Reverse Baby-step 2^{k} -ary Adult-step Method for ?(n) Decryption of Asymmetric-key RSA

When the public key e and the composite number n=pq are disclosed but not the private key d in an asymmetric-key RSA, message decryption is carried out by obtaining ?(n)=(p-1)(q-1)=n+1-(p+q) and subsequently computing d=e^{-1} (mod ?(n)). The most commonly used decryption algorithm is integer factorization of n/p=q or a^{2}≡b^{2} (mod n), a=(p+q)/ 2, b=(q-p)/2. But many of the RSA numbers remain unfactorable. This paper therefore applies baby-step giant-step discrete logarithm and 2^{k}-ary modular exponentiation to directly obtain ?(n). The proposed algorithm performs a reverse baby-step and 2^{k}-ary adult-step. As a results, it reduces the execution time of basic adult-step to 1/2^{k} times and the memory m=?sqrt {n}? to l, a^{l} >n, hence obtaining ?(n) by executing within l times.