Results from Multiple Precision Integer Multiplication Algorithms (c) Sunday, February 15, 2026, by James Pate Williams, Jr. Using Professor Emeritus Arjen K. Lenstra’s Large Integer Package (lip)

The following results illustrate the author’s and Professor Emeritus Arjen K. Lenstra’s multiplication algorithms. The methods in alphabetical order are:
1.	Bodrato’s modification of the Toom-Cook multiplication technique
2.	Lenstra’s large integer package (lip) built-in multiple precision method
3.	Long multiplication from pseudocode in Wikipedia
4.	Toom-Cook method from Cook’s 1966 PhD thesis
References:
1.	Papers by Marco Bodrato
2.	Multiplication - Wikipedia
3.	cr.yp.to/bib/1966/cook.html
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 4
Enter b = 2147483647
Enter M = 1234567890
Enter N = 1234567890
Number of repetitions = 1
M       = 971334442946674600221231647138533765049334517138898208814432641876416006\
400115736571
N       = 971334442946674600221231647138533765049334517138898208814432641876416006\
400115736571
Toom-Cook Multiplication
Average Runtime   = 0.000074200
Over 1 Repetitions
MN   = 943490600054526654019119035756125944918077526277676439897594263047388314\
836233229704805255157650098574056832103475286502442743200967269892687335\
748700513503753866838041
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 2
Enter b = 2147483647
Enter M = 1234567890
Enter N = 1234567890
Number of repetitions = 1
M       = 971334442946674600221231647138533765049334517138898208814432641876416006\
400115736571
N       = 971334442946674600221231647138533765049334517138898208814432641876416006\
400115736571
Lenstra lip Multiplication
Average Runtime   = 0.000007400
Over 1 Repetitions
MN   = 943490600054526654019119035756125944918077526277676439897594263047388314\
836233229704805255157650098574056832103475286502442743200967269892687335\
748700513503753866838041
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 4
Enter b = 2147483647
Enter M = 12345678901234567890
Enter N = 12345678901234567890
Number of repetitions = 1
M       = 2026130632828332098161978694183179835926961551850814936441274855035774\
533901237908301792067190081080660611853406806309172584227706257836105490\
238900767068890331810595558979010550
N       = 2026130632828332098161978694183179835926961551850814936441274855035774\
533901237908301792067190081080660611853406806309172584227706257836105490\
238900767068890331810595558979010550
Toom-Cook Multiplication
Average Runtime   = 0.000098600
Over 1 Repetitions
MN   = 41052053412853374997957659248772276766670464267610246450081646198672053\
134774107322853407211671548238594780193988795119754495077504801036773948\
450621659941905079488598768958918884281745778245753016460241195698241515\
318663783334584789872121452838087229440437832037084270117697010189272087\
88688141014237093579164596571668979636977920478671088735457011302500
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 2
Enter b = 2147483647
Enter M = 12345678901234567890
Enter N = 12345678901234567890
Number of repetitions = 1
M       = 2026130632828332098161978694183179835926961551850814936441274855035774\
533901237908301792067190081080660611853406806309172584227706257836105490\
238900767068890331810595558979010550
N       = 2026130632828332098161978694183179835926961551850814936441274855035774\
533901237908301792067190081080660611853406806309172584227706257836105490\
238900767068890331810595558979010550
Lenstra lip Multiplication
Average Runtime   = 0.000009300
Over 1 Repetitions
MN   = 41052053412853374997957659248772276766670464267610246450081646198672053\
134774107322853407211671548238594780193988795119754495077504801036773948\
450621659941905079488598768958918884281745778245753016460241195698241515\
318663783334584789872121452838087229440437832037084270117697010189272087\
88688141014237093579164596571668979636977920478671088735457011302500
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 5

D:\FLMultiplication\x64\Release\FLMultiplication.exe (process 20168) exited with code 0 (0x0).
Press any key to close this window . . .

== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 4
Enter b = 2147483647
Enter M = 123456789012345678901234567890
Enter N = 123456789012345678901234567890
Number of repetitions = 1
M       = 42263561959479590219289335715252528051946761885385061954793363463431018\
850749052611816045464227557359947957390833136631219765211655242009436305\
106409335048399270235917319942885139182317225885422762482775210395122870\
45968780928724406323190056412004806954312198921740353521
N       = 42263561959479590219289335715252528051946761885385061954793363463431018\
850749052611816045464227557359947957390833136631219765211655242009436305\
106409335048399270235917319942885139182317225885422762482775210395122870\
45968780928724406323190056412004806954312198921740353521
Toom-Cook Multiplication
Average Runtime   = 0.000183000
Over 1 Repetitions
MN   = 1786208669502770299576960480677491594428843761318376718810830214483008\
522779876058101551852828273108216537734660930558577510542169332383202300\
806637802232078747108185518603023831159502647581878153273000030245312879\
938941795477296487228094659168732765010051027312450437648026306828037634\
862435594225658630177203603798218997466111668748532465924932822495598422\
823492688498989692318248236025719132743625324592268073825418742555965769\
253184917848571722561153588394570162730241248387724570796710835550233766\
6905434432429374209450377625018057097441
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 2
Enter b = 2147483647
Enter M = 123456789012345678901234567890
Enter N = 123456789012345678901234567890
Number of repetitions = 1
M       = 42263561959479590219289335715252528051946761885385061954793363463431018\
850749052611816045464227557359947957390833136631219765211655242009436305\
106409335048399270235917319942885139182317225885422762482775210395122870\
45968780928724406323190056412004806954312198921740353521
N       = 42263561959479590219289335715252528051946761885385061954793363463431018\
850749052611816045464227557359947957390833136631219765211655242009436305\
106409335048399270235917319942885139182317225885422762482775210395122870\
45968780928724406323190056412004806954312198921740353521
Lenstra lip Multiplication
Average Runtime   = 0.000058300
Over 1 Repetitions
MN   = 1786208669502770299576960480677491594428843761318376718810830214483008\
522779876058101551852828273108216537734660930558577510542169332383202300\
806637802232078747108185518603023831159502647581878153273000030245312879\
938941795477296487228094659168732765010051027312450437648026306828037634\
862435594225658630177203603798218997466111668748532465924932822495598422\
823492688498989692318248236025719132743625324592268073825418742555965769\
253184917848571722561153588394570162730241248387724570796710835550233766\
6905434432429374209450377625018057097441
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 5

D:\FLMultiplication\x64\Release\FLMultiplication.exe (process 25124) exited with code 0 (0x0).
Press any key to close this window . . .

== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 4
Enter b = 1000000000000000000000000000000
Enter M = 123456789012345678901234567890
Enter N = 123456789012345678901234567890
Number of repetitions = 1
M       = 10000000000000000000000000000020000000000000000000000000000030000000000\
000000000000000000040000000000000000000000000000050000000000000000000000\
000000060000000000000000000000000000070000000000000000000000000000080000\
000000000000000000000000090000000000000000000000000000000000000000000000\
000000000000010000000000000000000000000000020000000000000000000000000000\
030000000000000000000000000000040000000000000000000000000000050000000000\
000000000000000000060000000000000000000000000000070000000000000000000000\
000000080000000000000000000000000000090000000000000000000000000000000000\
000000000000000000000000010000000000000000000000000000020000000000000000\
000000000000030000000000000000000000000000040000000000000000000000000000\
050000000000000000000000000000060000000000000000000000000000070000000000\
000000000000000000080000000000000000000000000000090000000000000000000000\
00000000
N       = 10000000000000000000000000000020000000000000000000000000000030000000000\
000000000000000000040000000000000000000000000000050000000000000000000000\
000000060000000000000000000000000000070000000000000000000000000000080000\
000000000000000000000000090000000000000000000000000000000000000000000000\
000000000000010000000000000000000000000000020000000000000000000000000000\
030000000000000000000000000000040000000000000000000000000000050000000000\
000000000000000000060000000000000000000000000000070000000000000000000000\
000000080000000000000000000000000000090000000000000000000000000000000000\
000000000000000000000000010000000000000000000000000000020000000000000000\
000000000000030000000000000000000000000000040000000000000000000000000000\
050000000000000000000000000000060000000000000000000000000000070000000000\
000000000000000000080000000000000000000000000000090000000000000000000000\
00000000
Toom-Cook Multiplication
Average Runtime   = 0.000399800
Over 1 Repetitions
MN   = 100000000000000000000000000000400000000000000000000000000001000000000\
000000000000000000002000000000000000000000000000003500000000000000000000\
000000005600000000000000000000000000008400000000000000000000000000012000\
000000000000000000000000016500000000000000000000000000020000000000000000\
000000000000022600000000000000000000000000024400000000000000000000000000\
025500000000000000000000000000026000000000000000000000000000026000000000\
000000000000000000025600000000000000000000000000024900000000000000000000\
000000024000000000000000000000000000033000000000000000000000000000040000\
000000000000000000000000045100000000000000000000000000048400000000000000\
000000000000050000000000000000000000000000050000000000000000000000000000\
048500000000000000000000000000045600000000000000000000000000041400000000\
000000000000000000036000000000000000000000000000049500000000000000000000\
000000060000000000000000000000000000067400000000000000000000000000071600\
000000000000000000000000072500000000000000000000000000070000000000000000\
000000000000064000000000000000000000000000054400000000000000000000000000\
041100000000000000000000000000024000000000000000000000000000033000000000\
000000000000000000040000000000000000000000000000044900000000000000000000\
000000047600000000000000000000000000048000000000000000000000000000046000\
000000000000000000000000041500000000000000000000000000034400000000000000\
000000000000024600000000000000000000000000012000000000000000000000000000\
016500000000000000000000000000020000000000000000000000000000022400000000\
000000000000000000023600000000000000000000000000023500000000000000000000\
000000022000000000000000000000000000019000000000000000000000000000014400\
000000000000000000000000008100000000000000000000000000000000000000000000\
0000000000000000
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 2
Enter b = 1000000000000000000000000000000
Enter M = 123456789012345678901234567890
Enter N = 123456789012345678901234567890
Number of repetitions = 1
M       = 10000000000000000000000000000020000000000000000000000000000030000000000\
000000000000000000040000000000000000000000000000050000000000000000000000\
000000060000000000000000000000000000070000000000000000000000000000080000\
000000000000000000000000090000000000000000000000000000000000000000000000\
000000000000010000000000000000000000000000020000000000000000000000000000\
030000000000000000000000000000040000000000000000000000000000050000000000\
000000000000000000060000000000000000000000000000070000000000000000000000\
000000080000000000000000000000000000090000000000000000000000000000000000\
000000000000000000000000010000000000000000000000000000020000000000000000\
000000000000030000000000000000000000000000040000000000000000000000000000\
050000000000000000000000000000060000000000000000000000000000070000000000\
000000000000000000080000000000000000000000000000090000000000000000000000\
00000000
N       = 10000000000000000000000000000020000000000000000000000000000030000000000\
000000000000000000040000000000000000000000000000050000000000000000000000\
000000060000000000000000000000000000070000000000000000000000000000080000\
000000000000000000000000090000000000000000000000000000000000000000000000\
000000000000010000000000000000000000000000020000000000000000000000000000\
030000000000000000000000000000040000000000000000000000000000050000000000\
000000000000000000060000000000000000000000000000070000000000000000000000\
000000080000000000000000000000000000090000000000000000000000000000000000\
000000000000000000000000010000000000000000000000000000020000000000000000\
000000000000030000000000000000000000000000040000000000000000000000000000\
050000000000000000000000000000060000000000000000000000000000070000000000\
000000000000000000080000000000000000000000000000090000000000000000000000\
00000000
Lenstra lip Multiplication
Average Runtime   = 0.000076500
Over 1 Repetitions
MN   = 100000000000000000000000000000400000000000000000000000000001000000000\
000000000000000000002000000000000000000000000000003500000000000000000000\
000000005600000000000000000000000000008400000000000000000000000000012000\
000000000000000000000000016500000000000000000000000000020000000000000000\
000000000000022600000000000000000000000000024400000000000000000000000000\
025500000000000000000000000000026000000000000000000000000000026000000000\
000000000000000000025600000000000000000000000000024900000000000000000000\
000000024000000000000000000000000000033000000000000000000000000000040000\
000000000000000000000000045100000000000000000000000000048400000000000000\
000000000000050000000000000000000000000000050000000000000000000000000000\
048500000000000000000000000000045600000000000000000000000000041400000000\
000000000000000000036000000000000000000000000000049500000000000000000000\
000000060000000000000000000000000000067400000000000000000000000000071600\
000000000000000000000000072500000000000000000000000000070000000000000000\
000000000000064000000000000000000000000000054400000000000000000000000000\
041100000000000000000000000000024000000000000000000000000000033000000000\
000000000000000000040000000000000000000000000000044900000000000000000000\
000000047600000000000000000000000000048000000000000000000000000000046000\
000000000000000000000000041500000000000000000000000000034400000000000000\
000000000000024600000000000000000000000000012000000000000000000000000000\
016500000000000000000000000000020000000000000000000000000000022400000000\
000000000000000000023600000000000000000000000000023500000000000000000000\
000000022000000000000000000000000000019000000000000000000000000000014400\
000000000000000000000000008100000000000000000000000000000000000000000000\
0000000000000000
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 5

D:\FLMultiplication\x64\Release\FLMultiplication.exe (process 22984) exited with code 0 (0x0).
Press any key to close this window . . .
 
