Some new results on binary polynomial multiplication

Abstract

This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernstein’s 3-way algorithm is introduced, followed by a new 5-way algorithm. Next, a new 3-way algorithm that improves asymptotic arithmetic complexity compared to Bernstein’s 3-way algorithm is introduced. This new algorithm uses three multiplications of one-third size polynomials over the binary field and one multiplication of one-third size polynomials over the finite field with four elements. Unlike Bernstein’s algorithm, which has a linear delay complexity with respect to input size, the delay complexity of the new algorithm is logarithmic. The number of bit operations for the multiplication of polynomials over the finite field with four elements is also computed. Finally, all these new results are combined to obtain improved complexities.

Appendices

Appendix A: New bounds for multiplication over $\mathbb {F}_2$

We give the new bounds for certain values of $n$ that are of interest for cryptographic applications. Note that the improvements can be further enhanced by obtaining the explicit algorithm and eliminating common operations as in [2, 3]. The results are shown in Table 2.

Appendix B: Algorithms for $n=9$ and $n=15$

For $n=9,\, A=\sum _{i=0}^{8}b[i]X^i,\, B=\sum _{i=0}^{8}b[i]X^i$ and $C=AB=\sum _{i=0}^{16}c[i]X^i$. The coefficients of $C$ are computed using the following algorithm:

Algorithm for $n=9$
$t1=a0*b0$	$t22=t20+t21$	$t43=b3+b6$	$t64=b2+b5$	$t85=t78*t82$	$t106=t26+t30$	$c0=t1$
$t2=a0*b1$	$t23=a4*b5$	$t44=b4+b7$	$t65=t59*t62$	$t86=t79*t81$	$t107=t99+t105$	$c1=t4$
$t3=a1*b0$	$t24=a5*b4$	$t45=b5+b8$	$t66=t59*t63$	$t87=t85+t86$	$t108=t100+t106$	$c2=t9$
$t4=t2+t3$	$t25=t23+t24$	$t46=t40*t43$	$t67=t60*t62$	$t88=t78*t83$	$t109=t101+t35$	$c3=t102$
$t5=a0*b2$	$t26=a5*b5$	$t47=t40*t44$	$t68=t66+t67$	$t89=t79*t82$	$t110=t76+t84$	$c4=t103$
$t6=a1*b1$	$t27=a6*b6$	$t48=t41*t43$	$t69=t59*t64$	$t90=t80*t81$	$t111=t77+t87$	$c5= t104$
$t7=a2*b0$	$t28=a6*b7$	$t49=t47+t48$	$t70=t60*t63$	$t91=t88+t89$	$t112=t107+t110$	$c6=t112$
$t8=t5+t6$	$t29=a7*b6$	$t50=t40*t45$	$t71=t61*t62$	$t92=t90+t91$	$t113=t108+t111$	$c7= t113$
$t9=t7+t8$	$t30=t28+t29$	$t51=t41*t44$	$t72=t69+t70$	$t93=t79*t83$	$t114=t109+t92$	$c8= t114$
$t10=a1*b2$	$t31=a6*b8$	$t52=t42*t43$	$t73=t71+t72$	$t94=t80*t82$	$t115=t105+t38$	$c9= t123$
$t11=a2*b1$	$t32=a7*b7$	$t53=t50+t51$	$t74=t60*t64$	$t95=t93+t94$	$t116=t106+t39$	$c10= t124$
$t12=t10+t11$	$t33=a8*b6$	$t54=t52+t53$	$t75=t61*t63$	$t96=t80*t83$	$t117=t115+t97$	$c11= t122$
$t13=a2*b2$	$t34=t31+t32$	$t55=t41*t45$	$t76=t74+t75$	$t97=t12+t14$	$t118=t116+t98$	$c12= t125$
$t14=a3*b3$	$t35=t33+t34$	$t56=t42*t44$	$t77=t61*t64$	$t98=t13+t17$	$t119=t35+t22$	$c13= t126$
$t15=a3*b4$	$t36=a7*b8$	$t57=t55+t56$	$t78=a0+a6$	$t99=t97+t1$	$t120=t46+t117$	$c14= t35$
$t16=a4*b3$	$t37=a8*b7$	$t58=t42*t45$	$t79=a1+a7$	$t100=t98+t4$	$t121=t49+t118$	$c15= t38$
$t17=t15+t16$	$t38=t36+t37$	$t59=a0+a3$	$t80=a2+a8$	$t101=t22+t9$	$t122=t54+t119$	$c16= t39$
$t18=a3*b5$	$t39=a8*b8$	$t60=a1+a4$	$t81=b0+b6$	$t102=t99+t65$	$t123=t95+t120$
$t19=a4*b4$	$t40=a3+a6$	$t61=a2+a5$	$t82=b1+b7$	$t103=t100+t68$	$t124=t96+t121$
$t20=a5*b3$	$t41=a4+a7$	$t62=b0+b3$	$t83=b2+b8$	$t104=t101+t73$	$t125=t57+t115$
$t21=t18+t19$	$t42=a5+a8$	$t63=b1+b4$	$t84=t78*t81$	$t105=t25+t27$	$t126=t58+t116$

