White Box Implementations Using Non-Commutative Cryptography.

Leandro Marin

doi:10.3390/s19051122

Abstract

In this paper, we present a method to create a safe arithmetic that can be used to obfuscate implementations that require operations over commutative groups. The method is based on the structure of the endomorphisms of certain extensions of the original commutative group. The endomorphisms of a commutative group are non-commutative (in general), thus we can use a non-commutative group to emulate the arithmetic of a commutative one. The techniques presented in this paper are very flexible and the programmer has a wide variety of options to obfuscate the algorithms. The system can be parameterized using conjugations, thus it is possible to generate a different arithmetic for each instance of the program with a change in the security parameters, even in cases in which this number is huge (for example, in IoT applications). The security of this method is based not only on the difficulty of the conjugacy search problem (in a harder version because only partial information about the groups is known by the attacker), but also in a number of extra options that can be chosen by the programmer. The paper explains the general method, analyzes its algebraic properties and provides detailed examples based on the vector spaces over and XOR operators.

Highlights

The most traditional framework for cryptography is a pair of users, Alice and Bob, where Alice wants to send a message m to Bob through a communication channel
Alice and Bob live in secure bubbles, but the outside world is insecure and other users can be listening to the channel, they agree to two functions E and D such that D ( E(m)) = m and instead of sending m, Alice sends
We show above that it is possible to generate a huge number of different arithmetics choosing some parameters, which makes it especially useful for the generation of code for IoT

Summary

Introduction

The most traditional framework for cryptography is a pair of users, Alice and Bob, where Alice wants to send a message m to Bob through a communication channel. Alice and Bob live in secure bubbles, but the outside world is insecure and other users can be listening to the channel, they agree to two functions E and D such that D ( E(m)) = m and instead of sending m, Alice sends. D can be parameterized by a value k (the key) such that the algorithms can be made public and, if the value k is kept safe, the communication is secure. This framework (the channel is insecure but the functions E and D are computed in a safe environment) is not valid in many real life situations.

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Conclusion