Galois Field Arithmetic Circuits Research Articles

An Algebraic Approach to Verifying Galois-Field Arithmetic Circuits with Multiple-Valued Characteristics

An Algebraic Approach to Verifying Galois-Field Arithmetic Circuits with Multiple-Valued Characteristics

Efficient Formal Verification of Galois-Field Arithmetic Circuits Using ZDD Representation of Boolean Polynomials

In this study, we present a new formal method for verifying the functionality of Galois-field (GF) arithmetic circuits. Assuming that the input–output relation (i.e., the specification of a GF arithmetic circuit) can be represented as polynomials over 2, the proposed method formally checks the equivalence between GF polynomials derived from a netlist and the specification. To efficiently verify the equivalence, we employ a zero-suppressed binary decision diagram (ZDD) to represent polynomials over 2. Even though polynomial reduction is the most time-consuming process of verification (i.e., equivalence checking), our new algorithm can efficiently reduce the GF polynomials in the form of a zero-suppressed binary decision diagram derived from the target netlist. The proposed algorithm derives the polynomials representing all intermediate nodes (i.e., the outputs of all gates) in the order from primary inputs to those primary outputs that are in accordance with the reverse topological traversal order. We demonstrated the efficiency and effectiveness of the proposed method via a set of experimental verifications. In particular, we confirmed that the proposed method can verify practical GF multipliers (including those used in standardized elliptic curve cryptography) approximately 30 times faster on average and at most 170 times faster than the best conventional method.

Formal Approach for Verifying Galois Field Arithmetic Circuits of Higher Degrees

This paper presents an efficient approach to verifying higher-degree Galois-field (GF) arithmetic circuits. The proposed method describes GF arithmetic circuits using a mathematical graph-based representation and verifies them by a combination of algebraic transformations and a new verification method based on natural deduction for first-order predicate logic with equal sign. The natural deduction method can verify one type of higher-degree GF arithmetic circuit efficiently while the existing methods require an enormous amount of time, if they can verify them at all. In this paper, we first apply the proposed method to the design and verification of various Reed-Solomon (RS) code decoders. We confirm that the proposed method can verify RS decoders with higher-degree functions while the existing method needs a lot of time or fail. In particular, we show that the proposed method can be applied to practical decoders with 8-bit symbols, which are performed with up to 2,040-bit operands. We then demonstrate the design and verification of the Advanced Encryption Standard (AES) encryption and decryption processors. As a result, the proposed method successfully verifies the AES decryption datapath while an existing method fails.

A Formal Approach to Designing Cryptographic Processors Based on $GF(2^m)$ Arithmetic Circuits

This paper proposes a formal approach to designing Galois-field (GF) arithmetic circuits, which are widely used in modern cryptographic processors. Our method describes GF arithmetic circuits in a hierarchical manner with high-level directed graphs associated with specific GFs and arithmetic functions. The proposed circuit description can be effectively verified by symbolic computations based on polynomial reduction using Grobner bases. The verified description is then translated into the equivalent hardware description language (HDL) codes, which are available for the conventional design flow. We first describe the proposed graph representation and present an example of the description and verification. The significant advantage of the proposed approach is demonstrated through experimental designs of parallel multipliers over GF(2m) for different word lengths and irreducible polynomials. The result shows that the proposed approach has a definite capability of formally verifying practical GF arithmetic circuits for which the conventional techniques fail. We also propose an application of this approach to cryptographic processor design. The target considered here is a 128-bit advanced encryption standard (AES) data path with a loop architecture. To the best of the authors' knowledge, this is the first verification of this type of practical AES data path. We present a detailed description of the AES data path and its verification. The proposed approach successfully verifies the AES data path description within 800 s.