N-Term Karatsuba Algorithm and its Application to Multiplier Designs for Special Trinomials

In this paper, we propose a new type of non-recursive Mastrovito multiplier for <inline-formula> <tex-math notation="LaTeX">$\text {GF}(2^{m})$ </tex-math></inline-formula> using an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-term Karatsuba algorithm (KA), where <inline-formula> <tex-math notation="LaTeX">$\text {GF}(2^{m})$ </tex-math></inline-formula> is defined by an irreducible trinomial, <inline-formula> <tex-math notation="LaTeX">$x^{m}+x^{k}+1, m=nk$ </tex-math></inline-formula>. We show that such a type of trinomial combined with the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-term KA can fully exploit the spatial correlation of entries in related Mastrovito product matrices and lead to a low-complexity architecture. The optimal parameter <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is further studied. As the main contribution of this paper, the lower bound of the space complexity of our proposal is about <inline-formula> <tex-math notation="LaTeX">$O({m^{2}}/{2})+m^{3/2})$ </tex-math></inline-formula>. Meanwhile, the time complexity matches the best Karatsuba multiplier known to date. To the best of our knowledge, it is the first time that Karatsuba-based multiplier has reached such a space complexity bound while maintaining a relatively low time delay.

[1]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[2]  Arash Reyhani-Masoleh,et al.  Bit-Serial and Bit-Parallel Montgomery Multiplication and Squaring over GF(2^m) , 2009, IEEE Transactions on Computers.

[3]  Harald Niederreiter,et al.  Finite fields: Preface , 1996 .

[4]  Huapeng Wu,et al.  Bit-Parallel Finite Field Multiplier and Squarer Using Polynomial Basis , 2002, IEEE Trans. Computers.

[5]  Alessandro Cilardo.  Fast Parallel GF(2^m) Polynomial Multiplication for All Degrees , 2013, IEEE Transactions on Computers.

[6]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[7]  Christophe Nègre.  Efficient parallel multiplier in shifted polynomial basis , 2007, J. Syst. Archit..

[8]  Michele Elia,et al.  Low Complexity Bit-Parallel Multipliers for GF(2^m) with Generator Polynomial x^m+x^k+1 , 1999 .

[9]  A. Menezes,et al.  Applications of Finite Fields , 1992 .

[10]  Yin Li,et al.  Speedup of bit-parallel Karatsuba multiplier in GF(2m) generated by trinomials , 2011, Inf. Process. Lett..

[11]  Dowon Hong,et al.  Low complexity bit-parallel multiplier for GF(2/sup m/) defined by all-one polynomials using redundant representation , 2005, IEEE Transactions on Computers.

[12]  Huapeng Wu.  Montgomery Multiplier and Squarer for a Class of Finite Fields , 2002, IEEE Trans. Computers.

[13]  Yin Li,et al.  Novel bit-parallel multiplier for GF(2m) defined by all-one polynomial using generalized Karatsuba algorithm , 2014, Inf. Process. Lett..

[14]  Ming Gu,et al.  Overlap-free Karatsuba-Ofman polynomial multiplication algorithms , 2010 .

[15]  Yin Li,et al.  New bit-parallel Montgomery multiplier for trinomials using squaring operation , 2016, Integr..

[16]  M. Anwar Hasan,et al.  A survey of some recent bit-parallel GF(2n) multipliers , 2015, Finite Fields Their Appl..

[17]  Christof Paar,et al.  Generalizations of the Karatsuba Algorithm for Efficient Implementations , 2006, IACR Cryptol. ePrint Arch..

[18]  Haibin Shen,et al.  Low complexity bit parallel multiplier for GF(2m) generated by equally-spaced trinomials , 2008, Inf. Process. Lett..

[19]  Yin Li,et al.  Mastrovito Form of Non-Recursive Karatsuba Multiplier for All Trinomials , 2017, IEEE Transactions on Computers.

[20]  Ian F. Blake,et al.  Elliptic curves in cryptography , 1999 .

[21]  Peter L. Montgomery,et al.  Five, six, and seven-term Karatsuba-like formulae , 2005, IEEE Transactions on Computers.

[22]  Rudolf Lide,et al.  Finite fields , 1983 .

[23]  Yin Li,et al.  Efficient Nonrecursive Bit-Parallel Karatsuba Multiplier for a Special Class of Trinomials , 2018, VLSI Design.

[24]  Haining Fan.  A Chinese Remainder Theorem Approach to Bit-Parallel $GF(2^{n})$ Polynomial Basis Multipliers for Irreducible Trinomials , 2016, IEEE Transactions on Computers.

[25]  Tong Zhang,et al.  Systematic Design of Original and Modified Mastrovito Multipliers for General Irreducible Polynomials , 2001, IEEE Trans. Computers.

[26]  Guochu Shou,et al.  Low complexity architecture of bit parallel multipliers for GF(2 m ) , 2010 .

[27]  Ming Gu,et al.  Obtaining more Karatsuba-like formulae over the binary field , 2012, IET Inf. Secur..

[28]  Berk Sunar,et al.  Mastrovito Multiplier for All Trinomials , 1999, IEEE Trans. Computers.

[29]  Yiqi Dai,et al.  Fast Bit-Parallel GF(2^n) Multiplier for All Trinomials , 2005, IEEE Trans. Computers.