High-speed cryptography in characteristic 2

Introduction
Tools:
Linear maps
Polynomial multiplication:
Bit operations
Field multiplication:
Type-II bases
Elliptic curves:
Binary Edwards
Challenges
Error-correcting codes:
McBits

Minimum number of bit operations for multiplication

Define M(n) as the minimum number of bit operations (ANDs and XORs) needed to multiply an n-bit polynomial f[0] + f[1] t + ... + f[n-1] t^(n-1) by another n-bit polynomial g[0] + g[1] t + ... + g[n-1] t^(n-1), producing a (2n-1)-bit polynomial h[0] + h[1] t + ... + h[2n-2] t^(2n-2).

Schoolbook multiplication implies that M(n) ≤ 2n^2-2n+1. However, better bounds are known for all n ≥ 6. For example:

  • Schoolbook: M(64) ≤ 8065.
    2006 Weimerskirch--Paar: M(64) ≤ 4593 (3864 XORs and 729 ANDs).
    2007 Peter--Langendoerfer "classic Karatsuba multiplication": M(64) ≤ 4593.
    2007 Peter--Langendoerfer "improved iterative Karatsuba": M(64) ≤ 4250.
    2006 von zur Gathen--Shokrollahi Table 1: M(64) ≤ 3945.
    2003 Rodriguez-Henriquez--Koc Table 1: M(64) ≤ 3945.
    2000.08 Bernstein: M(64) ≤ 3725.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(64) ≤ 3682.
  • Schoolbook: M(113) ≤ 25313.
    2006 Weimerskirch--Paar: M(113) ≤ 12657 (10665 XORs and 1992 ANDs).
    2006 von zur Gathen--Shokrollahi Table 1: M(113) ≤ 11522.
    2005 Chang--Kim--Park--Lim "non-redundant Karatsuba-Ofman algorithm": M(113) ≤ 11138.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(113) ≤ 9534.
  • Schoolbook: M(128) ≤ 32513.
    2006 Weimerskirch--Paar: M(128) ≤ 14287 (12100 XORs and 2187 ANDs).
    2007 Peter--Langendoerfer "classic Karatsuba multiplication": M(128) ≤ 14287.
    2007 Peter--Langendoerfer "improved iterative Karatsuba": M(128) ≤ 13146.
    2006 von zur Gathen--Shokrollahi Table 1: M(128) ≤ 12343.
    2003 Rodriguez-Henriquez--Koc Table 1: M(128) ≤ 12343.
    2000.08 Bernstein: M(128) ≤ 11620.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(128) ≤ 11486.
  • Schoolbook: M(131) ≤ 34061.
    2005 Chang--Kim--Park--Lim "non-redundant Karatsuba-Ofman algorithm": M(131) ≤ 15939.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(131) ≤ 11961.
  • Schoolbook: M(163) ≤ 52813.
    2005 Chang--Kim--Park--Lim "non-redundant Karatsuba-Ofman algorithm": M(163) ≤ 21791.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(163) ≤ 16923.
  • Schoolbook: M(193) ≤ 74113.
    2003 Rodriguez-Henriquez--Koc Section 4.1: M(193) ≤ 29725 (20524 XORs and 9201 ANDs).
    2005 Chang--Kim--Park--Lim "non-redundant Karatsuba-Ofman algorithm": M(193) ≤ 27803.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(193) ≤ 21865.
  • Schoolbook: M(194) ≤ 74885.
    2006 von zur Gathen--Shokrollahi Section 3: M(194) ≤ 26575.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(194) ≤ 21906.
  • Schoolbook: M(251) ≤ 125501.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(251) ≤ 33096.
  • Schoolbook: M(256) ≤ 130561.
    2007 Peter--Langendoerfer "classic Karatsuba multiplication": M(256) ≤ 43881 (37320 XORs and 6561 ANDs).
    2007 Peter--Langendoerfer "improved iterative Karatsuba": M(256) ≤ 40415.
    2003 Rodriguez-Henriquez--Koc Table 1: M(256) ≤ 38049.
    2000.08 Bernstein: M(256) ≤ 35753.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(256) ≤ 34079.
  • Schoolbook: M(283) ≤ 159613.
    2005 Chang--Kim--Park--Lim "non-redundant Karatsuba-Ofman algorithm": M(283) ≤ 52828.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(283) ≤ 38735.
  • Schoolbook: M(512) ≤ 523265.
