**Beauty of Cryptography: the Cryptographic Sequences and the Golden Ratio**

*Shenghui Su and Jianhua Zheng and Shuwang Lv*

**Abstract: **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 term difference ratio (z[n+1] - z[n]) / z[n] is more smooth and expeditious than through the term 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.

**Category / Keywords: **foundations / Minimal extra-super increasing sequence, Fibonacci sequence, Golden ratio, Golden ratio conjugate, Term ratio difference

**Date: **received 4 Sep 2021, last revised 7 Sep 2021

**Contact author: **idology98 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20210907:140934 (All versions of this report)

**Short URL: **ia.cr/2021/1129

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