Integer Factorization: Trial Division Algorithm
Small review of Trial Division algorithm
Contents
- Background
- Introduction
- Trial Division: The Classic Variants
- Benchmarks
- Checking Correctness of Variants
- Points of Interest
- Links
- History
Background
This article is part of a set of articles. Each article highlights an aspect around Integer Factorization.
- Integer Factorization: Trial Division Algorithm[^] : Focuses on variants of Trial Division algorithm and their respective efficiency.
- Integer Factorization: Dreaded list of primes[^] : Focuses on a method to handle a large list of Primes by compression.
- Integer Factorization: Optimizing Small Factors Checking[^] : Focuses on a method to check small factors faster than division.
- Integer factorization: Reversing the Multiplication[^] : Focuses on another approach to optimize the Trial Division algorithm.
The xPrompt Download and prg Files
xPrompt.zip contains the binary of the xHarbour Scripting Engine for Windows. This is the app needed to run the prg code.
Usage:
- Copy prg file in same directory than xPrompt
- Launch xPrompt
- Type "
do FileName.prg" to run the code
xHarbour is freeware and can be downloaded at xHarbour.org[^]
xHarbour is part of the xBase family languages: xBase - Wikipedia[^]
Flowcharts
The flowcharts are related to this article : TopDown Analyze Method with Tree Graphs as Support[^]
If image is too small, open in a new tab and zoom, it is a svg file.
Introduction
When starting to play with Integer Factorization, trying all possible factors is the first idea, that algorithm is named Trial Division.
The algorithm has two purposes - finding a prime factor or finding if an integer is a prime by not by finding a prime factor.
Since the algorithm is about finding a factor, the worst case is when the integer to factorize is a prime.
Trial Division: The Classic Variants
It looks obvious that the most efficient method is to check only prime numbers, but handling the list of primes is a problem by itself. The classical variants exist as solutions to avoid that problem. As methods get more sophisticated, they remove more useless non prime numbers, thus being more efficient.
Brute Force
Some newbies are testing every single number below n. That's overkill. The inefficiency remains hidden as long the integer to factorize is not a prime.
The maximum cost for n is n-2 divisions. Complexity is O(n).
// Trial Division: Brute Force 1
// Check all numbers until Cand - 1
long long TD_BF1(long long Cand) {
Count = 0;
long long Top = Cand - 1;
for (long long Div = 2; Div <= Top; Div++) {
Count++;
if (Cand % Div == 0) {
return Div;
}
}
return Cand;
}
// Trial Division Brute Force 1
// May 2019
function TD_BF1(Prod)
Local D, Top
Top= Prod-1
for D= 2 to Top
if Prod % D = 0
return D
endif
next
return Prod
In TrialDiv.cpp and TrialDiv.prg in TrialDiv.zip.
TD_BF1.graphml and TD_BF1.svg in TrialDiv.zip.
Testing all numbers until n/2 is better, but is also overkill. Just like with previous method, the inefficiency remains hidden as long the integer to factorize is not a prime.
The maximum cost for n is n / 2 divisions. Complexity is O(n).
// Trial Division: Brute Force 2
// Check all numbers until Cand / 2
long long TD_BF2(long long Cand) {
Count = 0;
long long Top = Cand / 2;
for (long long Div = 2; Div <= Top; Div++) {
Count++;
if (Cand % Div == 0) {
return Div;
}
}
return Cand;
}
// Trial Division Brute Force 2
// May 2019
function TD_BF2(Prod)
Local D, Top
Top= Prod/2
for D= 2 to Top
if Prod % D = 0
return D
endif
next
return Prod
In TrialDiv.cpp and TrialDiv.prg in TrialDiv.zip.
TD_BF2.graphml and TD_BF2.svg in TrialDiv.zip.
Basic Version: Square Root
A little analysis shows that there is no factor to check after the square root.
The maximum cost for n is √n divisions. Complexity id O(√n).
// Trial Division: Square Root
// Check all numbers until Square Root
long long TD_SR(long long Cand) {
Count = 0;
long long Top = sqrt(Cand);
for (long long Div = 2; Div <= Top; Div++) {
Count++;
if (Cand % Div == 0) {
return Div;
}
}
return Cand;
}
// Trial Division Square Root
// May 2019
function TD_SR(Prod)
Local D, Top
Top= int(sqrt(Prod))
for D= 2 to Top
if Prod % D = 0
return D
endif
next
return Prod
In TrialDiv.cpp and TrialDiv.prg in TrialDiv.zip.
