For example, I would like to use N/2x as the Generator base. so, Gx=(N/2)x ,2P= Gx,  3P=.....,  would and so on.

PVK1+PVK2+PVK3 =N
PVK4+PVK5+PVK6 = 2N
x+x1+x2= P ya 2P
y+y1= P
vase hame private key ha in group shekl migire. ekhtelaf beyn kole in ablien group ya jameshono mishe hesab kard. in sabet mikone curve constant hast. hala az samt N to P ye constant hast ke mesl bridge N to P map mikone.
ye constant hast mesl lmda va beta k N be P vasl mikone . shah kelid curve.

in taze 5% chizayi hast k mitonam bet inja begam . man omidvar boodam betoonam ba shoma ya pooya87 private sohbat konam . fekr mikardam yeki inja komakam kone . hamaro namitonam inja bezaram. plz understand.

movafegham , mishasam kasayi k halesh karde bashan . ama fesad male on admayi hast k bedon zahmat ye chiz peyda mikonan . kasayi k sal ha tool keshide k ino peyda kardan wise mishan automatedly . man khodam joeat nadaram inaro jayi begam ama majbooram kardi inja begam, loool.

پابلیک کی ۲، ۴ ۶ در واقع نماینده عدد ۲ به توان ۲۵۵ در منحنی بیت کوین هستن چه چیز جالبی در این مورد وجود داره؟
ini k alan gofti ye hint bozorg bood vasam. 2^255 k nesf curve .1 ta N/2.  plz bishtar begoo manzooret chiye az namayande ?

aval sabet kon Ke  P=N.

any value^(N-1)%N =1
any value^(P-1)%P =1

P=N yani 1=1

cube root 1 = 1 (cube root har addadi = ba khode on addad) , P and N prime hastan.

cube root N va P hesab kon:
vase N :
3^(N-1)/3 % N =lmda
2^(N-1)/3 % N =lmda2

 vase P :
3^(P-1)/3 % P =beta
2^(P-1)/3 % P =beta

python:

print pow (3,(N-1)/3,N) = lmda
print pow (2,(N-1)/3,N) = lmda2
lmda+lmda2=N-1
print pow (2,(P-1)/3,P) = beta
print pow (3,(P-1)/3,P) = beta2
lmda+lmda2=N-1
beta+beta2=P-1

inja shoma sabet mikone Ke P=N . (  اما وقتی با اسکالار کار کنید متوجه میشید که اصلا پی هیچ تاثیری در محاسبات شما نداره یا به عبارتی دیگر ، پی وجود خارجی نداره در معادلات مربوط به محاسبه پرایوت کی)
daram mohasebato ba addad va algorithm neshonet midam.

har scalar k eshghet mikeshe entekhab kon az 1 ta N-1 .

scalaro k dari x coordinate ham jolosh bezar.

print (scalar*lmda) % N = scalar2  ,,,,,,,,, az N map mishe to  P,,,,,,,, print (x*beta)% P =x2

print (scalar 2 * lmda) % N = scalar3  ,,,,,,,,, az N map mishe to  P,,,,,,,,  print (x2*beta)% P =x3

print (scalar 3 * lmda) % N = scalar  ,,,,,,,,, az N map mishe to  P,,,,,,,,  print (x2*beta)% P =x

scalar = x
scalar2 =x2
scalar3 =x3

hamin lotfan to python emtehan bokon , man codesho dishab gozashtam, aval khodet emtehan bokon, va rabete ro bebin . man bishtar tozih namidam ta ghati nakoni . ta alan motevajeh beshi k vojood khareji dare . shoma migi nadare chon emtehan nakardi ya baghiye k migan nadare ro ghabool dari.
try it.

injoori fekr nakon , kheyli ha hastan k curvo shekondan faghat publish namishe. man chizayi k migam ba example hastan theory nistan . faghat yekam komak mikham . age soali dashti begoo ba example bet begam.


code payino to python 2 run kon bebin .ablein groups 6 set of pk ,3x ,2y






pvk_hex = 40360748372915677739060455046407153445907911874643188801639465548522 # input privatekey

# 6 Pubkeys are
# Pubkey = [x,y]  [x*beta%p, y]  [x*beta2%p, y] [x,p-y]  [x*beta%p, p-y]  [x*beta2%p, p-y]

# 6 Privatekeys are
# pvk, pvk*lmda%N, pvk*lmda2%N, N-pvk, N-pvk*lmda%N, N-pvk*lmda2%N

# Field parameters
Pcurve = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
N = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141

# Constants Based on Cube root of 1
beta = 0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee
beta2 = 0x851695d49a83f8ef919bb86153cbcb16630fb68aed0a766a3ec693d68e6afa40  # beta*beta
lmda = 0x5363ad4cc05c30e0a5261c028812645a122e22ea20816678df02967c1b23bd72
lmda2 = 0xac9c52b33fa3cf1f5ad9e3fd77ed9ba4a880b9fc8ec739c2e0cfc810b51283ce  # lmda*lmda

