This document discusses investigating the efficiency of supersingular isogeny-based cryptography, which has been proposed as an alternative to current public-key cryptosystems that may be vulnerable to quantum computers. The author examines previous work on efficiency, identifies a gap around parameter generation in SIDH, and determines that supersingular curve construction takes most time. Experiments are conducted to study the impact of curve construction efficiency. Results provide hypotheses on efficiency factors and suggestions for improving the cryptosystem. The author implements supersingular curve construction and theorizes certain prime inputs may boost performance. This research aims to further analyze the efficiency of a post-quantum candidate cryptosystem.

1. Abstract
The public-key cryptosystems currently in use today like RSA and Elliptic Curve Cryptography (ECC)
are said to be threatened by the use of quantum computers in the near future. In an effort to
address this issue, many post quantum cryptosystems have been developed over the years like the
supersingular isogeny based cryptography invented by De Feo, Jao, and Plût (2014). However, as for
any recently developed cryptosystem, its efficiency is not yet well understood and not fully analysed.
We propose to investigate the efficiency of the supersingular isogeny based cryptosystem and
determine which factors may have an impact on the efficiency of the system.
For this purpose, we first examine the previous efforts made to investigate and improve the
efficiency of this system. Then, we identify a gap consisting in the parameters generation phase of
the Supersingular Isogeny Diffie–Hellman key exchange protocol (SIDH) for which the efficiency is
not yet investigated. Afterwards, we determine that the component responsible for most of the
execution time within that phase is the supersingular curve construction. A novel approach along
with experiments to investigate the efficiency of that component are then proposed and the results
of the experiments are recorded and analysed.
These results enable the formulation of several hypotheses about the factors that might influence
the efficiency of the system. Drawing from these hypotheses, we finally give suggestions about the
efficiency of the supersingular Isogeny based Cryptosystem.
The main contributions of our research are: a theoretical result as for choosing the input prime 𝑝 of
the supersingular isogeny based protocol. Primes satisfying certain conditions are suspected to
improve the performance of the supersingular curve construction and hence the performance of the
entire protocol. Also, we give the complete implementation of the Supersingular curve construction
algorithm in C with multiple precision integer support using the library GMP.
Our research is considered a step forward in investigating the efficiency of one of the postquantum
cryptosystem candidates for replacing the current public-key cryptosystems in use today. This
investigation could be used in future research to improve the efficiency of this system and to
compare between the efficiency of the different postquantum cryptosystems in development. Such
comparison would help researchers in the field choose the most practical cryptosystem to be used in
the real world in the future against quantum computers. One such cryptosystem would allow
computers’ users to protect their data and communications on the internet against quantum attacks
able to violate users’ data confidentiality and integrity.