TD_SR.graphml and TD_SR.svg in TrialDiv.zip.
Get Rid of Even Factors
With further analysis, one can see that there is no even factor beyond 2.
The maximum cost for n is (√n) * 1 / 2 => (√n) * 0.50 divisions. Complexity of is O(√n).
// Trial Division: Square Root + Even Factors
// Check all numbers until Square Root and Skip Even Factors
long long TD_SREF(long long Cand) {
Count = 0;
// check 2 the only Even Prime
Count++;
if (Cand % 2 == 0) {
return 2;
}
long long Top = sqrt(Cand);
for (long long Div = 3; Div <= Top; Div += 2) {
Count++;
if (Cand % Div == 0) {
return Div;
}
}
return Cand;
}
// Trial Division Square Root + Even Factors
// May 2019
function TD_SREF(Prod)
Local D, Top
if Prod % 2 = 0
return 2
endif
Top= int(sqrt(Prod))
for D= 3 to Top step 2
if Prod % D = 0
return D
endif
next
return Prod
In TrialDiv.cpp and TrialDiv.prg in TrialDiv.zip.
TD_SREF.graphml and TD_SREF.svg in TrialDiv.zip.
The Wheel
The wheel is an extension of the previous optimization. One builds the wheel from small primes, say 2 and 3, the size of the wheel is 2 * 3 = 6. When one writes a list of numbers in a 6 columns table and removes 2 and 3, one can see that primes are only in the first column and in the fifth column. That is the wheel, we only have to check numbers of columns 1 and 5.
| 1 | 2 | 3 | 4 | 5 | 6 |
| 7 | 8 | 9 | 10 | 11 | 12 |
| 13 | 14 | 15 | 16 | 17 | 18 |
| 19 | 20 | 21 | 22 | 23 | 24 |
| 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 |
| 37 | 38 | 39 | 40 | 41 | 42 |
| 43 | 44 | 45 | 46 | 47 | 48 |
| 49 | 50 | 51 | 52 | 53 | 54 |
| 55 | 56 | 57 | 58 | 59 | 60 |
The maximum cost for n is (√n) * (1 / 2) * (2 / 3) => (√n) * (1 / 3) => (√n) * 0.33 divisions. Complexity is O(√n).
// Trial Division Square Root + Wheel
// May 2019
<pre lang="c++">
// Trial Division: Square Root + Wheel
// Check all numbers until Square Root and Skip useless factors with a wheel
long long TD_SRW(long long Cand) {
long long SPrimes[] = { 2, 3, 0 };
long long Wheel[] = { 4, 2, 0 };
long long Div;
Count = 0;
long long Top = sqrt(Cand);
// Check small primes
for (long long Ind = 0; SPrimes[Ind] != 0; Ind++) {
Div = SPrimes[Ind]; // for debug purpose
if (SPrimes[Ind] > Top) {
return Cand;
}
Count++;
if (Cand % SPrimes[Ind] == 0) {
return SPrimes[Ind];
}
}
// Start the Wheel
Div = 1;
while (Div <= Top) {
for (long long Ind = 0; Wheel[Ind] != 0; Ind++) {
Div += Wheel[Ind];
if (Div > Top) {
break;
}
Count++;
if (Cand % Div == 0) {
return Div;
}
}
}
return Cand;
}
function TD_SRW(Prod)
local D, Top, SPrimes, Wheel, W
// Check small primes
SPrimes= {2, 3}
Wheel= {4, 2}
for each D in SPrimes
if Prod % D = 0
return D
endif
next
// Start the wheel
D= 1
Top= int(sqrt(Prod))
while D <= Top
for each W in wheel
D += W
if Prod % D = 0
return D
endif
next
enddo
return Prod
In TrialDiv.cpp and TrialDiv.prg in TrialDiv.zip.
TD_SRW.graphml and TD_SRW.svg in TrialDiv.zip.
