Finding linear building-blocks for RTL synthesis of polynomial datapaths with fixed-size bit-vectors

Polynomial computations over fixed-size bit-vectors are found in many practical datapath designs. For efficient RTL synthesis, it is important to identify good decompositions of the polynomial into smaller/simpler units. Symbolic computer algebra algorithms and tools have been used for this purpose. However, fixed-size (m) bit-vector arithmetic is polynomial algebra over the finite integer ring Z₂m, which is a non-unique factorization domain (non-UFD). While non-UFDs provide an extra freedom to search for decompositions, they complicate polynomial manipulation as traditional division-based algorithms are inapplicable. This paper presents new mathematical concepts for polynomial decomposition over Z₂m, for RTL synthesis over fixed-size m-bit vectors. Given a polynomial, we identify a specific set of linear expressions and compute the Grobner bases of their ideal (over non-UFD Z₂m) using syzygies. This basis serves as good building-blocks for the given computation. A decomposition is identified by subsequent Grobner basis reduction. Experimental results demonstrate significant area savings due to our approach, as compared against contemporary datapath synthesis techniques.