Acurve = 0
Bcurve = 7  # These two define the elliptic curve. y^2 = x^3 + Acurve * x + Bcurve
Gx = 55066263022277343669578718895168534326250603453777594175500187360389116729240  # generator point x
Gy = 32670510020758816978083085130507043184471273380659243275938904335757337482424  # generator point y
GPoint = (Gx, Gy)

def OnCurve(x, y):
    A = (y * y) % Pcurve
    B = (x * x * x + Acurve * x + Bcurve) % Pcurve
    return A == B

def legendre_symbol(a, p):
    ls = pow(a, (p - 1) // 2, p)
    return -1 if ls == p - 1 else ls

def modsqrt(a, p):
    if legendre_symbol(a, p) != 1:
        return 0
    elif a == 0:
        return 0
    elif p == 2:
        return p
    elif p % 4 == 3:
        return pow(a, (p + 1) // 4, p)
    s = p - 1
    e = 0
    while s % 2 == 0:
        s //= 2
        e += 1
    n = 2
    while legendre_symbol(n, p) != -1:
        n += 1
    x = pow(a, (s + 1) // 2, p)
    b = pow(a, s, p)
    g = pow(n, s, p)
    r = e
    while True:
        t = b
        m = 0
        for m in range(r):
            if t == 1:
                break
            t = pow(t, 2, p)
        if m == 0:
            return x
        gs = pow(g, 2 ** (r - m - 1), p)
        g = (gs * gs) % p
        x = (x * gs) % p
        b = (b * g) % p
        r = m

def modinv(a, n):
    lm, hm = 1, 0
    low, high = a % n, n
    while low > 1:
        ratio = high // low
        nm, new = hm - lm * ratio, high - low * ratio
        lm, low, hm, high = nm, new, lm, low
    return lm % n

def ECadd(a, b):
    LamAdd = ((b[1] - a[1]) * modinv(b[0] - a[0], Pcurve)) % Pcurve
    x = (LamAdd * LamAdd - a[0] - b[0]) % Pcurve
    y = (LamAdd * (a[0] - x) - a[1]) % Pcurve
    return (x, y)

def ECdouble(a):
    Lam = ((3 * a[0] * a[0] + Acurve) * modinv(2 * a[1], Pcurve)) % Pcurve
    x = (Lam * Lam - 2 * a[0]) % Pcurve
    y = (Lam * (a[0] - x) - a[1]) % Pcurve
    return (x, y)

def one_to_6privatekey(pvk_hex):
    pvk = int(pvk_hex)
    print('PVK1 :', pvk)
    print('PVK2 :', (pvk * lmda % N))
    print('PVK3 :', (pvk * lmda2 % N))
    print('PVK4 :', (N - pvk))
    print('PVK5 :', (N - pvk * lmda % N))
    print('PVK6 :', (N - pvk * lmda2 % N))
    print('Pubkey1 :', EccMultiply(GPoint, pvk))
    print('Pubkey2 :', EccMultiply(GPoint, pvk * lmda % N))
    print('Pubkey3 :', EccMultiply(GPoint, pvk * lmda2 % N))
    print('Pubkey4 :', EccMultiply(GPoint, N - pvk))
    print('Pubkey5 :', EccMultiply(GPoint, N - pvk * lmda % N))
    print('Pubkey6 :', EccMultiply(GPoint, N - pvk * lmda2 % N))
    print()
    print()
    print('pvk++ :', (N - pvk) + (N - pvk * lmda % N) + (N - pvk * lmda2 % N))
    print('pvk-- :', (N - pvk) - (N - pvk * lmda % N) - (N - pvk * lmda2 % N))
    print('pvk--/2 :', ((N - pvk) - (N - pvk * lmda % N) - (N - pvk * lmda2 % N)) / 2)

def EccMultiply(GenPoint, ScalarHex):
    if ScalarHex == 0 or ScalarHex >= N:
        raise Exception("Invalid Scalar/Private Key")
    ScalarBin = bin(ScalarHex)[2:]
    Q = GenPoint
    for i in range(1, len(ScalarBin)):
        Q = ECdouble(Q)
        if ScalarBin == "1":
            Q = ECadd(Q, GenPoint)
    return Q

# Call the function
one_to_6privatekey(pvk_hex)

bebin be ezay har point in N ye x to P hast. ma vaghti cyclic group curve peyda konim (P+2P.......) ye K hast Ke vaghti P be khodesh add mishe mirese be point to infinity ya 0. in K constant hast va G base mitone harchizi bashe .

shoma vaghti cyclic group curvo peyda kardi , be vasile privatekey k dari mitoni curve trace koni va hame poinato touch koni. Order secp256k bozorge va security ziyadi dare.

https://asecuritysite.com/ecc/ecc_points_mult

age be mesal lmda va beta tavajoh karde bashi , lmda va beta cube root 1 hastan . ye constant dg inja hast k man besh migam ( zeta) ke private key map mikone be X. onja mod (N+P)/2 estefade mishe. bridge N to P.