The wheel can include more primes. With 2, 3 and 5, the size of wheel is 30.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
The maximum cost for n is (√n) * (1 / 2) * (2 / 3) * (4 / 5) => (√n) * (4 / 15) => (√n) * 0.27 divisions. Complexity is O(√n).
long long SPrimes[] = { 2, 3, 5, 0 };
long long Wheel[] = { 6, 4, 2, 4, 2, 4, 6, 2, 0 };
// only those 2 lines need change
SPrimes= {2, 3, 5}
Wheel= {6, 4, 2, 4, 2, 4, 6, 2}
Or get even bigger. With 2, 3, 5 and 7, the size of wheel is 210.
The maximum cost for n is (√n) * (1 / 2) * (2 / 3) * (4 / 5) * (6 / 7) => (√n) * (24 / 105) => (√n) * 0.23 divisions. Complexity is O(√n).
The List of Primes
Basically, it is a wheel variant, where the list of small primes exceeds the ones needed for the wheel, and after the end of list, we fall back on the wheel. The advantage over the simple wheel is that it avoids testing non prime factors not filtered by the wheel. As long as we are in the list of primes, the workload is optimum.
Just have to take care about setting the wheel correctly at the end of prime list.
The maximum cost for n is slightly better than the wheel, the list primes just save non prime divisors that are in the wheel. So the longer the list of primes, the better it gets. Complexity is O(√n).
// Trial Division: Square Root + Prime list + Wheel
// Check all numbers until Square Root, start with a list of primes
// and Skip useless factors with a wheel
long long TD_SRPW(long long Cand) {
long long SPrimes[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191,
193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263,
269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347,
349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421,
431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593,
599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661,
673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757,
761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853,
857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941,
947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021,
1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093,
1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181,
1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259,
1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321,
1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433,
1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493,
1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579,
1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657,
1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741,
1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831,
1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913,
1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003,
2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087,
2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161,
2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269,
2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347,
2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417,
2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531,
2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621,
2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693,
2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767,
2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851,
2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953,
2957, 2963, 2969, 2971, 2999, 0 };
long long Wheel[] = { 6, 4, 2, 4, 2, 4, 6, 2, 0 };
long long Div;
Count = 0;
long long Top = sqrt(Cand);
// Check small primes
for (long long Ind = 0; SPrimes[Ind] != 0; Ind++) {
Div = SPrimes[Ind]; // for debug purpose
if (SPrimes[Ind] > Top) {
return Cand;
}
Count++;
if (Cand % SPrimes[Ind] == 0) {
return SPrimes[Ind];
}
}
// Start the Wheel
Div = 3001;
while (Div <= Top) {
for (long long Ind = 0; Wheel[Ind] != 0; Ind++) {
if (Div > Top) {
return Cand;
}
Count++;
if (Cand % Div == 0) {
return Div;
}
Div += Wheel[Ind];
}
}
return Cand;
}
// Trial Division Square Root + Wheel + list of primes
// with a large list of primes
// May 2019
function TD_SRW2(Prod)
local D, Top, SPrimes, Wheel, W
// Check small primes
SPrimes= {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,
149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211,
223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293}
Wheel= {6, 4, 2, 4, 2, 4, 6, 2}
for each D in SPrimes
if Prod % D = 0
return D
endif
next
// Start the wheel
D= 301
Top= int(sqrt(Prod))
while D <= Top
for each W in wheel
D += W
if Prod % D = 0
return D
endif
next
enddo
return Prod
In TrialDiv.cpp and TrialDiv.prg in TrialDiv.zip.
Benchmarks
Benchmarks are here to highlight efficiency of variants.
As divisions/modulos are what cost time, counting them is enough for the benchmark.
Benchmark 1
This first benchmark is here to highlight the inefficiency of Brute Force variant.
