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 Predict(P, Q, byte, p, b, curve, r, d):

    result = [];
    req = (byte/30).ceil();
    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)))):
                    r = 0;
                    for j in range(req):
                        s_cand = (s_cand*P)[0].lift();
                        r_cand = (s_cand*Q)[0].lift();
                        r_cand = r_cand & bitmask;
                        r = (r << (30*8)) | r_cand
                    r = r >> ((30*req - byte)*8)
                    result.append(r);
            except:
                continue;
    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 "r = ", r;

%time rp = Predict(P, Q, 120, p, b, curve, r, d);
for i in range(len(rp)):
    print "rp = ", hex(rp[i]);

r = Random_Generator(P, Q, 60);
print "r = ", hex(r);

r = Random_Generator(P, Q, 60);
print "r = ", 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) r = 77520781625289714994182336106159725694469664721153374789457932012102615587015 CPU time: 884.02 s, Wall time: 1756.95 s rp = 253a5e4cbfa9d0f1a4e188b55802909c70876893d3af49cf824003dfb63617e722c0fd85515ee0133c69ed739af0a1533037cbf73374058135b704d446ff2a9e98a8b64c69b65c633998f8b42a421c50cc95d461c978c8117caf35a435aa806dad260e4189f54739c9acd6028b7bc68f73d3101296b5c04e rp = 24f0a075580d6649084c5e7668510b7e7e44af52bbb447371dc2d4277c0eb2f9cd87488e516629af63cc507f67b9a6915ec250e4f3ea93e85b8684e5279eb4b3840cc61203d6b502bfdd701a5c8d37477faee34d526d43ef44a223de8e34372b1b02dbf4cf30e5d002d4d91080fec3a031e182b5d5989498 r = 253a5e4cbfa9d0f1a4e188b55802909c70876893d3af49cf824003dfb63617e722c0fd85515ee0133c69ed739af0a1533037cbf73374058135b704d4 r = 46ff2a9e98a8b64c69b65c633998f8b42a421c50cc95d461c978c8117caf35a435aa806dad260e4189f54739c9acd6028b7bc68f73d3101296b5c04e