Twisted Mind Trio (A / An Exceptional / An Evil Twisted Mind)

cryptohack

Task: three CryptoHack ECC oracles using an x-only XZ Montgomery-style ladder over Z/modulus that return x([d]P) for a submitted x-coordinate. Solution: CH1 twist + smooth-order Pohlig-Hellman; CH2 Smart-style anomalous lift over Z/q^2 where the x-only formal log scales as d^2 (square-root to finish); CH3 free gcd factoring of a squarefree modulus then dual-twist CRT+Pohlig-Hellman recovering a 191-bit key from one query.