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This article presents an investigation of asymptotic time complexities of several algorithms related to Megrelishvili protocol. The analysis are carried out for the private keys computations and public exchange of values, public key constructions, as well as an elementary exhaustive search attack algorithm. We show that the complexities of these algorithms are higher than the complexities of elementary algorithms involved in the conventional Diffie - Hellman protocol (DHP) or its variant on elliptic curves (ECDHP). This condition also implies that Megrelishvili protocol is more secure than DHP and ECDHP against exhaustive search attack.

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This article presents an extension of our previous research in [1] where we propose two variants of Megrelishvili key distribution schemes and investigate some of their elementary theoretical security analysis. We briefly discuss the two protocols in [1] and propose another two schemes which are more efficient than the preceding ones. Additionally, we also devise effective procedures for constructing a new mutual key if the group membership is altered. Furthermore, we discuss the security of the protocols rigorously and we provide a sufficient condition for breaking the protocols by way of solving several instances of Megrelishvili vector-matrix problems (MVMP). We prove that the secret group key can be recovered easily if an attacker can express the sum of the secret exponents of the participants as a linear combination of the secret exponents excerpted from the transmission. Based on this result, we reason that our Megrelishvili key distribution schemes are theoretically at least as secure as the standard two-party Megrelishvili key exchange procedure.Keywords: Megrelishvili protocol, Megrelishvili key distribution scheme, key distribution scheme, protocol extension Abstrak Artikel ini menyajikan sebuah pengembangan dari penelitian kami sebelumnya di [1] yang berisi pengajuan dua varian skema distribusi kunci Megrelishvili dan pembahasan analisis teori keamanan dasar mereka. Kami membahas secara singkat dua protokol di [1] dan mengajukan dua skema lain yang lebih efisien daripada yang sebelumnya. Selain itu, kami juga merancang prosedur yang efektif untuk membangun kunci bersama yang baru jika keanggotaan kelompok diubah. Selanjutnya, kami membahas keamanan protokol-protokol secara sistematis dan menyajikan suatu kondisi cukup (sufficient condition) untuk memecahkan protokol-protokol tersebut dengan cara memecahkan beberapa kasus masalah vektormatriks Megrelishvili (Megrelishvili vector-matrix problem, MVMP). Kami membuktikan bahwa kunci rahasia suatu kelompok dapat ditemukan dengan mudah bila penyerang dapat mengekspresikan jumlah pangkat rahasia dari seluruh peserta sebagai kombinasi linier dari pangkat rahasia yang diekstraksi dari transmisi. Berdasarkan hasil ini, kami beranggapan bahwa skema distribusi kunci Megrelishvili secara teori setidaknya sama amannya dengan prosedur pertukaran kunci Megrelishvili standar dua pihak.Kata Kunci: protokol Megrelishvili, skema distribusi kunci Megrelishvili, skema distribusi kunci, pengembangan protokol Received on xxx, accepted on xxx, published on xxx

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