Recently, we present a novel Mastrovito form of nonrecursive Karatsuba multiplier for all trinomials. Specifically, we found that related Mastrovito matrix is very simple for equally spaced trinomial (EST) combined with classic Karatsuba algorithm (KA), which leads to a highly efficient Karatsuba multiplier. In this paper, we consider a new special class of irreducible trinomial, namely,
Efficient hardware implementation of the finite field arithmetic, especially for
The field elements are usually represented by a certain basis such as polynomial basis (PB), normal basis (NB), and dual basis (DB). In PB representation, the multiplication consists of multiplying two polynomials and reducing the result modulo an irreducible polynomial. The choice of such an irreducible polynomial is critical to perform the reduction operation efficiently. Irreducible trinomial is one of the most common considerations [
The Karatsuba algorithm (KA) works recursively by breaking down one big multiplication into two or more submultiplications. It is a typical divideandconquer algorithm. Please note that the classic KA starts with a way to multiply two 2term polynomials using three scalar multiplications. Some other variations are also investigated. More details can be found in [
Recently, we proposed a novel nonrecursive Karatsuba multiplier that is based on Mastrovito approach [
In this paper, we explore another special case of our former scheme to obtain even more efficient nonrecursive Karatsuba multipliers. Our main idea is analogous to Shen and Jin [
The rest of this paper is organized as follows: In Section
In this section, we briefly review some related notations and algorithms used throughout this paper. Consider the finite field
Let
Generally speaking, the optimal choice of
We then reduce the above matrix in view to obtain the field product expressed in SPB representation. The reduced matrix, denoted by
Moreover, one can check that the irreducible trinomial in the form of
In the end, we also introduce some notations pertaining to matrices and vectors, which are already proposed in [
The Karatsuba algorithm [
In general, the Mastrovito multiplication utilizing the KA will increase the time complexity. Our former result shows that a Mastrovito multiplier using classic KA costs one more
Let
We first consider the computation of
The Mastrovito matrix
The proof is analogous with the proof of observation
According to the reduction rule in (
Based on Proposition
By swapping and combining some overlapped entries, expression (
Perform rowvector products:
in parallel. The symbol “
Sum up all the
using binary XOR tree firstly and then add these results together.
It is easy to see that the rowvector products (
Space and time complexities of
Operation  #AND  #XOR  Time delay 

Inner products in ( 

  
Partial addition in ( 
  


 


Then we consider the computation of
The reduction of
Analogous with the computation of
Space and time complexities of
Operation  #AND  #XOR  Time delay 


 



 


Inner products in ( 

  
Partial addition in ( 
  


 


From Tables
Table
Comparison of bitparallel multipliers for
Multiplier  #AND  #XOR  Time delay 

Sunar and Koç [ 



Wu [ 



Wu [ 



Fan and Dai [ 



Elia et al. [ 



Negre [ 



Fan [ 



Fan [ 



Li et al. [ 



This paper 



In Table
Complexity for practical field
Basis  #AND  #XOR  Time 

PB [ 
21609  21608  
PB [ 
16280  16838  
SPB [ 
21609  21608  
SPB [ 
21609  21608 

SPB [ 
21609  27391 

PBCRT Type A [ 
21560  21560 

PBCRT Type B [ 
21560  21658 

SPB [ 
16280  17394 

SPB (this paper)  14406  14748  
In this paper, a new Mastrovito multiplier architecture for trinomial of the form
The authors declare that they have no conflicts of interest.
This work is supported by the Natural Science Foundation of China (nos. 61402393, 61601396) and Shanghai Key Laboratory of Integrated Administration Technologies for Information Security (no. AGK201607).