In this paper we introduce a general class of stream and block ciphers that are defined by means of systems of (ordinary) explicit difference equations over a finite field. We call this class "difference ciphers". Many important ciphers such as systems of LFSRs, Trivium/Bivium and Keeloq are difference ciphers. To the purpose of studying their underlying explicit difference systems, we introduce key notions as state transition endomorphisms and show conditions for their invertibility. Reducible and periodic systems are also considered. We then propose general algebraic attacks to difference ciphers which are experimented by means of Bivium and Keeloq.

中文翻译：

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流/分组密码、差分方程和代数攻击

在本文中，我们介绍了一类通用的流和分组密码，它们是通过有限域上的（普通）显式差分方程系统定义的。我们称此类为“差异密码”。许多重要的密码，例如 LFSR、Trivium/Bivium 和 Keeloq 系统都是差分密码。为了研究它们潜在的显式差分系统，我们引入了作为状态转换自同态的关键概念，并展示了它们可逆的条件。还考虑了可约系统和周期系统。然后，我们提出了对差异密码的一般代数攻击，这些密码是通过 Bivium 和 Keeloq 进行实验的。