    2003 Rodriguez-Henriquez--Koc Table 1: M(512) ≤ 116191 (81199 XORs and 34992 ANDs).
    2000.08 Bernstein: M(512) ≤ 109048.
    2009.05 Bernstein "Batch binary Edwards" Section 2: M(512) ≤ 98018.

The following table presents upper bounds on M(1), M(2), ..., M(1000). The links in the table are to straight-line code justifying the upper bounds.

nM(n) ≤Notes
11
25
313
425
541
6572000.08 Bernstein
781
81002000.08 Bernstein
9132
101552000.08 Bernstein
111892009.05 Bernstein
122082009.05 Bernstein
13256
142892000.08 Bernstein
153292009.05 Bernstein
163502009.05 Bernstein
174142009.05 Bernstein
184562000.08 Bernstein
195022009.05 Bernstein
205282009.05 Bernstein
216022009.05 Bernstein
226412009.05 Bernstein
236782009.05 Bernstein
247042009.05 Bernstein
258002009.05 Bernstein
268562009.05 Bernstein
279222009.05 Bernstein
289562009.05 Bernstein
2910442009.05 Bernstein
3010852009.05 Bernstein
3111292009.05 Bernstein
3211582009.05 Bernstein
3312862009.05 Bernstein
3413582009.05 Bernstein
3514412009.05 Bernstein
3614832009.05 Bernstein
3715852009.05 Bernstein
3816362009.05 Bernstein
3916872009.05 Bernstein
4017202009.05 Bernstein
4118712009.05 Bernstein
4219502009.05 Bernstein
4320202009.05 Bernstein
4420642009.05 Bernstein
4521502009.05 Bernstein
4621922009.05 Bernstein
4722392009.05 Bernstein
4822682009.05 Bernstein
4924602009.05 Bernstein
5025722009.05 Bernstein
5126772009.05 Bernstein
5227352009.05 Bernstein
5328812009.05 Bernstein
5429482009.05 Bernstein
5530172009.05 Bernstein
5630602009.05 Bernstein
5732392009.05 Bernstein
5833202009.05 Bernstein
5934062009.05 Bernstein
6034562009.05 Bernstein
6135522009.05 Bernstein
6235952009.05 Bernstein
6336512009.05 Bernstein
6436822009.05 Bernstein
6539382009.05 Bernstein
6640502009.05 Bernstein
6741342009.05 Bernstein
6841832009.05 Bernstein
6944032009.05 Bernstein
7044522009.05 Bernstein
7144992009.05 Bernstein
7246422009.05 Bernstein
7348282009.05 Bernstein
7448642009.05 Bernstein
7550972009.05 Bernstein
7651332009.05 Bernstein
7752392009.05 Bernstein
7853222009.05 Bernstein
7953842009.05 Bernstein
8054202009.05 Bernstein
8157402009.05 Bernstein
8257992009.05 Bernstein
8358752009.05 Bernstein
8459962009.05 Bernstein
8561582009.05 Bernstein
8662022009.05 Bernstein
8763532009.05 Bernstein
8863972009.05 Bernstein
8964952009.05 Bernstein
9065682009.05 Bernstein
9166662009.05 Bernstein
9267172009.05 Bernstein
9369912009.05 Bernstein
9470432009.05 Bernstein
9570962009.05 Bernstein
9671322009.05 Bernstein
9775162009.05 Bernstein
9875742009.05 Bernstein
9978702009.05 Bernstein
10079092009.05 Bernstein
10180472009.05 Bernstein
10281842009.05 Bernstein
10383222009.05 Bernstein
10484042009.05 Bernstein
10586352009.05 Bernstein
10687172009.05 Bernstein
10788102009.05 Bernstein
10889592009.05 Bernstein
10991412009.05 Bernstein
11091932009.05 Bernstein
11193702009.05 Bernstein
11294222009.05 Bernstein
11395342009.05 Bernstein
11496152009.05 Bernstein
11597272009.05 Bernstein
11697882009.05 Bernstein
117101202009.05 Bernstein
118101812009.05 Bernstein
119102422009.05 Bernstein
120104382009.05 Bernstein
121107272009.05 Bernstein
122107702009.05 Bernstein
123110502009.05 Bernstein
124110932009.05 Bernstein
125112542009.05 Bernstein
126113462009.05 Bernstein
127114472009.05 Bernstein
128114862009.05 Bernstein
129117922009.05 Bernstein
130118812009.05 Bernstein
131119612009.05 Bernstein
132121152009.05 Bernstein
133122492009.05 Bernstein
134122922009.05 Bernstein
135124532009.05 Bernstein
136124992009.05 Bernstein
137125952009.05 Bernstein
138126762009.05 Bernstein
