Insecure primitive elements in an ElGamal signature protocol

Consider the classical ElGamal digital signature scheme based on the modular relation α ≡ y r [p]. In this work, we prove that if we can compute a natural integer i such that α mod p is smooth and divides p − 1, then it is possible to sign any given document without knowing the secret key. Therefore we extend and reinforce Bleichenbacher’s attack presented at Eurocrypt’96.