N+ trace = P in kole land scape curve hast. Privatekey 1 + privatekey2 = N .
mesal k bala zadam y coodinateshono check kon .y+y1 = P

curve nesf hast yani private key 1 va N-1 joftesh 1 x dare.

privatekey * lmda % N =private key 2
privatekey2 * lmda % N =private key 3
private key 3  * lmda % N = privatekey


vase x.
x*beta% P=x1
x1*beta% P=x2
x2*beta% P=x

inja ye group shekl migire har private key be x khodesh to p map mishe .

lmda=37718080363155996902926221483475020450927657555482586988616620542887997980018
beta =55594575648329892869085402983802832744385952214688224221778511981742606582254

farz kon private key 1 :
1* lmda % N =37718080363155996902926221483475020450927657555482586988616620542887997980018
         
37718080363155996902926221483475020450927657555482586988616620542887997980018* lmda % N =78074008874160198520644763525212887401909906723592317393988542598630163514318

78074008874160198520644763525212887401909906723592317393988542598630163514318  * lmda % N = 1

x= 55066263022277343669578718895168534326250603453777594175500187360389116729240
.

55066263022277343669578718895168534326250603453777594175500187360389116729240*beta% P=85340279321737800624759429340272274763154997815782306132637707972559913914315

85340279321737800624759429340272274763154997815782306132637707972559913914315*beta% P=91177636130617246552803821781935006617134368061721227770777272682868638699771

91177636130617246552803821781935006617134368061721227770777272682868638699771*beta% P=x

3 ta private key k sum up to N or 2N  , 3ta x k sum up to P or 2P.

emtehan kon har addad k to N hast private key to P hast . lmda va beta constent hastan . vase hame curve kar mikone.

hala man mikham N to P vasl konam ba mod (N+P)/2

yani (privatekey * constant)%(N+P)/2 = x

55066264608740580925902835263552626277158644419429811251322144102771898930868,
35000000000000000000000000000007170096612204497355648312511576682020357129521

55066263631910704684474776255832186390255162907064863539845550919319586704978
,24000000000000000000000000000008466067031270411703362315978512594550332060655

[55066262766467582071571635661170610675903420581719622959945228556230635485737,
25000000000000000000000000000001053356757443811285325568782149486290088691860,

[55066261391271670393551703935422686260596584915928171895725576461530908325110L, 32000000000000000000000000000008585700376150222373625767868339803759359429013

55066261358892075685633026474628047109284635249574981731025630372163025729062,
21000000000000000000000000000000312114129260327938340681542296352939738025415

[55066260368248682851554064236072039061459798623034309182252341704048790061988,
36000000000000000000000000000002759037579743314259767605825403634427767092149

x coordinate , private key counting points

PUZZLE brute force hast . man daram trap door curve migam . algorithm k curve beshkane . P=NP hard.
vase hame chizayi ham k migam proof daram with example . hamash math hast namitoni ba hex befahmi .

bache ha kasi inja mitoone ye hint be man bede vase koblitz curve ya hamoon secp256k .

fasele beyn 2ta x coordinate dakhel range 50*10^77 ta 59*10^77 sabet hast.

mesal :  1 , 55066263022277343669578718895168534326250603453777594175500187360389116729240

privatekey ,55066263022277343669578718895168534326250603453777594175500187360389116729241
privatekey ,55066263022277343669578718895168534326250603453777594175500187360389116729239

fasle 2ta private key sabet hast vase hame addad ha to in range.

endomorphism ha sabet mikone k curve contant hast.
N + trace = P
N1 + N2 = N
P = Y1 +Y2
X coordinate generates Y1 and Y2
2 to 1 rabete beyn N va  X coordinate

Base point privKey = 1
Base point privkey = N -1
base point avaz namishe va faghi namikone Gx koja entekhab konim.
private key va ( N-private key )  to ye point hamo ghat mikonan k mishe x .

private key + ( N-private key ) = N point infinity

vaghti fasele 2 private key x va x+1 dashte basham , mitonam fasle hameye private key to rang ro hesab konam .

curve nesf hast , hameye privatekey az N/2+1 ta N-1  tekrar mishe , mesl ayne .

N to P mapping az addad (N+P)/2 estefade mishe.

lmda+lmda2 =N-1
beta + beta 2 = P-1
P- N = trace-1

kheyli sade hast age kasi chizi midoone k mitoone be man komak kone lotfan bege. man deta daram k harfamo sabet mikone .

do you know the exact release date ?