139127722009.05 Bernstein
140128242009.05 Bernstein
141132392009.05 Bernstein
142132912009.05 Bernstein
143133482009.05 Bernstein
144136062009.05 Bernstein
145139812009.05 Bernstein
146140202009.05 Bernstein
147144272009.05 Bernstein
148144792009.05 Bernstein
149146812009.05 Bernstein
150148302009.05 Bernstein
151150322009.05 Bernstein
152151542009.05 Bernstein
153154652009.05 Bernstein
154155872009.05 Bernstein
155157022009.05 Bernstein
156158992009.05 Bernstein
157161532009.05 Bernstein
158162212009.05 Bernstein
159164622009.05 Bernstein
160165182009.05 Bernstein
161166742009.05 Bernstein
162167672009.05 Bernstein
163169232009.05 Bernstein
164170002009.05 Bernstein
165173922009.05 Bernstein
166174692009.05 Bernstein
167175482009.05 Bernstein
168177582009.05 Bernstein
169180932009.05 Bernstein
170181462009.05 Bernstein
171184522009.05 Bernstein
172185052009.05 Bernstein
173186942009.05 Bernstein
174188022009.05 Bernstein
175189912009.05 Bernstein
176190822009.05 Bernstein
177193082009.05 Bernstein
178193992009.05 Bernstein
179194952009.05 Bernstein
180196452009.05 Bernstein
181197912009.05 Bernstein
182198442009.05 Bernstein
183200332009.05 Bernstein
184200812009.05 Bernstein
185201872009.05 Bernstein
186202802009.05 Bernstein
187203862009.05 Bernstein
188204392009.05 Bernstein
189209852009.05 Bernstein
190210382009.05 Bernstein
191211042009.05 Bernstein
192213642009.05 Bernstein
193218652009.05 Bernstein
194219062009.05 Bernstein
195222822009.05 Bernstein
196223232009.05 Bernstein
197225732009.05 Bernstein
198226142009.05 Bernstein
199228802009.05 Bernstein
200230022009.05 Bernstein
201234432009.05 Bernstein
202235652009.05 Bernstein
203236592009.05 Bernstein
204238112009.05 Bernstein
205242232009.05 Bernstein
206242812009.05 Bernstein
207244392009.05 Bernstein
208244982009.05 Bernstein
209247282009.05 Bernstein
210249032009.05 Bernstein
211251332009.05 Bernstein
212251922009.05 Bernstein
213256162009.05 Bernstein
214258752009.05 Bernstein
215259322009.05 Bernstein
216261522009.05 Bernstein
217264012009.05 Bernstein
218265542009.05 Bernstein
219269642009.05 Bernstein
220271172009.05 Bernstein
221273132009.05 Bernstein
222274262009.05 Bernstein
223278362009.05 Bernstein
224279222009.05 Bernstein
225281852009.05 Bernstein
226282312009.05 Bernstein
227284742009.05 Bernstein
228286342009.05 Bernstein
229287752009.05 Bernstein
230289282009.05 Bernstein
231291242009.05 Bernstein
232291702009.05 Bernstein
233293542009.05 Bernstein
234294592009.05 Bernstein
235295752009.05 Bernstein
236296682009.05 Bernstein
237303122009.05 Bernstein
238304052009.05 Bernstein
239304772009.05 Bernstein
240306362009.05 Bernstein
241312602009.05 Bernstein
242313062009.05 Bernstein
243315552009.05 Bernstein
244316012009.05 Bernstein
245319312009.05 Bernstein
246320202009.05 Bernstein
247323502009.05 Bernstein
248324192009.05 Bernstein
249328242009.05 Bernstein
250330102009.05 Bernstein
251330962009.05 Bernstein
252333102009.05 Bernstein
253335332009.05 Bernstein
254336642009.05 Bernstein
255339482009.05 Bernstein
256340792009.05 Bernstein
257342512009.05 Bernstein
258343882009.05 Bernstein
259346722009.05 Bernstein
260347202009.05 Bernstein
261349752009.05 Bernstein
262350292009.05 Bernstein
263351902009.05 Bernstein
264353852009.05 Bernstein
265355572009.05 Bernstein
266356112009.05 Bernstein
267359242009.05 Bernstein
268359782009.05 Bernstein
269360862009.05 Bernstein
270362662009.05 Bernstein
271364092009.05 Bernstein
272364922009.05 Bernstein
273370842009.05 Bernstein
274371672009.05 Bernstein