For $n=15,\, A=\sum _{i=0}^{14}a[i]X^i,\, B=\sum _{i=0}^{14}a[i]X^i$ and $C=AB=\sum _{i=0}^{28}c[i]X^i$. The coefficients of $C$ are computed using the following algorithm:

Algorithm for $n=15$
$t1=a[0]*b[0]$	$t59=a[14]*b[12]$	$t117=t114+t115$	$t175=t174+t173$	$t233=t230+t220$	$t291=t276+t288$
$t2=a[0]*b[1]$	$t60=t57+t58$	$t118=t117+t116$	$t176=t162*t166$	$t234=t231+t221$	$t292=t277+t289$
$t3=a[1]*b[0]$	$t61=t60+t59$	$t119=t105*t109$	$t177=t163*t165$	$t235=t232+t222$	$t293=t233+t265$
$t4=t2+t3$	$t62=a[13]*b[14]$	$t120=t106*t108$	$t178=t176+t177$	$t236=t226+t218$	$t294=t234+t266$
$t5=a[0]*b[2]$	$t63=a[14]*b[13]$	$t121=t119+t120$	$t179=t163*t166$	$t237=t227+t219$	$t295=t235+t61$
$t6=a[1]*b[1]$	$t64=t62+t63$	$t122=t106*t109$	$t180=t123+t66$	$t238=t35+t9$	$t296=t284+t293$
$t7=a[2]*b[0]$	$t65=a[14]*b[14]$	$t123=a[12]+a[9]$	$t181=t124+t67$	$t239=t40+t38$	$t297=t285+t294$
$t8=t5+t6$	$t66=a[3]+a[0]$	$t124=a[13]+a[10]$	$t182=t125+t68$	$t240=t43+t39$	$t298=t286+t295$
$t9=t8+t7$	$t67=a[4]+a[1]$	$t125=a[14]+a[11]$	$t183=t126+t69$	$t241=t239+t236$	$t299=t178+t186$
$t10=a[1]*b[2]$	$t68=a[5]+a[2]$	$t126=b[12]+b[9]$	$t184=t127+t70$	$t242=t240+t237$	$t300=t179+t189$
$t11=a[2]*b[1]$	$t69=b[3]+b[0]$	$t127=b[13]+b[10]$	$t185=t128+t71$	$t243=t48+t238$	$t301=t296+t299$
$t12=t10+t11$	$t70=b[4]+b[1]$	$t128=b[14]+b[11]$	$t186=t180*t183$	$t244=t53+t241$	$t302=t297+t300$
$t13=a[2]*b[2]$	$t71=b[5]+b[2]$	$t129=t123*t126$	$t187=t180*t184$	$t245=t56+t242$	$t303=t298+t194$
$t14=a[3]*b[3]$	$t72=t66*t69$	$t130=t123*t127$	$t188=t181*t183$	$t246=t61+t243$	$t304=t1+t244$
$t15=a[3]*b[4]$	$t73=t66*t70$	$t131=t124*t126$	$t189=t187+t188$	$t247=t110+t102$	$t305=t4+t245$
$t16=a[4]*b[3]$	$t74=t67*t69$	$t132=t130+t131$	$t190=t180*t185$	$t248=t113+t103$	$t306=t9+t246$
$t17=t15+t16$	$t75=t73+t74$	$t133=t123*t128$	$t191=t181*t184$	$t249=t247+t244$	$t307=t64+t304$
