EllipticCurvesPart5-TheMagic-SigningAndVerifying.py

# Python 2.7.6 - Super simple Elliptic Curve Presentation. No imported libraries, wrappers, nothing. # For educational purposes only
# Below are the public specs for Bitcoin's curve - the secp256k1
Pcurve = 2**256 - 2**32 - 2**9 - 2**8 - 2**7 - 2**6 - 2**4 -1 # The proven prime
N=0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 # Number of points in the field
Acurve = 0; Bcurve = 7 # This defines the curve. y^2 = x^3 + Acurve * x + Bcurve
Gx = 55066263022277343669578718895168534326250603453777594175500187360389116729240
Gy = 32670510020758816978083085130507043184471273380659243275938904335757337482424
GPoint = (Gx,Gy) # This is our generator point. Tillions of dif ones possible

#Individual Transaction/Personal Information
privKey = 75263518707598184987916378021939673586055614731957507592904438851787542395619 #replace with any private key
RandNum = 28695618543805844332113829720373285210420739438570883203839696518176414791234 #replace with a truly random number
HashOfThingToSign = 86032112319101611046176971828093669637772856272773459297323797145286374828050 # the hash of your message/transaction

def modinv(a,n=Pcurve): #Extended Euclidean Algorithm/'division' in elliptic curves
    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(xp,yp,xq,yq): # Not true addition, invented for EC. It adds Point-P with Point-Q.
    m = ((yq-yp) * modinv(xq-xp,Pcurve)) % Pcurve
    xr = (m*m-xp-xq) % Pcurve
    yr = (m*(xp-xr)-yp) % Pcurve
    return (xr,yr)

def ECdouble(xp,yp): # EC point doubling,  invented for EC. It doubles Point-P.
    LamNumer = 3*xp*xp+Acurve
    LamDenom = 2*yp
    Lam = (LamNumer * modinv(LamDenom,Pcurve)) % Pcurve
    xr = (Lam*Lam-2*xp) % Pcurve
    yr = (Lam*(xp-xr)-yp) % Pcurve
    return (xr,yr)

def EccMultiply(xs,ys,Scalar): # Double & add. EC Multiplication, Not true multiplication
    if Scalar == 0 or Scalar >= N: raise Exception("Invalid Scalar/Private Key")
    ScalarBin = str(bin(Scalar))[2:]
    Qx,Qy=xs,ys
    for i in range (1, len(ScalarBin)): # This is invented EC multiplication.
        Qx,Qy=ECdouble(Qx,Qy); # print "DUB", Qx; print
        if ScalarBin[i] == "1":
            Qx,Qy=ECadd(Qx,Qy,xs,ys); # print "ADD", Qx; print
    return (Qx,Qy)

print; print "******* Public Key Generation *********"
xPublicKey, yPublicKey = EccMultiply(Gx,Gy,privKey)
print "the private key (in base 10 format):"; print privKey; print
print "the uncompressed public key (starts with '04' & is not the public address):"; print "04",xPublicKey,yPublicKey

print; print "******* Signature Generation *********"
xRandSignPoint, yRandSignPoint = EccMultiply(Gx,Gy,RandNum)
r = xRandSignPoint % N; print "r =", r
s = ((HashOfThingToSign + r*privKey)*(modinv(RandNum,N))) % N; print "s =", s

print; print "******* Signature Verification *********>>"
w = modinv(s,N)
xu1, yu1 = EccMultiply(Gx,Gy,(HashOfThingToSign * w)%N)
xu2, yu2 = EccMultiply(xPublicKey,yPublicKey,(r*w)%N)
x,y = ECadd(xu1,yu1,xu2,yu2)
print r==x; print