275372752009.05 Bernstein
276374352009.05 Bernstein
277379522009.05 Bernstein
278380132009.05 Bernstein
279381872009.05 Bernstein
280382482009.05 Bernstein
281383902009.05 Bernstein
282384512009.05 Bernstein
283387352009.05 Bernstein
284387972009.05 Bernstein
285396002009.05 Bernstein
286396622009.05 Bernstein
287397252009.05 Bernstein
288398822009.05 Bernstein
289406342009.05 Bernstein
290406802009.05 Bernstein
291413672009.05 Bernstein
292414132009.05 Bernstein
293418072009.05 Bernstein
294419482009.05 Bernstein
295424412009.05 Bernstein
296424872009.05 Bernstein
297428812009.05 Bernstein
298429492009.05 Bernstein
299432552009.05 Bernstein
300435242009.05 Bernstein
301438302009.05 Bernstein
302438792009.05 Bernstein
303442982009.05 Bernstein
304444542009.05 Bernstein
305446022009.05 Bernstein
306449082009.05 Bernstein
307450672009.05 Bernstein
308452142009.05 Bernstein
309457182009.05 Bernstein
310458652009.05 Bernstein
311460132009.05 Bernstein
312462642009.05 Bernstein
313466642009.05 Bernstein
314467562009.05 Bernstein
315470392009.05 Bernstein
316471312009.05 Bernstein
317473722009.05 Bernstein
318476402009.05 Bernstein
319478812009.05 Bernstein
320479732009.05 Bernstein
321484352009.05 Bernstein
322486602009.05 Bernstein
323487632009.05 Bernstein
324490212009.05 Bernstein
325492562009.05 Bernstein
326494152009.05 Bernstein
327497392009.05 Bernstein
328498982009.05 Bernstein
329500902009.05 Bernstein
330502512009.05 Bernstein
331505752009.05 Bernstein
332506352009.05 Bernstein
333509262009.05 Bernstein
334509882009.05 Bernstein
335511752009.05 Bernstein
336513942009.05 Bernstein
337515862009.05 Bernstein
338516482009.05 Bernstein
339519972009.05 Bernstein
340520592009.05 Bernstein
341521812009.05 Bernstein
342523892009.05 Bernstein
343525522009.05 Bernstein
344526432009.05 Bernstein
345533632009.05 Bernstein
346534542009.05 Bernstein
347535762009.05 Bernstein
348537642009.05 Bernstein
349543872009.05 Bernstein
350544582009.05 Bernstein
351546562009.05 Bernstein
352547272009.05 Bernstein
353550692009.05 Bernstein
354552662009.05 Bernstein
355556082009.05 Bernstein
356556792009.05 Bernstein
357563232009.05 Bernstein
358566742009.05 Bernstein
359567452009.05 Bernstein
360570322009.05 Bernstein
361574342009.05 Bernstein
362576402009.05 Bernstein
363581052009.05 Bernstein
364583112009.05 Bernstein
365586102009.05 Bernstein
366587442009.05 Bernstein
367592092009.05 Bernstein
368593342009.05 Bernstein
369599012009.05 Bernstein
370597672009.05 Bernstein
371601512009.05 Bernstein
372603852009.05 Bernstein
373604562009.05 Bernstein
374606622009.05 Bernstein
375609612009.05 Bernstein
376610142009.05 Bernstein
377613042009.05 Bernstein
378613572009.05 Bernstein
379615282009.05 Bernstein
380616302009.05 Bernstein
381622432009.05 Bernstein
382623452009.05 Bernstein
383624562009.05 Bernstein
384626782009.05 Bernstein
385632712009.05 Bernstein
386633202009.05 Bernstein
387636342009.05 Bernstein
388636832009.05 Bernstein
389639992009.05 Bernstein
390641762009.05 Bernstein
391644922009.05 Bernstein
392645912009.05 Bernstein
393649442009.05 Bernstein
394651562009.05 Bernstein
395652822009.05 Bernstein
396655272009.05 Bernstein
397656612009.05 Bernstein
398658252009.05 Bernstein
399660942009.05 Bernstein
400662582009.05 Bernstein
401664022009.05 Bernstein
402665422009.05 Bernstein
403668412009.05 Bernstein
404668942009.05 Bernstein
405671502009.05 Bernstein
406672032009.05 Bernstein
407673742009.05 Bernstein
408675822009.05 Bernstein
409677532009.05 Bernstein
410678092009.05 Bernstein
