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# elliptic-shiho/solve_sssa_attack.py

Created February 6, 2016 19:39
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Sharif University CTF 2016: Crypto 350 British Elevator Solver
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 # original file name: solve_sssa_attack.sage from sage.all import * p = 16857450949524777441941817393974784044780411511252189319 A = 16857450949524777441941817393974784044780411507861094535 B = 77986137112576 E = EllipticCurve(GF(p), [A, B]) print E.order() == p g = E(5732560139258194764535999929325388041568732716579308775, 14532336890195013837874850588152996214121327870156054248) v = E(2609506039090139098835068603396546214836589143940493046, 8637771092812212464887027788957801177574860926032421582) def hensel_lift(curve, p, point): A, B = map(long, (E.a4(), E.a6())) x, y = map(long, point.xy()) fr = y**2 - (x**3 + A*x + B) t = (- fr / p) % p t *= inverse_mod(2 * y, p) # (y**2)' = 2 * y t %= p new_y = y + p * t return x, new_y # lift points x1, y1 = hensel_lift(E, p, g) x2, y2 = hensel_lift(E, p, v) # calculate new A, B (actually, they will be the same here) mod = p ** 2 A2 = y2**2 - y1**2 - (x2**3 - x1**3) A2 = A2 * inverse_mod(x2 - x1, mod) A2 %= mod B2 = y1**2 - x1**3 - A2 * x1 B2 %= mod # new curve E2 = EllipticCurve(IntegerModRing(p**2), [A2, B2]) # calculate dlog g2s = (p - 1) * E2(x1, y1) v2s = (p - 1) * E2(x2, y2) x1s, y1s = map(long, g2s.xy()) x2s, y2s = map(long, v2s.xy()) dx1 = (x1s - x1) / p % p dx2 = (y1s - y1) / p dy1 = (x2s - x2) dy2 = (y2s - y2) % p print "%d, %d, %d, %d, %d" % (dx1, dy1, dx2, dy2, p) m = dy1 * inverse_mod(dx1, p) * dx2 * inverse_mod(dy2, p) m %= p print m

### elliptic-shiho commented Feb 8, 2016

Writeup:
Given Elliptic Curve is Anomalous Elliptic Curve. so, I apply SSSA-Attack.

Flag is `6418297401790414611703852603267852625498215178707956450`