On Algebraic Property of T-Functions

Ruilin LI  Bing SUN  Chao LI  Shaojing FU  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E95-A   No.1   pp.267-269
Publication Date: 2012/01/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E95.A.267
Print ISSN: 0916-8508
Type of Manuscript: Special Section LETTER (Special Section on Cryptography and Information Security)
Category: 
Keyword: 
cryptographic functions,  single-cycle T-function,  modular addition,  algebraic degree,  

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Summary: 
T-function is a kind of cryptographic function which is shown to be useful in various applications. It is known that any function f on F2n or Z2n automatically deduces a unique polynomial fF ∈ F2n[x] with degree ≤ 2n-1. In this letter, we study an algebraic property of fF while f is a T-function. We prove that for a single cycle T-function f on F2n or Z2n, deg fF=2n-2 which is optimal for a permutation. We also consider a kind of widely used T-function in many cryptographic algorithms, namely the modular addition function Ab(x)=x+b ∈ Z2n[x]. We demonstrate how to calculate deg Ab F from the constant value b. These results can facilitate us to evaluate the immunity of the T-function based cryptosystem against some known attacks such as interpolation attack and integral attack.