On the Complexity of Hybrid $n$ -Term Karatsuba Multiplier for Trinomials

Efficient Bit-Parallel Multiplier for All Trinomials Based on n-Term Karatsuba Algorithm

“…We evaluate the explicit space and time complexities of the proposed multiplier. The complexity of the proposed multiplier is comparable with those in [18] and [19]. However, more flexible options for trinomials and values for n, m 0 and r may yield better complexities.…”

“…In [19], such limitation for m in [18] is loosened, that is, m satisfies m = nm 0 + r with 0 ≤ r < n, m 0 , but there is the limitation m ≥ 2k for trinomial. The multiplier in [18] or [19] achieves the lowest space complexity among similar bit-parallel multipliers. The proposed multiplier in this paper further eases limitations for trinomials and a decomposition of m in [18], [19].…”

“…The complexity of this multiplier depends on the choices of values n, m 0 , and k. Although an arbitrary integer m can be decomposed into m = nm 0 or m = nm 0 + 1 with n, m 0 > 1, the options for values of n, m 0 are limited. Such limitation for the decomposition of m is loosened in [19]. Li.…”

“…Li. et al in [19] utilize alternative approach to perform modular reduction instead of Mastrovito approach and presented nterm Karatsuba multiplier for a trinomial x m + x k + 1, where m = nm 0 + r with 0 ≤ r < n, m 0 . However, there is the limitation m ≥ 2k for the choice of trinomial.…”

“…In this paper, we would like to further ease limitations on trinomials and a decomposition of m in [18], [19]. We present a hybrid multiplier for any irreducible trinomial x m + x k + 1 (0 < k < m) with a decomposition m = nm 0 + r with 0 < r < m 0 and 1 < n. (Since the case r = 0 is already dealt with in [18, Section III], we do not address the case.)…”