long long Test[] = { 11, 31, 53, 71, 97, 0, 101, 1009, 2003, 3001, 4001,
5003, 6007, 7001, 8009, 9001, 10007, 20011, 30011, 40009, 49999,
1000003, 4000021, 9000011, 0 };
cout << "Comparaison Variantes avec Brute Force" << endl;
cout << "Number\tTD_BF1\tTD_BF2\tTD_SR\tTD_SREF\tTD_SRW\tTD_SRPW" << endl;
for (Ind = 0; Test[Ind] != 0; Ind++) {
cout << Test[Ind] << "\t";
TD_BF1(Test[Ind]);
cout << Count << "\t";
TD_BF2(Test[Ind]);
cout << Count << "\t";
TD_SR(Test[Ind]);
cout << Count << "\t";
TD_SREF(Test[Ind]);
cout << Count << "\t";
TD_SRW(Test[Ind]);
cout << Count << "\t";
TD_SRPW(Test[Ind]);
cout << Count << endl;
}
cout << endl;
Test = { 11, 31, 53, 71, 97, 101, 1009, 2003, 3001, 4001, 5003, 6007, 7001, 8009, 9001, 10007, 20011, 30011, 40009, 49999, 1000003, 4000021, 9000011 }
? "Comparaison Variantes avec Brute Force"
? "Number","TD_BF1","TD_BF2","TD_SR","TD_SREF","TD_SRW","TD_SRPW"
for Ind = 1 to 5
? Test[Ind]
TD_BF1(Test[Ind])
?? Count
TD_BF2(Test[Ind])
?? Count
TD_SR(Test[Ind])
?? Count
TD_SREF(Test[Ind])
?? Count
TD_SRW(Test[Ind])
?? Count
TD_SRPW(Test[Ind])
?? Count
next
?
Number TD_BF1 TD_BF2 TD_SR TD_SREF TD_SRW TD_SRPW
11 9 4 2 2 2 2
31 29 14 4 3 3 3
53 51 25 6 4 4 4
71 69 34 7 4 4 4
97 95 47 8 5 4 4
Benchmark 2
This extended benchmark is for variants other than Brute Force.
cout << "Comparaison Variantes sans Brute Force" << endl;
cout << "Number\tTD_SR\tTD_SREF\tTD_SRW\tTD_SRPW\tDelta" << endl;
// test after first 0
for (Ind++; Test[Ind] != 0; Ind++) {
cout << Test[Ind] << "\t";
TD_SR(Test[Ind]);
cout << Count << "\t";
TD_SREF(Test[Ind]);
cout << Count << "\t";
TD_SRW(Test[Ind]);
int C1 = Count;
cout << Count << "\t";
TD_SRPW(Test[Ind]);
cout << Count << "\t";
cout << C1 - Count << endl;
}
cout << endl;
? "Number","TD_SR","TD_SREF","TD_SRW","TD_SRPW"
for Ind = 6 to len(Test)
? Test[Ind]
TD_SR(Test[Ind])
?? Count
TD_SREF(Test[Ind])
?? Count
TD_SRW(Test[Ind])
?? Count
TD_SRPW(Test[Ind])
?? Count
next
?
Comparison Variants sans Brute Force
Number TD_SR TD_SREF TD_SRW TD_SRPW Delta
101 9 5 4 4 0
1009 30 16 11 11 0
2003 43 22 14 14 0
3001 53 27 17 16 1
4001 62 32 19 18 1
5003 69 35 20 19 1
6007 76 39 23 21 2
7001 82 42 25 23 2
8009 88 45 26 24 2
9001 93 47 27 24 3
10007 99 50 28 25 3
20011 140 71 40 34 6
30011 172 87 49 40 9
40009 199 100 56 46 10
49999 222 112 62 48 14
1000003 999 500 268 168 100
4000021 1800 901 483 279 204
9000011 2999 1500 802 430 372
Delta shows how the list of primes improves things, and it gets better with long list of primes.
Checking Correctness of Variants
Rgis code checks correctness of code by comparing results of variants.
cout << "Vérification Variantes" << endl;
for (long long Cand = 3; Cand < 10000000; Cand += 10) {
long long d1 = TD_SREF(Cand);
long long d2 = TD_SRPW(Cand);
if (d1 != d2) {
cout << Cand << "\t" << d1 << "\t" << d2 << endl;
}
}
Points of Interest
Trial Division being brute force, one can see that there is brute force and brute force.
Links
- Integer factorization: Integer factorization - Wikipedia[^]
- Trial division: Trial division - Wikipedia[^]
- Brute-force search: Brute-force search - Wikipedia[^]
History
- 1st April, 2019: First version
- 27th November, 2020: Second version
- 20th December, 2020: Third version: some cleaning, moved download to top
- 25th December, 2020: Fourth version: some cleaning and corrections
- 31st January, 2021: Corrections
- 30th January, 2022: Added table of contents and little update
- 16th September, 2023: Added flowcharts and corrections.
- 15th December, 2020: Typos