Galois field arithmetic is a critical component in communication and security-related hardware, requiring dedicated arithmetic circuit architectures for greater performance. In many Galois field applications, such as cryptography, the datapath size in the circuits can be very large. Formal verification of such circuits is beyond the capabilities of contemporary verification techniques. This paper addresses formal verification of combinational arithmetic circuits over Galois fields of the type F 2 k using a computer-algebra/algebraic-geometry based approach.The verification problem is formulated as membership testing of a given specification polynomial in a corresponding ideal generated by the circuit constraints. Ideal membership testing requires the computation of a Gröbner basis, which is computationally very expensive. To overcome this limitation, we analyze the circuit topology and derive a term order to represent the polynomials. Subsequently, using the theory Gröbner bases over F 2 k , we show that this term order renders the set of polynomials itself a minimal Gröbner basis of this ideal. Consequently, the verification test reduces to a much simpler case of Gröbner basis reduction via polynomial division, significantly enhancing verification efficiency.To further improve our approach, we exploit the concepts presented in the F 4 algorithm for Gröbner basis, and show that our verification test can be formulated as Gaussian elimination on a matrix representation of the problem. Finally, we demonstrate the ability of our approach to verify the correctness of, and detect bugs in, up to 163-bit circuits in F 2 163 -whereas verification utilizing contemporary techniques proves infeasible.
Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition.
For a reduced F -finite ring R of characteristic p > 0 and q = p e one can write R 1/q = R a q ⊕ M q , where M q has no free direct summands over R. We investigate the structure of F -finite, F -pure rings R by studying how the numbers a q grow with respect to q. This growth is quantified by the splitting dimension and the splitting ratios of R which we study in detail. We also prove the existence of a special prime ideal P(R) of R, called the splitting prime, that has the property that R/P(R) is strongly F -regular. We show that this ideal captures significant information with regard to the F -purity of R.
Custom arithmetic circuits designed over Galois fields F 2 k are prevalent in cryptography, where the field size k is very large (e.g. k = 571-bits). Equivalence checking of such large custom arithmetic circuits against baseline golden models is beyond the capabilities of contemporary techniques. This paper addresses the problem by deriving word-level canonical polynomial representations from gatelevel circuits as Z = F (A) over F 2 k , where Z and A represent the output and input bit-vectors of the circuit, respectively. Using algebraic geometry, we show that the canonical polynomial abstraction can be derived by computing a Gröbner basis of a set of polynomials extracted from the circuit, using a specific elimination (abstraction) term order. By efficiently applying these concepts, we can derive the canonical abstraction in hierarchically designed, custom arithmetic circuits with up to 571-bit datapath, whereas contemporary techniques can verify only up to 163-bit circuits.
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