New attacks on prime power N = prq using good approximation of ϕ(N)

This paper proposes three new attacks. our first attack is based on the rsa key equation ed - kϕ(n) = 1 where ϕ(n) = pr-1(p - 1)(q - 1). let q < p < 2q. if δ < 1-γ/2 we shows that k/d can be recov- ered among the convergents of the continued fractions expansions of we furthered our analysis on j prime power moduli ni = pi rqi satisfying a variant of the above mentioned condition. we utilized the lll algorithm on j prime power public keys (ni,ei) with ni = pi rqi and we were able to factorize the j prime power moduli ni = pi rqi simultaneously in polynomial time.