In this paper, the authors construct a new type of cryptographic sequence which is named an extra-super increasing sequence, and give the definitions of the minimal super increasing sequence {a[1], a[2], …, a[n]} and minimal extra-super increasing sequence {z[1], z[2], …, z[n]}. Prove that the minimal extra-super increasing sequence is the odd-positioned subsequence of the Fibonacci sequence, namely {z[1], z[2], …, z[n], …} = {F[1], F[3], …, F[2n-1], …}, which indicates that the approach to the golden ratio phi through the difference (z[n+1] / z[n] – 1]) is more smooth and expeditious than through the ratio (F[n+1] / F[n]). Further prove that the limit of the term ratio difference between the two cryptographic sequences equals the golden ratio conjugate PHI, namely lim (n to infinity) (z[n+1] / z[n] – a[n+1] / a[n]) = PHI, which reveals the beauty of cryptography.

By admin