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An algebraic approach to circulant column parity mixers
1Digital Security, Radboud University, Nijmegen, The Netherlands.
Designs, Codes, and Cryptography
|November 25, 2024
Summary
Circulant Column Parity Mixers (CCPMs), key in cryptography, are now analyzed using module theory. This new approach simplifies understanding their algebraic properties and explains Xoodoo
Area of Science:
- Cryptography
- Abstract Algebra
- Module Theory
Background:
- Circulant Column Parity Mixers (CCPMs) are essential linear maps in cryptographic primitives like Keccak-f (SHA3) and Xoodoo.
- Existing knowledge on CCPMs is limited, primarily focusing on their invertibility and group properties.
- The complexity of describing CCPMs via linear algebra hinders deeper algebraic analysis.
Purpose of the Study:
- To introduce module theory as a novel framework for studying CCPMs.
- To uncover and explain the algebraic properties of CCPMs using this new approach.
- To analyze the linear layer of Xoodoo and similar structures (DCD-compositions).
Main Methods:
- Application of module theory from commutative algebra to analyze CCPMs.
- Investigation of algebraic properties through the lens of module theory.
- Examination of DCD-compositions, including the linear layer of Xoodoo.
Main Results:
- Module theory simplifies the deduction of complex algebraic properties for CCPMs.
- Known results on CCPMs are shown to be straightforward consequences of module theoretic concepts.
- A mathematical explanation for the low order (32) of Xoodoo's linear layer is provided.
- The study demonstrates the design of a DCD-composition with a higher order.
Conclusions:
- Module theory offers a powerful and simplified approach to understanding CCPMs and related cryptographic components.
- This framework provides a solid mathematical foundation for analyzing the algebraic properties of linear layers in ciphers.
- The findings pave the way for designing new cryptographic components with desired algebraic characteristics.
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