Dual_EC_DRBG_basic_with_backdoor.sagews

bitmask = (2^(30*8) - 1);

#internal state
s_i = None;

def Dual_EC_DRBG(P, Q, h_adin=0, s_0=None):

    global s_i;

    if(s_0 == None):
        s_0 = int(floor((2^16-1)*random()));

    if(s_i == None):
        s_i = s_0;

    t_i = s_i ^^ h_adin;
    s_i = (t_i*P)[0].lift();
    r_i = (s_i*Q)[0].lift();
    r_i = r_i & bitmask;

    return r_i;

def Random_Generator(P, Q, byte, h_adin=0):

    result = 0;
    req = (byte/30).ceil();

    for i in range(req):
        if(i == 0):
            result = (result << (30*8)) | Dual_EC_DRBG(P, Q, h_adin)
        else:
            result = (result << (30*8)) | Dual_EC_DRBG(P, Q)

    result = result >> ((30*req - byte)*8)

    return result;

def Get_Internal_State(P, Q, p, b, curve, r, d):

    result = [];
    r_1 = r >> (len(hex(r))*4 - 30*8);
    r_2 = r & (2^(len(hex(r))*4 - 30*8) - 1);

    for i in range(2^16):
        mb = i << (30*8);
        x_cand = mb | r_1;
        y = Mod(x_cand^3 - 3*x_cand + b, p);
        if(y.is_square()):
            y_cand = y.sqrt();
            try:
                R = curve(x_cand, y_cand);
                s_cand = (d*R)[0].lift();
                r_cand = (s_cand*Q)[0].lift();
                r_cand = r_cand & bitmask;
                if((hex(r_cand).startswith(hex(r_2))) or (hex(r_2).startswith(hex(r_cand)))):
                    result.append(s_cand);
            except:
                continue;
    return result;

def Predict_Next(P, Q, byte, s_cand, h_adin=0):

    result = 0;
    req = (byte/30).ceil();

    for i in range(req):
        if(i == 0):
            t_cand = s_cand ^^ h_adin;
            s_cand = (t_cand*P)[0].lift();
            r_cand = (s_cand*Q)[0].lift();
            r_cand = r_cand & bitmask;
            result = (result << (30*8)) | r_cand
        else:
            s_cand = (s_cand*P)[0].lift();
            r_cand = (s_cand*Q)[0].lift();
            r_cand = r_cand & bitmask;
            result = (result << (30*8)) | r_cand

    result = result >> ((30*req - byte)*8)

    return result;

#Curve P-256
p = 115792089210356248762697446949407573530086143415290314195533631308867097853951;
n = 115792089210356248762697446949407573529996955224135760342422259061068512044369;
b = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b;
Px = 0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296;
Py = 0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5;
#Qx = 0xc97445f45cdef9f0d3e05e1e585fc297235b82b5be8ff3efca67c59852018192;
#Qy = 0xb28ef557ba31dfcbdd21ac46e2a91e3c304f44cb87058ada2cb815151e610046;

# y^2 = x^3 - 3*x + b (mod p)
curve = EllipticCurve(GF(p), [0, 0, 0, -3, b]);
print curve;

P = curve(Px, Py);
#Q = curve(Qx, Qy);

#Backdoor
d = 5;
order = P.additive_order();
e = inverse_mod(d, order);
#P = d*Q;
Q = e*P;

print "P = ", P;
print "Q = ", Q;

r = Random_Generator(P, Q, 32);
print "r1 = ", r;

%time s = Get_Internal_State(P, Q, p, b, curve, r, d);
for i in range(len(s)):
    print "s = ", hex(s[i]);
    %time rp = Predict_Next(P, Q, 60, s[i]);
    print "predict = ", hex(rp);

r = Random_Generator(P, Q, 30);
print "r2 = ", hex(r);

r = Random_Generator(P, Q, 30);
print "r3 = ", hex(r);

Elliptic Curve defined by y^2 = x^3 + 115792089210356248762697446949407573530086143415290314195533631308867097853948*x + 41058363725152142129326129780047268409114441015993725554835256314039467401291 over Finite Field of size 115792089210356248762697446949407573530086143415290314195533631308867097853951 P = (48439561293906451759052585252797914202762949526041747995844080717082404635286 : 36134250956749795798585127919587881956611106672985015071877198253568414405109 : 1) Q = (100222093819885759857726245131128697024676897724593576735535145416600847521071 : 112705950327624587511154978849178363127000253898669394213565898066000545039919 : 1) r1 = 84951740873355541902580988446661854287990194777557492186266755538146265974056 CPU time: 452.00 s, Wall time: 502.07 s s = 1be9412c483156d40d095d88d77237e18d621cb31d9369d1c9d71a6995185aa7 CPU time: 0.05 s, Wall time: 0.05 s predict = c7f0c01124f8539d1be0456b9a776367b40971776654b7fb119f9a3f938413e2792ab0b77962294ff3e352072d9a11c8160dc9426b86cc43b49b7c18 r2 = c7f0c01124f8539d1be0456b9a776367b40971776654b7fb119f9a3f9384 r3 = 13e2792ab0b77962294ff3e352072d9a11c8160dc9426b86cc43b49b7c18