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Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А. Бонч-Бруевича

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Guide to Elliptic Curve Cryptography.pdf

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APPENDIX A

Sample Parameters

This appendix presents elliptic curve domain parameters D = (q, FR, S, a, b, P, n, h) that are suitable for cryptographic use; see §4.2 for a review of the notation. In §A.1, an algorithm for testing irreducibility of a polynomial is presented. This algorithm can be used to generate a reduction polynomial for representing elements of the ﬁnite ﬁeld F pm . Also included in §A.1 are tables of irreducible binary polynomials that are recommended by several standards including ANSI X9.62 and ANSI X9.63 as reduction polynomials for representing the elements of binary ﬁelds F2m . The 15 elliptic curves recommended by NIST in the FIPS 186-2 standard for U.S. federal government use are listed in §A.2.

A.1 Irreducible polynomials

A polynomial f (z) = am zm + · · · + a1z + a0 F p[z] of degree m ≥ 1 is irreducible over F p if f (z) cannot be factored as a product of polynomials in F p[z] each of degree less than m. Since f (z) is irreducible if and only if am−1 f (z) is irreducible, it sufﬁces to only consider monic polynomials (i.e., polynomials with leading coefﬁcient am = 1).

For any prime p and integer m ≥ 1, there exists at least one monic irreducible polynomial of degree m in F p[z]. In fact, the exact number of such polynomials is

Np (m) = 1 µ(d) pm/d ,

m d|m

258 A. Sample Parameters

where the summation index d ranges over all positive divisors of m, and the Mobius¨ function µ is deﬁned as follows:

		1,	if d = 1,
µ(d)	=	0,	if d is divisible by the square of a prime,
	=	( 1)l ,	if d is the product of l distinct primes.
It has been shown that		−
			1			Np (m)	1
			1		≤	Np (m)	1
							≈		.
				2m		pm	≈	m	.

Thus, if polynomials in F p[z] can be efﬁciently tested for irreducibility, then irreducible polynomials of degree m can be efﬁciently found by selecting random monic polynomials of degree m in F p[z] until an irreducible one is found—the expected number of trials is approximately m.

Algorithm A.1 is an efﬁcient test for deciding irreducibility. It is based on the fact

that a polynomial f (z) of	degree m is irreducible over			F p	if and only if gcd( f (z), z pi	−
		m	.
z) = 1 for each i, 1 ≤ i ≤		2

Algorithm A.1 Testing a polynomial for irreducibility

INPUT: A prime p and a polynomial f (z) F p[z] of degree m ≥ 1.
OUTPUT: Irreducibility of		f (z).

1.u(z) ← z.

2.For i from 1 to m2 do:

2.1u(z) ← u(z) p mod f (z).

2.2d(z) ← gcd( f (z), u(z) − z).

2.3If d(z) = 1 then return(“reducible”).

3.Return(“irreducible”).

For each m, 2 ≤ m ≤ 600, Tables A.1 and A.2 list an irreducible trinomial or pentanomial f (z) of degree m over F2. The entries in the column labeled “T ” are the degrees of the nonzero terms of the polynomial excluding the leading term zm and the constant term 1. For example, T = k represents the trinomial zm + zk + 1, and T = (k3 , k2, k1) represents the pentanomial zm + zk3 + zk2 + zk1 + 1. The following criteria from the ANSI X9.62 and ANSI X9.63 standards were used to select the reduction polynomials:

(i)If there exists an irreducible trinomial of degree m over F2, then f (z) is the irreducible trinomial zm + zk + 1 for the smallest possible k.

(ii)If there does not exist an irreducible trinomial of degree m over F2, then f (z) is the irreducible pentanomial zm + zk3 + zk2 + zk1 + 1 for which (a) k3 is the smallest possible; (b) for this particular value of k3, k2 is the smallest possible; and (c) for these particular values of k3 and k2, k1 is the smallest possible.

A.1.

