692360644145829622209833684329913297188986597=
81292460333046534861896783477920749818876442289795948381273441455000762414306


c = 20512110721064986655115276517793644548315905493617615099140032380968965828494/
81292460333046534861896783477920749818876442289795948381273441455000762414306=
23578750110654438173404407907450265080473019639451825850605815020978465167024


c = 23578750110654438173404407907450265080473019639451825850605815020978465167024


rx = 23578750110654438173404407907450265080473019639451825850605815020978465167024^2 -
89565891926547004231252920425935692360644145829622209833684329913297188986597-
55066263022277343669578718895168534326250603453777594175500187360389116729240

rx = 25759636913902563110438328477658084082469757293658084474899962813078412260632-
34499628904269660561674201530767158034393542375844615658184142552908072257357

rx = 107052097246949097972335111955578833901346199583454032856173404268079174674938


rx hex = 0xecad56ff86123a68f47514b195ab6837ba69c1fbdf0beb6339e6f3caead069fa


Now I will try to calculate the private key 0000000000000000000000000000000000000000000000000000000000000005

[BASE16]

Private key 3 is 3 * G

Add the Generator 3 times.

0x02F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9

Point 1 {x: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
            y: 32670510020758816978083085130507043184471273380659243275938904335757337482424n}

X:  0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
Y:  0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

Point 2 {x: 89565891926547004231252920425935692360644145829622209833684329913297188986597n,
            y: 12158399299693830322967808612713398636155367887041628176798871954788371653930n}

X:  0xc6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
Y:  0x1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a

Point 3 {x: 112711660439710606056748659173929673102114977341539408544630613555209775888121n,
            y: 25583027980570883691656905877401976406448868254816295069919888960541586679410n}

X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

[ricardosuperx]

Private key 3 is 3 * G

Add the Generator 3 times.

0x02F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9

Point 1 {x: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
            y: 32670510020758816978083085130507043184471273380659243275938904335757337482424n}

X:  0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
Y:  0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

Point 2 {x: 89565891926547004231252920425935692360644145829622209833684329913297188986597n,
            y: 12158399299693830322967808612713398636155367887041628176798871954788371653930n}

X:  0xc6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
Y:  0x1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a

Point 3 {x: 112711660439710606056748659173929673102114977341539408544630613555209775888121n,
            y: 25583027980570883691656905877401976406448868254816295069919888960541586679410n}

X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

Corrected, thanks!
 
X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

EDITED:THIS IS THE KEY PUBLIC OF THE PRIVATE KEY
0000000000000000000000000000000000000000000000000000000000000003

[BASE16]

Private key 3 is 3 * G

Add the Generator 3 times.

0x02F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9

Point 1 {x: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
            y: 32670510020758816978083085130507043184471273380659243275938904335757337482424n}

X:  0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
Y:  0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

Point 2 {x: 89565891926547004231252920425935692360644145829622209833684329913297188986597n,
            y: 12158399299693830322967808612713398636155367887041628176798871954788371653930n}

X:  0xc6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
Y:  0x1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a

Point 3 {x: 112711660439710606056748659173929673102114977341539408544630613555209775888121n,
            y: 25583027980570883691656905877401976406448868254816295069919888960541586679410n}

X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

Corrected, thanks!
Now What do I do with:
X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672
?

What do you want to do with it ?
You can turn it into uncompressed public key:

publicKey = '04' + X + Y
publicKey = 04f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9388f7b0f632de 8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

Or turn it into a even compressed public key:
publicKey = '02' + X
publicKey = 02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9

Or turn it into a odd compressed public key:
publicKey = '03' + X
publicKey = 03f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9

One of the last two is invalid.
You have to look at the least significant bit of Y to see if its odd or even.

Y = 0x388F7B0F632DE8140FE337E62A37F3566500A99934C2231B6CB9FD7584B8E672
Y = 0b11100010001111011110110000111101100011001011011110100000010100000011111110001 1001101111110011000101010001101111111001101010110011001010000000010101001100110 0100110100110000100010001100011011011011001011100111111101011101011000010010111 0001110011001110010
The last bit is even so it's '02'

You can also convert it to WIF:

5HpHagT65TZzG1PH3CSu63k8DbpvD8s5ip4nEB3kEsreB1FQ8BZ uncompressed

KwDiBf89QgGbjEhKnhXJuH7LrciVrZi3qYjgd9M7rFU74sHUHy8S compressed

And you can calculate various addresses but that is far past private to pubkey  

[ricardosuperx]

Private key 3 is 3 * G

Add the Generator 3 times.

0x02F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9

Point 1 {x: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
            y: 32670510020758816978083085130507043184471273380659243275938904335757337482424n}

X:  0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
Y:  0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8

Point 2 {x: 89565891926547004231252920425935692360644145829622209833684329913297188986597n,
            y: 12158399299693830322967808612713398636155367887041628176798871954788371653930n}

X:  0xc6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5
Y:  0x1ae168fea63dc339a3c58419466ceaeef7f632653266d0e1236431a950cfe52a

Point 3 {x: 112711660439710606056748659173929673102114977341539408544630613555209775888121n,
            y: 25583027980570883691656905877401976406448868254816295069919888960541586679410n}

X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

Corrected, thanks!
Now What do I do with:
X:  0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
Y:  0x388f7b0f632de8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672
?