$t18=a[3]*b[5]$	$t76=t66*t71$	$t134=t124*t127$	$t192=t182*t183$	$t250=t248+t245$	$t308=t65+t305$
$t19=a[4]*b[4]$	$t77=t67*t70$	$t135=t125*t126$	$t193=t190+t191$	$t251=t118+t246$	$t309=t247+t307$
$t20=a[5]*b[3]$	$t78=t68*t69$	$t136=t133+t134$	$t194=t193+t192$	$t252=t186+t148$	$t310=t248+t308$
$t21=t18+t19$	$t79=t76+t77$	$t137=t136+t135$	$t195=t181*t185$	$t253=t189+t151$	$t311=t118+t306$
$t22=t21+t20$	$t80=t79+t78$	$t138=t124*t128$	$t196=t182*t184$	$t254=t194+t156$	$t312=t178+t309$
$t23=a[4]*b[5]$	$t81=t67*t71$	$t139=t125*t127$	$t197=t195+t196$	$t255=t252+t205$	$t313=t179+t310$
$t24=a[5]*b[4]$	$t82=t68*t70$	$t140=t125*t128$	$t198=t182*t185$	$t256=t253+t208$	$t314=t197+t312$
$t25=t23+t24$	$t83=t81+t82$	$t141=t138+t139$	$t199=t180+a[6]$	$t257=t254+t213$	$t315=t198+t313$
$t26=a[5]*b[5]$	$t84=t68*t71$	$t142=a[9]+t85$	$t200=t181+a[7]$	$t258=t249+t255$	$t316=t216+t314$
$t27=a[6]*b[6]$	$t85=a[6]+a[0]$	$t143=a[10]+t86$	$t201=t182+a[8]$	$t259=t250+t256$	$t317=t217+t315$
$t28=a[6]*b[7]$	$t86=a[7]+a[1]$	$t144=a[11]+t87$	$t202=t183+b[6]$	$t260=t251+t257$	$c0=t1$
$t29=a[7]*b[6]$	$t87=a[8]+a[2]$	$t145=b[9]+t88$	$t203=t184+b[7]$	$t261=t53+t51$	$c1=t4$
$t30=t28+t29$	$t88=b[6]+b[0]$	$t146=b[10]+t89$	$t204=t185+b[8]$	$t262=t56+t52$	$c2=t9$
$t31=a[6]*b[8]$	$t89=b[7]+b[1]$	$t147=b[11]+t90$	$t205=t199*t202$	$t263=t261+t64$	$c3=t223$
$t32=a[7]*b[7]$	$t90=b[8]+b[2]$	$t148=t142*t145$	$t206=t199*t203$	$t264=t262+t65$	$c4=t224$
$t33=a[8]*b[6]$	$t91=t85*t88$	$t149=t142*t146$	$t207=t200*t202$	$t265=t263+t141$	$c5=t225$
$t34=t31+t32$	$t92=t85*t89$	$t150=t143*t145$	$t208=t206+t207$	$t266=t264+t140$	$c6=t233$
$t35=t34+t33$	$t93=t86*t88$	$t151=t149+t150$	$t209=t199*t204$	$t267=t263+t239$	$c7=t234$
$t36=a[7]*b[8]$	$t94=t92+t93$	$t152=t142*t147$	$t210=t200*t203$	$t268=t264+t240$	$c8=t235$
$t37=a[8]*b[7]$	$t95=t85*t90$	$t153=t143*t146$	$t211=t201*t202$	$t269=t61+t48$	$c9=t258$
$t38=t36+t37$	$t96=t86*t89$	$t154=t144*t145$	$t212=t209+t210$	$t270=t121+t129$	$c10=t259$