Irreducible polynomials

259

−	51	6, 3, 1	101	7, 6, 1	151	3	201	14	251	7, 4, 2
1	52	3	102	29	152	6, 3, 2	202	55	252	15
1	53	6, 2, 1	103	9	153	1	203	8, 7, 1	253	46
1	54	9	104	4, 3, 1	154	15	204	27	254	7, 2, 1
2	55	7	105	4	155	62	205	9, 5, 2	255	52
1	56	7, 4, 2	106	15	156	9	206	10, 9, 5	256	10, 5, 2
1	57	4	107	9, 7, 4	157	6, 5, 2	207	43	257	12
4, 3, 1	58	19	108	17	158	8, 6, 5	208	9, 3, 1	258	71
1	59	7, 4, 2	109	5, 4, 2	159	31	209	6	259	10, 6, 2
3	60	1	110	33	160	5, 3, 2	210	7	260	15
2	61	5, 2, 1	111	10	161	18	211	11, 10, 8	261	7, 6, 4
3	62	29	112	5, 4, 3	162	27	212	105	262	9, 8, 4
4, 3, 1	63	1	113	9	163	7, 6, 3	213	6, 5, 2	263	93
5	64	4, 3, 1	114	5, 3, 2	164	10, 8, 7	214	73	264	9, 6, 2
1	65	18	115	8, 7, 5	165	9, 8, 3	215	23	265	42
5, 3, 1	66	3	116	4, 2, 1	166	37	216	7, 3, 1	266	47
3	67	5, 2, 1	117	5, 2, 1	167	6	217	45	267	8, 6, 3
3	68	9	118	33	168	15, 3, 2	218	11	268	25
5, 2, 1	69	6, 5, 2	119	8	169	34	219	8, 4, 1	269	7, 6, 1
3	70	5, 3, 1	120	4, 3, 1	170	11	220	7	270	53
2	71	6	121	18	171	6, 5, 2	221	8, 6, 2	271	58
1	72	10, 9, 3	122	6, 2, 1	172	1	222	5, 4, 2	272	9, 3, 2
5	73	25	123	2	173	8, 5, 2	223	33	273	23
4, 3, 1	74	35	124	19	174	13	224	9, 8, 3	274	67
3	75	6, 3, 1	125	7, 6, 5	175	6	225	32	275	11, 10, 9
4, 3, 1	76	21	126	21	176	11, 3, 2	226	10, 7, 3	276	63
5, 2, 1	77	6, 5, 2	127	1	177	8	227	10, 9, 4	277	12, 6, 3
1	78	6, 5, 3	128	7, 2, 1	178	31	228	113	278	5
2	79	9	129	5	179	4, 2, 1	229	10, 4, 1	279	5
1	80	9, 4, 2	130	3	180	3	230	8, 7, 6	280	9, 5, 2
3	81	4	131	8, 3, 2	181	7, 6, 1	231	26	281	93
7, 3, 2	82	8, 3, 1	132	17	182	81	232	9, 4, 2	282	35
10	83	7, 4, 2	133	9, 8, 2	183	56	233	74	283	12, 7, 5
7	84	5	134	57	184	9, 8, 7	234	31	284	53
2	85	8, 2, 1	135	11	185	24	235	9, 6, 1	285	10, 7, 5
9	86	21	136	5, 3, 2	186	11	236	5	286	69
6, 4, 1	87	13	137	21	187	7, 6, 5	237	7, 4, 1	287	71
6, 5, 1	88	7, 6, 2	138	8, 7, 1	188	6, 5, 2	238	73	288	11, 10, 1
4	89	38	139	8, 5, 3	189	6, 5, 2	239	36	289	21
5, 4, 3	90	27	140	15	190	8, 7, 6	240	8, 5, 3	290	5, 3, 2
3	91	8, 5, 1	141	10, 4, 1	191	9	241	70	291	12, 11, 5
7	92	21	142	21	192	7, 2, 1	242	95	292	37
6, 4, 3	93	2	143	5, 3, 2	193	15	243	8, 5, 1	293	11, 6, 1
5	94	21	144	7, 4, 2	194	87	244	111	294	33
4, 3, 1	95	11	145	52	195	8, 3, 2	245	6, 4, 1	295	48
1	96	10, 9, 6	146	71	196	3	246	11, 2, 1	296	7, 3, 2
5	97	6	147	14	197	9, 4, 2	247	82	297	5
5, 3, 2	98	11	148	27	198	9	248	15, 14, 10	298	11, 8, 4
9	99	6, 3, 1	149	10, 9, 7	199	34	249	35	299	11, 6, 4
4, 3, 2	100	15	150	53	200	5, 3, 2	250	103	300	5

Table A.1. Irreducible binary polynomials of degree m, 2 ≤ m ≤ 300.