What do you want to do with it ?
You can turn it into uncompressed public key:

publicKey = '04' + X + Y
publicKey = 04f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9388f7b0f632de 8140fe337e62a37f3566500a99934c2231b6cb9fd7584b8e672

Or turn it into a even compressed public key:
publicKey = '02' + X
publicKey = 02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9

Or turn it into a odd compressed public key:
publicKey = '03' + X
publicKey = 03f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9

One of the last two is invalid.
You have to look at the least significant bit of Y to see if its odd or even.

Y = 0x388F7B0F632DE8140FE337E62A37F3566500A99934C2231B6CB9FD7584B8E672
Y = 0b11100010001111011110110000111101100011001011011110100000010100000011111110001 1001101111110011000101010001101111111001101010110011001010000000010101001100110 0100110100110000100010001100011011011011001011100111111101011101011000010010111 0001110011001110010
The last bit is even so it's '02'

You can also convert it to WIF:

5HpHagT65TZzG1PH3CSu63k8DbpvD8s5ip4nEB3kEsreB1FQ8BZ uncompressed

KwDiBf89QgGbjEhKnhXJuH7LrciVrZi3qYjgd9M7rFU74sHUHy8S compressed

And you can calculate various addresses but that is far past private to pubkey  

SORRY! I convert in decimal: 02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9,
instead of: 0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
 
Now I will try to calculate the private key 0000000000000000000000000000000000000000000000000000000000000005

[BASE16]

SORRY! I convert in decimal: 02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9,
instead of: 0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
 

It's both the same in BASE16.
The first one is public key with prefix ready for processing while the second has 0x prefix to tell that it's hexadecimal but you can tell that easily by looking at the symbols.

You can easily convert them if you want.
Just use whatever works best for you 

I still don't know what you are trying to do exactly.

[ricardosuperx]

SORRY! I convert in decimal: 02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9,
instead of: 0xf9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9
 

It's both the same in BASE16.
The first one is public key with prefix ready for processing while the second has 0x prefix to tell that it's hexadecimal but you can tell that easily by looking at the symbols.

You can easily convert them if you want.
Just use whatever works best for you 

I still don't know what you are trying to do exactly.

I'm just trying to learn the math behind bitcoin! I was always curious to understand how public keys are created and how Bitcoin works etc ... I was able to understand how to double points and now I understand  addition points
I'm feeling like a SATOSHI NAKAMOTO LOL

[bytcoin]

If the formula is always the same, All public keys will also always be the same! "Where am I wrong?

(dx, dy, c, R.x, R.y, Q.x Q.y, P.x, P.y)Can someone explain to me what are this?

I think Rx and Ry are the coordinates of the public key, right?

ok, I will try to explain as simply as possible because it is true that I myself struggled to understand the system.
So it is a question here of adding 2 points (not the same). In this example we use :

first point: (all values are in décimal for a better comprehension)

X coordinate: 21262057306151627953595685090280431278183829487175876377991189246716355947009 (it is Qx)
Y coordinate: 41749993296225487051377864631615517161996906063147759678534462689479575333124 (it is Qy)
The Private key for this point is 0000000000000000000000000000000000000000000000000000000000000008

second point to add:

X coordinate: 89565891926547004231252920425935692360644145829622209833684329913297188986597 (it is Px)
Y coordinate: 12158399299693830322967808612713398636155367887041628176798871954788371653930 (it is Py)
The Private key for this point is 0000000000000000000000000000000000000000000000000000000000000002

So to add the 1st point to the second we use the formula : (here modulo is 115792089237316195423570985008687907853269984665640564039457584007908834671663 )

dx = (Q.x - P.x) % modulo
dy = (Q.y - P.y) % modulo
c = dy * invert(dx) % modulo
R.x = (c*c - P.x - Q.x) % modulo
R.y = (c*(P.x - R.x) - P.y) % modulo

in our example we have

dx = (21262057306151627953595685090280431278183829487175876377991189246716355947009 - 89565891926547004231252920425935692360644145829622209833684329913297188986597) % modulo
dx = 47488254616920819145913749673032646770809668323194230583764443341328001632075

dy = (41749993296225487051377864631615517161996906063147759678534462689479575333124 - 12158399299693830322967808612713398636155367887041628176798871954788371653930) % modulo
dy = 29591593996531656728410056018902118525841538176106131501735590734691203679194

invert of dx = 70279122268919195963430815486314537773961171454828771794853116552210630553734

c = dy * invert(dx) % modulo
c = 16132032934385503768504319366562120314980927452732756733183380715276156205226

So the new point (8 + 2)

R.x = (c*c - P.x - Q.x) % modulo
R.x --> X coordinate of (8+2) = 72488970228380509287422715226575535698893157273063074627791787432852706183111

R.y = (c*(P.x - R.x) - P.y) % modulo
R.y --> Y coordinate of (8+2) = 62070622898698443831883535403436258712770888294397026493185421712108624767191

If we check these coordinates, we find that it corresponds to the private key: 000000000000000000000000000000000000000000000000000000000000000a (10)

There you go, I hope I was as clear as possible and apologies for my broken English ^^





[/quote]

Good explanation ... I understood perfectly