Abstract
Commitment schemes are important tools in cryptography and used as building blocks in many cryptographic protocols. We propose two commitment schemes by using Rubik’s groups. Our proposals do not lay the security on the taken-for-granted hardness of the word problem over Rubik’s groups. Instead, our first proposal is based on a symmetric encryption algorithm that is secure based on the hardness of the conjugacy search problem over Rubik’s groups, while our second proposal is based on the hardness of a newly derived problem—the functional towering conjugacy search problem over Rubik’s groups. The former is proved secure in the sense of both computational hiding and binding, while the latter is proved even secure in the sense of perfect hiding and computational binding. Furthermore, the proposed schemes have a remarkable performance advantage: a linear commitment/opening speed. We also evaluate the efficiency of the commitment schemes and show that they are considerably fast.
Highlights
The commitment scheme is one of the most important cryptographic primitives and has been widely used as an essential building block in many cryptographic protocols, such as the verifiable secret sharing scheme [1] and zero-knowledge protocols [2]
The other scheme can be regarded as a non-Abelian variant of the Pedersen commitment scheme, the security of which depends on the intractability assumption of the FT-CSP problem over the Rubik’s group
We evaluate the efficiency of the schemes and show that they are more efficient than the Pedersen commitment scheme in terms of the computational cost
Summary
The commitment scheme is one of the most important cryptographic primitives and has been widely used as an essential building block in many cryptographic protocols, such as the verifiable secret sharing scheme [1] and zero-knowledge protocols [2]. To the best of our knowledge, the security of almost all explicit constructions for commitment schemes is based on number-theoretic hardness assumptions, such as the IFP-based scheme due to Goldreich [6]. This paper proposes two new fast commitment schemes that use Rubik’s cube rotation operations. To the best of our knowledge, one of them is the first commitment scheme based on a symmetric encryption algorithm from a non-Abelian group. The other scheme can be regarded as a non-Abelian variant of the Pedersen commitment scheme, the security of which depends on the intractability assumption of the FT-CSP problem over the Rubik’s group. The rest of the paper is organized as follows: in Section 2, we introduce necessary notations and preliminaries, including the relevant concepts of commitment scheme, Rubik’s group, and intractability assumptions.
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