# On the uniqueness of a type of cascade connection representations for NFSRs

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## Abstract

Cascade connection architectures of nonlinear feedback shift registers (NFSRs) have been widely used in cryptography. In particular, the Grain family of stream ciphers uses the cascade connection architecture of an LFSR into an NFSR. A cascade connection representation is not always unique. The nonuniqueness of the representation may threat the security of a cipher. Inspired by the Grain family of stream ciphers, in this paper, we focus on cascade connections of an LFSR into an NFSR. A necessary and sufficient condition for the uniqueness of this class of cascade connection representations is provided under a reasonable condition that the involved NFSR has only trivial cascade connection decompositions. In particular, as a direct application of new results, it is theoretically proved that the cascade connection representation of a Grain-like structure, an *n*-bit primitive LFSR into an *n*-bit NFSR with a positive integer *n*, is unique not considering some trivial distinct representations if the involved *n*-bit NFSR satisfies the condition. Besides, it is verified that all the main registers used in the Grain family of stream ciphers satisfy the condition.

## Keywords

Nonlinear feedback shift registers Cascade connection Grain-like structure## Mathematics Subject Classification

94A55 94A60## Notes

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