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 1
Enter e in 10^e = 50
      b = 100000000000000000000000000000000000000000000000000
Enter U = 12345678901234567890123456789012345678901234567890
Enter V = 12345678901234567890123456789012345678901234567890
Enter R = 1
Bodrato Multiplication
Average Runtime   = 0.005534500
Over 1 Repetitions
W    = 15241578753238836750495351562566681945008382873375704923650053345576253\
6198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 2
Enter b = 10
Enter M = 12345678901234567890123456789012345678901234567890
Enter N = 12345678901234567890123456789012345678901234567890
Number of repetitions = 1
M       = 12345678901234567890123456789012345678901234567890
N       = 12345678901234567890123456789012345678901234567890
Lenstra lip Multiplication
Average Runtime   = 0.000005700
Over 1 Repetitions
MN   = 15241578753238836750495351562566681945008382873375704923650053345576253\
6198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 3
Enter b = 100000000000000000000000000000000000000000000000000
Enter A = 12345678901234567890123456789012345678901234567890
Enter B = 12345678901234567890123456789012345678901234567890
Enter # = 1
Long Multiplication
Average Runtime   = 0.000029500
Over 1 Repetitions
x    = 15241578753238836750495351562566681945008382873375704923650053345576253\
6198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 4
Enter b = 10
Enter M = 12345678901234567890123456789012345678901234567890
Enter N = 12345678901234567890123456789012345678901234567890
Number of repetitions = 1
M       = 12345678901234567890123456789012345678901234567890
N       = 12345678901234567890123456789012345678901234567890
Toom-Cook Multiplication
Average Runtime   = 0.000278200
Over 1 Repetitions
MN   = 15241578753238836750495351562566681945008382873375704923650053345576253\
6198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 5

D:\FLMultiplication\x64\Release\FLMultiplication.exe (process 9856) exited with code 0 (0x0).
Press any key to close this window . . .

== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 1
Enter e in 10^e = 100
      b = 100000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000
Enter U = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Enter V = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Enter R = 1
Bodrato Multiplication
Average Runtime   = 0.007581100
Over 1 Repetitions
W    = 15241578753238836750495351562566681945008382873376009755225118122311263\
526910001524158887669562677515622630876390795200121932731260478594250876\
39153757049236500533455762536198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 2
Enter b = 10
Enter M = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Enter N = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Number of repetitions = 1
M       = 123456789012345678901234567890123456789012345678901234567890123456789012\
3456789012345678901234567890
N       = 123456789012345678901234567890123456789012345678901234567890123456789012\
3456789012345678901234567890
Lenstra lip Multiplication
Average Runtime   = 0.000004300
Over 1 Repetitions
MN   = 15241578753238836750495351562566681945008382873376009755225118122311263\
526910001524158887669562677515622630876390795200121932731260478594250876\
39153757049236500533455762536198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 3
Enter b = 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Enter A = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Enter B = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Enter # = 1
Long Multiplication
Average Runtime   = 0.000070000
Over 1 Repetitions
x    = 15241578753238836750495351562566681945008382873376009755225118122311263\
526910001524158887669562677515622630876390795200121932731260478594250876\
39153757049236500533455762536198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 4
Enter b = 10
Enter M = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Enter N = 1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890
Number of repetitions = 1
M       = 123456789012345678901234567890123456789012345678901234567890123456789012\
3456789012345678901234567890
N       = 123456789012345678901234567890123456789012345678901234567890123456789012\
3456789012345678901234567890
Toom-Cook Multiplication
Average Runtime   = 0.000396200
Over 1 Repetitions
MN   = 15241578753238836750495351562566681945008382873376009755225118122311263\
526910001524158887669562677515622630876390795200121932731260478594250876\
39153757049236500533455762536198787501905199875019052100
== Menu ==
1 Bodrato Multiplication
2 Lenstra lip Multiplication
3 Long Multiplication
4 Toom-Cook Algorithm
5 Exit
Option (1 - 5) = 5

D:\FLMultiplication\x64\Release\FLMultiplication.exe (process 20640) exited with code 0 (0x0).
Press any key to close this window . . .
References:
Author: jamespatewilliamsjr

My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises. View all posts by jamespatewilliamsjr
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