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On the Resistance of Boolean Functions Against Algebraic Attacks Using Univariate Polynomial Representation

機(jī)譯：基于單變量多項(xiàng)式表示的布爾函數(shù)對(duì)代數(shù)攻擊的抵抗力

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In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function $f$ was introduced, defined as the minimum degree of the annihilators of $f$ and $f+ 1$. An annihilator of $f$ is a nonzero Boolean function $g$ , such that $fcdot g=0$. While several constructions of Boolean functions with optimal algebraic immunity have been proposed, there is no significant progress concerning the resistance against the so-called fast algebraic attacks. In this paper, we provide a framework to assess the resistance of Boolean functions against the new algebraic attacks, including fast algebraic attacks. The analysis is based on the univariate polynomial representation of Boolean functions and necessary and sufficient conditions are presented for a Boolean function to have optimal behavior against all the new algebraic attacks. Finally, we introduce a new infinite family of balanced Boolean functions described by their univariate polynomial representation. By applying the new framework, we prove that all the members of the family have optimal algebraic immunity and we efficiently evaluate their behavior against fast algebraic attacks.
機(jī)譯：在過(guò)去的幾年中，針對(duì)具有線性反饋功能的流密碼的代數(shù)攻擊已得到顯著改善。作為對(duì)新攻擊的回應(yīng)，引入了布爾函數(shù)$ f $的代數(shù)免疫概念，定義為$ f $和$ f + 1 $的零化子的最小程度。 $ f $的an滅者是一個(gè)非零布爾函數(shù)$ g $，因此$ fcdot g = 0 $。盡管已經(jīng)提出了具有最佳代數(shù)免疫性的布爾函數(shù)的幾種構(gòu)造，但是在抵抗所謂的快速代數(shù)攻擊方面沒(méi)有取得重大進(jìn)展。在本文中，我們提供了一個(gè)評(píng)估布爾函數(shù)對(duì)新代數(shù)攻擊（包括快速代數(shù)攻擊）的抵抗力的框架。該分析基于布爾函數(shù)的單變量多項(xiàng)式表示，并給出了布爾函數(shù)針對(duì)所有新的代數(shù)攻擊具有最佳行為的必要和充分條件。最后，我們介紹了一個(gè)新的無(wú)窮系列平衡布爾函數(shù)，這些函數(shù)由它們的單變量多項(xiàng)式表示來(lái)描述。通過(guò)應(yīng)用新框架，我們證明了該家族的所有成員都具有最佳的代數(shù)免疫性，并且我們有效地評(píng)估了他們對(duì)快速代數(shù)攻擊的行為。

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