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This paper proposes a compact and efficient \(GF(2^8)\) inversion circuit design based on a combination of non-redundant and redundant Galois Field (GF) arithmetic. The proposed design utilizes redundant GF representations, called Polynomial Ring Representation (PRR) and Redundantly Represented Basis (RRB), to implement \(GF(2^8)\) inversion using a tower field \(GF((2^4)^2)\). In addition to the redundant representations, we introduce a specific normal basis that makes it possible to map the former components for the 16th and 17th powers of input onto logic gates in an efficient manner. The latter components for \(GF(2^4)\) inversion and \(GF(2^4)\) multiplication are then implemented by PRR and RRB, respectively. The flexibility of the redundant representations provides efficient mappings from/to the \(GF(2^8)\). This paper also evaluates the efficacy of the proposed circuit by means of gate counts and logic synthesis with a 65 nm CMOS standard cell library and comparisons with conventional circuits, including those with tower fields \(GF(((2^2)^2)^2)\). Consequently, we show that the proposed circuit achieves approximately 40 % higher efficiency in terms of area-time product than the conventional best \(GF(((2^2)^2)^2)\) circuit excluding isomorphic mappings. We also demonstrate that the proposed circuit achieves the best efficiency (i.e., area-time product) for an AES encryption S-Box circuit including isomorphic mappings.

Highly Efficient $$GF(2^8)$$ G F ( 2 8 ) Inversion Circuit Based on Redundant GF Arithmetic and Its Application to AES Design