UNB ECE4253 Digital Communications
Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada

Addition and Multiplication Tables in Galois Fields GF(2m)

* See GF(2m) calculator tool.

This page presents addition and multiplication tables for Galois fields GF(2m).


Using the Galois Field GF(25) = GF(32) based on the primitive P(x) = x5 + x2 + 1 = (100101) = 37 (decimal)

Addition Table

Values in GF(25) are 5-bits each, spanning the decimal range [0..31]. Addition takes place on these 5-bit binary values using bitwise XOR.

For example: 4 + 9 = (00100) + (01001) = (01101) = 13   (highlighted below)

The choice of polynomial P(x) plays no role in the addition operation.

+ 0 1 2 3 4 5 6 7 89 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
0012345678910111213141516171819202122232425262728293031
1103254769811101312151417161918212023222524272629283130
2230167451011891415121318191617222320212627242530312829
3321076541110981514131219181716232221202726252431302928
4456701231213141589101120212223161718192829303124252627
5547610321312151498111021202322171619182928313025242726
6674523011415121310118922232021181916173031282926272425
7765432101514131211109823222120191817163130292827262524
8891011121314150123456724252627282930311617181920212223
9981110131215141032547625242726292831301716191821202322
10101189141512132301674526272425303128291819161722232021
11111098151413123210765427262524313029281918171623222120
12121314158910114567012328293031242526272021222316171819
13131215149811105476103229283130252427262120232217161918
14141512131011896745230130312829262724252223202118191617
15151413121110987654321031302928272625242322212019181716
16161718192021222324252627282930310123456789101112131415
17171619182120232225242726292831301032547698111013121514
18181916172223202126272425303128292301674510118914151213
19191817162322212027262524313029283210765411109815141312
20202122231617181928293031242526274567012312131415891011
21212023221716191829283130252427265476103213121514981110
22222320211819161730312829262724256745230114151213101189
23232221201918171631302928272625247654321015141312111098
24242526272829303116171819202122238910111213141501234567
25252427262928313017161918212023229811101312151410325476
26262724253031282918191617222320211011891415121323016745
27272625243130292819181716232221201110981514131232107654
28282930312425262720212223161718191213141589101145670123
29292831302524272621202322171619181312151498111054761032
30303128292627242522232021181916171415121310118967452301
31313029282726252423222120191817161514131211109876543210

Multiplication Table

Values in GF(25) are 5-bits each, spanning the decimal range [0..31]. Multiplication takes place on 5-bit binary values (with modulo 2 addition) and then the result is computed modulo P(x) = (100101) = 37 (decimal).

For example: 6 × 4 = (00110) × (00100) =  (11000)  = (11000)  mod (100101) = 24   (highlighted below)

The specific polynomial P(x) provides the modulus for the multiplication results.

× 1 2 34 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
112345678910111213141516171819202122232425262728293031
224681012141618202224262830571313159112123171929312527
336512151092427302920231817212219162526312813141181274
448121620242851139211729251014262630182215117331272319
551015201730271387225281922263116211411412318292436912
661210243020182119253113111715935231727292628221624148
771492827182129261920161583124172234131025121130251623
881624513212910226181573123202841217251930221462719113
991827181926211162531017244132231512233061520297142128
1010203013725192616144232939172751528228211131216121824
1111222992312018254152716136110232883302119245142617127
1212242021251311532327262221430186101171931172995482816
1313262317281167102916222712114320253118589419302421215
1414281829191153117313212301627217968262041024222523511
1515301725227823249614116311142126182912328192135102720
1616521102615312041713014271113298247232182592812193226
1717722143192428132710183214291226111922051166231530825
1818119216317422523620721826927102411251230133114281529
1919316621522123115281025926241127830132914207234181172
2020132526142331752881131618719103029916422227151224121
2121152630111742512223718829232241392861914271201653110
2222931184271312383019526122201129166251532110281772414
2323112822129109302213182031852514419152427121671326617
2424211315232623061119179428251122022143277311810816295
2525231411182852215124294101991630722721123168172013326
2626171172922121420315919242286132327110161883252115430
2727198324161162921145302213122331415202871017252918261
2828291313230277626424255191514181216171382021923111022
2929312276425191412178212310330281245726161315181122209
3030257239141611211812282527228151713124629342610201913
3131274191282332824716151120625292211014175263012291318

Select a primitive polynomial P(x)

2020-07-06 00:12:27 ADT
Last Updated: 2011-02-02
Richard Tervo [ tervo@unb.ca ] Back to the course homepage...