Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations an...Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.展开更多
The advent of development of high-performance, low-power digital circuits is achieved by a suitable emerging nanodevice called quantum-dot cellular automata（QCA）. Even though many efficient arithmetic circuits were ...The advent of development of high-performance, low-power digital circuits is achieved by a suitable emerging nanodevice called quantum-dot cellular automata（QCA）. Even though many efficient arithmetic circuits were designed using QCA, there is still a challenge to implement high-speed circuits in an optimized manner.Among these circuits, one of the essential structures is a parallel multi-digit decimal adder unit with significant speed which is very attractive for future environments. To achieve high speed, a new correction logic formulation method is proposed for single and multi-digit BCD adder. The proposed enhanced single-digit BCD adder（ESDBA）is 26% faster than the carry flow adder（CFA）-based BCD adder. The multi-digit operations are also performed using the proposed ESDBA, which is cascaded innovatively. The enhanced multi-digit BCD adder（EMDBA） performs two 4-digit and two 8-digit BCD addition 50% faster than the CFA-based BCD adder with the nominal overhead of the area. The EMDBA performs two 4-digit BCD addition 24% faster with 23% decrease in the area, similarly for 8-digit operation the EMDBA achieves 36% increase in speed with 21% less area compared to the existing carry look ahead（CLA）-based BCD adder design. The proposed multi-digit adder produces significantly less delay of（N – 1）＋3.5 clock cycles compared to the N＊One digit BCD adder delay required by the conventional BCD adder method. It is observed that as per our knowledge this is the first innovative proposal for multi-digit BCD addition using QCA.展开更多
文摘Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.
文摘The advent of development of high-performance, low-power digital circuits is achieved by a suitable emerging nanodevice called quantum-dot cellular automata（QCA）. Even though many efficient arithmetic circuits were designed using QCA, there is still a challenge to implement high-speed circuits in an optimized manner.Among these circuits, one of the essential structures is a parallel multi-digit decimal adder unit with significant speed which is very attractive for future environments. To achieve high speed, a new correction logic formulation method is proposed for single and multi-digit BCD adder. The proposed enhanced single-digit BCD adder（ESDBA）is 26% faster than the carry flow adder（CFA）-based BCD adder. The multi-digit operations are also performed using the proposed ESDBA, which is cascaded innovatively. The enhanced multi-digit BCD adder（EMDBA） performs two 4-digit and two 8-digit BCD addition 50% faster than the CFA-based BCD adder with the nominal overhead of the area. The EMDBA performs two 4-digit BCD addition 24% faster with 23% decrease in the area, similarly for 8-digit operation the EMDBA achieves 36% increase in speed with 21% less area compared to the existing carry look ahead（CLA）-based BCD adder design. The proposed multi-digit adder produces significantly less delay of（N – 1）＋3.5 clock cycles compared to the N＊One digit BCD adder delay required by the conventional BCD adder method. It is observed that as per our knowledge this is the first innovative proposal for multi-digit BCD addition using QCA.