$t39=a[8]*b[8]$	$t97=t87*t88$	$t155=t152+t153$	$t213=t212+t211$	$t271=t122+t132$	$c11=t260$
$t40=a[9]*b[9]$	$t98=t95+t96$	$t156=t155+t154$	$t214=t200*t204$	$t272=t267+t270$	$c12=t290$
$t41=a[9]*b[10]$	$t99=t98+t97$	$t157=t143*t147$	$t215=t201*t203$	$t273=t268+t271$	$c13=291$
$t42=a[10]*b[9]$	$t100=t86*t90$	$t158=t144*t146$	$t216=t214+t215$	$t274=t269+t137$	$c14=t292$
$t43=t41+t42$	$t101=t87*t89$	$t159=t157+t158$	$t217=t201*t204$	$t275=t272+t223$	$c15=t301$
$t44=a[9]*b[11]$	$t102=t100+t101$	$t160=t144*t147$	$t218=t12+t1$	$t276=t273+t224$	$c16=t302$
$t45=a[10]*b[10]$	$t103=t87*t90$	$t161=t104+a[3]$	$t219=t13+t4$	$t277=t274+t225$	$c17=t303$
$t46=a[11]*b[9]$	$t104=a[12]+a[6]$	$t162=t105+a[4]$	$t220=t14+t218$	$t278=t159+t167$	$c18=t316$
$t47=t44+t45$	$t105=a[13]+a[7]$	$t163=t106+a[5]$	$t221=t17+t219$	$t279=t160+t170$	$c19=t317$
$t48=t47+t46$	$t106=a[14]+a[8]$	$t164=t107+b[3]$	$t222=t22+t9$	$t280=t205+t216$	$c20=t311$
$t49=a[10]*b[11]$	$t107=b[12]+b[6]$	$t165=t108+b[4]$	$t223=t72+t220$	$t281=t208+t217$	$c21=t272$
$t50=a[11]*b[10]$	$t108=b[13]+b[7]$	$t166=t109+b[5]$	$t224=t75+t221$	$t282=t148+t197$	$c22=t273$
$t51=t49+t50$	$t109=b[14]+b[8]$	$t167=t161*t164$	$t225=t80+t222$	$t283=t151+t198$	$c23=t274$
$t52=a[11]*b[11]$	$t110=t104*t107$	$t168=t161*t165$	$t226=t27+t25$	$t284=t278+t280$	$c24=t265$
$t53=a[12]*b[12]$	$t111=t104*t108$	$t169=t162*t164$	$t227=t30+t26$	$t285=t279+t281$	$c25=t266$
$t54=a[12]*b[13]$	$t112=t105*t107$	$t170=t168+t169$	$t228=t91+t83$	$t286=t175+t213$	$c26=t61$
$t55=a[13]*b[12]$	$t113=t111+t112$	$t171=t161*t166$	$t229=t94+t84$	$t287=t282+t284$	$c27=t64$
$t56=t54+t55$	$t114=t104*t109$	$t172=t162*t165$	$t230=t228+t226$	$t288=t283+t285$	$c28=t65$
$t57=a[12]*b[14]$	$t115=t105*t108$	$t173=t163*t164$	$t231=t229+t227$	$t289=t156+t286$
$t58=a[13]*b[13]$	$t116=t106*t107$	$t174=t171+t172$	$t232=t99+t35$	$t290=t275+t287$