411681322009.05 Bernstein
412681882009.05 Bernstein
413682942009.05 Bernstein
414684842009.05 Bernstein
415686152009.05 Bernstein
416687062009.05 Bernstein
417695742009.05 Bernstein
418696652009.05 Bernstein
419697712009.05 Bernstein
420699322009.05 Bernstein
421707302009.05 Bernstein
422707922009.05 Bernstein
423709732009.05 Bernstein
424710352009.05 Bernstein
425714602009.05 Bernstein
426716902009.05 Bernstein
427721152009.05 Bernstein
428721772009.05 Bernstein
429729892009.05 Bernstein
430734682009.05 Bernstein
431735352009.05 Bernstein
432738762009.05 Bernstein
433743882009.05 Bernstein
434746562009.05 Bernstein
435753052009.05 Bernstein
436755732009.05 Bernstein
437759582009.05 Bernstein
438761062009.05 Bernstein
439767812009.05 Bernstein
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441774342009.05 Bernstein
442774832009.05 Bernstein
443779002009.05 Bernstein
444782102009.05 Bernstein
445786272009.05 Bernstein
446786892009.05 Bernstein
447793002009.05 Bernstein
448794162009.05 Bernstein
449796282009.05 Bernstein
450800262009.05 Bernstein
451803012009.05 Bernstein
452804602009.05 Bernstein
453811922009.05 Bernstein
454813512009.05 Bernstein
455815632009.05 Bernstein
456819342009.05 Bernstein
457824542009.05 Bernstein
458825862009.05 Bernstein
459829532009.05 Bernstein
460830852009.05 Bernstein
461834062009.05 Bernstein
462837342009.05 Bernstein
463840552009.05 Bernstein
464841872009.05 Bernstein
465848152009.05 Bernstein
466851142009.05 Bernstein
467852392009.05 Bernstein
468855772009.05 Bernstein
469859082009.05 Bernstein
470861152009.05 Bernstein
471865512009.05 Bernstein
472867582009.05 Bernstein
473870222009.05 Bernstein
474872072009.05 Bernstein
475876192009.05 Bernstein
476877192009.05 Bernstein
477880902009.05 Bernstein
478881682009.05 Bernstein
479884192009.05 Bernstein
480886662009.05 Bernstein
481889422009.05 Bernstein
482890082009.05 Bernstein
483894652009.05 Bernstein
484895312009.05 Bernstein
485896972009.05 Bernstein
486899492009.05 Bernstein
487901882009.05 Bernstein
488902912009.05 Bernstein
489911512009.05 Bernstein
490912542009.05 Bernstein
491914202009.05 Bernstein
492916562009.05 Bernstein
493923832009.05 Bernstein
494924702009.05 Bernstein
495927202009.05 Bernstein
496928072009.05 Bernstein
497932092009.05 Bernstein
498934102009.05 Bernstein
499938122009.05 Bernstein
500938992009.05 Bernstein
501946532009.05 Bernstein
502950142009.05 Bernstein
503951032009.05 Bernstein
504954242009.05 Bernstein
505959042009.05 Bernstein
506961242009.05 Bernstein
507966472009.05 Bernstein
508968672009.05 Bernstein
509972122009.05 Bernstein
510973762009.05 Bernstein
511978992009.05 Bernstein
512980182009.05 Bernstein
513984642009.05 Bernstein
514985272009.05 Bernstein
515988432009.05 Bernstein
516991172009.05 Bernstein
517994622009.05 Bernstein
518995252009.05 Bernstein
5191000812009.05 Bernstein
5201001442009.05 Bernstein
5211003432009.05 Bernstein
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5241010462009.05 Bernstein
5251015872009.05 Bernstein
5261017052009.05 Bernstein
5271019042009.05 Bernstein
5281021822009.05 Bernstein
5291025632009.05 Bernstein
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5461060422009.05 Bernstein
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5511069822009.05 Bernstein
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5551078282009.05 Bernstein
5561078862009.05 Bernstein
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5581082062009.05 Bernstein
5591083452009.05 Bernstein
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