260		A. Sample Parameters

	m	T	m	T	m	T	m	T	m	T	m	T

	301	9, 5, 2	351	34	401	152	451	16, 10, 1	501	5, 4, 2	551	135
	302	41	352	13, 11, 6	402	171	452	6, 5, 4	502	8, 5, 4	552	19, 16, 9
	303	1	353	69	403	9, 8, 5	453	15, 6, 4	503	3	553	39
	304	11, 2, 1	354	99	404	65	454	8, 6, 1	504	15, 14, 6	554	10, 8, 7
	305	102	355	6, 5, 1	405	13, 8, 2	455	38	505	156	555	10, 9, 4
	306	7, 3, 1	356	10, 9, 7	406	141	456	18, 9, 6	506	23	556	153
	307	8, 4, 2	357	11, 10, 2	407	71	457	16	507	13, 6, 3	557	7, 6, 5
	308	15	358	57	408	5, 3, 2	458	203	508	9	558	73
	309	10, 6, 4	359	68	409	87	459	12, 5, 2	509	8, 7, 3	559	34
	310	93	360	5, 3, 2	410	10, 4, 3	460	19	510	69	560	11, 9, 6
	311	7, 5, 3	361	7, 4, 1	411	12, 10, 3	461	7, 6, 1	511	10	561	71
	312	9, 7, 4	362	63	412	147	462	73	512	8, 5, 2	562	11, 4, 2
	313	79	363	8, 5, 3	413	10, 7, 6	463	93	513	26	563	14, 7, 3
	314	15	364	9	414	13	464	19, 18, 13	514	67	564	163
	315	10, 9, 1	365	9, 6, 5	415	102	465	31	515	14, 7, 4	565	11, 6, 1
	316	63	366	29	416	9, 5, 2	466	14, 11, 6	516	21	566	153
	317	7, 4, 2	367	21	417	107	467	11, 6, 1	517	12, 10, 2	567	28
	318	45	368	7, 3, 2	418	199	468	27	518	33	568	15, 7, 6
	319	36	369	91	419	15, 5, 4	469	9, 5, 2	519	79	569	77
	320	4, 3, 1	370	139	420	7	470	9	520	15, 11, 2	570	67
	321	31	371	8, 3, 2	421	5, 4, 2	471	1	521	32	571	10, 5, 2
	322	67	372	111	422	149	472	11, 3, 2	522	39	572	12, 8, 1
	323	10, 3, 1	373	8, 7, 2	423	25	473	200	523	13, 6, 2	573	10, 6, 4
	324	51	374	8, 6, 5	424	9, 7, 2	474	191	524	167	574	13
	325	10, 5, 2	375	16	425	12	475	9, 8, 4	525	6, 4, 1	575	146
	326	10, 3, 1	376	8, 7, 5	426	63	476	9	526	97	576	13, 4, 3
	327	34	377	41	427	11, 6, 5	477	16, 15, 7	527	47	577	25
	328	8, 3, 1	378	43	428	105	478	121	528	11, 6, 2	578	23, 22, 16
	329	50	379	10, 8, 5	429	10, 8, 7	479	104	529	42	579	12, 9, 7
	330	99	380	47	430	14, 6, 1	480	15, 9, 6	530	10, 7, 3	580	237
	331	10, 6, 2	381	5, 2, 1	431	120	481	138	531	10, 5, 4	581	13, 7, 6
	332	89	382	81	432	13, 4, 3	482	9, 6, 5	532	1	582	85
	333	2	383	90	433	33	483	9, 6, 4	533	4, 3, 2	583	130
	334	5, 2, 1	384	12, 3, 2	434	12, 11, 5	484	105	534	161	584	14, 13, 3
	335	10, 7, 2	385	6	435	12, 9, 5	485	17, 16, 6	535	8, 6, 2	585	88
	336	7, 4, 1	386	83	436	165	486	81	536	7, 5, 3	586	7, 5, 2
	337	55	387	8, 7, 1	437	6, 2, 1	487	94	537	94	587	11, 6, 1
	338	4, 3, 1	388	159	438	65	488	4, 3, 1	538	195	588	35
	339	16, 10, 7	389	10, 9, 5	439	49	489	83	539	10, 5, 4	589	10, 4, 3
	340	45	390	9	440	4, 3, 1	490	219	540	9	590	93
	341	10, 8, 6	391	28	441	7	491	11, 6, 3	541	13, 10, 4	591	9, 6, 4
	342	125	392	13, 10, 6	442	7, 5, 2	492	7	542	8, 6, 1	592	13, 6, 3
	343	75	393	7	443	10, 6, 1	493	10, 5, 3	543	16	593	86
	344	7, 2, 1	394	135	444	81	494	17	544	8, 3, 1	594	19
	345	22	395	11, 6, 5	445	7, 6, 4	495	76	545	122	595	9, 2, 1
	346	63	396	25	446	105	496	16, 5, 2	546	8, 2, 1	596	273
	347	11, 10, 3	397	12, 7, 6	447	73	497	78	547	13, 7, 4	597	14, 12, 9
	348	103	398	7, 6, 2	448	11, 6, 4	498	155	548	10, 5, 3	598	7, 6, 1
	349	6, 5, 2	399	26	449	134	499	11, 6, 5	549	16, 4, 3	599	30
	350	53	400	5, 3, 2	450	47	500	27	550	193	600	9, 5, 2

Table A.2. Irreducible binary polynomials of degree m, 301 ≤ m ≤ 600.

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