First online: September 15, 2020,
Last updated: October 5, 2020.

Remarks on the version of September 13, 2020 of eprint 2019/485 (Barbulescu, El Mrabet and Ghammam)

Differences in the model of cost

As explained in the 2020 version of eprint 2019/485, the model of cost is slightly different from eprint 2019/885. The preprint 2019/485 makes use of the general formula (equation 1 Section 3.2)

c_sieve × (1/𝒜) × (number of (a, b) pairs) + c_{linear algebra} × ((number of prime ideals on both sides)/(𝒜×c_filter))²

There are two sets of differences:

Differences in the choices of c_sieve, c_filter and c_{linear algebra};
Differences in the choices of approximations (in the implementation).

In eprint 2017/334 and 2019/485 the values are c_sieve = 1, c_filter = log₂B and c_{linear algebra} = 128;

In eprint 2019/431 and 2019/885 the values are c_sieve = ln ln B, c_filter = 20 and c_{linear algebra} = 200⌈(log₂r)/64⌉.

The preprint 2019/485 follows the estimates from Barbulescu–Duquesne first designed in eprint 2017/334. The implementation choices are the following.

The number of pairs (a,b) is upper bounded by (2A + 1)^2deg h/(2w) where w is the number of units up to sign in K_h, and lower bounded by
(number of prime ideals on both sides) × ρ((log₂N_f)/(log₂B))^− 1ρ((log₂N_g)/(log₂B))^− 1
An approximation of the number of distinct prime ideals of norm up to B is the number of prime integers up to B, which is B/ln B where B is the smoothness bound
The (a, b) pairs are randomly sampled with randint(-A,A+1) in Python
The norms N_f, N_g are averaged before computing the Dickman-rho function on the average norms: ρ(log₂(averaged N_f)/log₂B)ρ(log₂(averaged N_g)/log₂B)
It is considered that α(f, h) = 0.0 and α(g, h) = 0.0
The smoothness probability of the norms is approximated with the first order term ρ(log₂N_f/log₂B)ρ(log₂N_g/log₂B)
25600 random pairs are sampled to compute the average norms (3000 in eprint 2019/485)

Later in 2019 the implementation choices were refined with the following tricks.

The number of (a, b) pairs is estimated to be (2A + 1)^2deg h/(2wζ_{K_h}(2)) where 1/ζ_{K_h}(2) allows to account for the duplicates due to non-coprime ideals
The number of distinct prime ideals of norm up to B is approximated with the formula LogIntegral(B) that is more accurate than B/ln B for large values of B
The (a, b) pairs are randomly sampled with randint(-A,A) or randrange(-A,A+1) in Python
The average is made on the Dickman-rho values instead of the norms
The exact value of α(f, h) and α(g, h) is computed, and polynomials are chosen to optimise α
Two terms are involved in the smoothness estimate as in Murphy’s thesis: u_f = (ln N_f + α(f, h))/ln B and Pr_f = ρ(u_f) + (1 − γ)ρ(u_f − 1)/ln N_f and the same on g side
10⁵ to 10⁶ random samples are used to compute the averages.

Tables of seeds

The Tables 10 to 17 at the end of the paper are mostly taken from the version of 2019 despite typos, wrong seeds and wrong security levels reported in eprint 2019/1371 and communicated to the authors on August 20, 2019. The following data is generated with this SageMath script. Because the tables below are generated from a script, the seeds are given in hexadecimal and 2-NAF form, and sparse form if it is short. Sometimes the 2-NAF form does not match exactly the sparse form given in eprint 2019/485 because it is not a 2-NAF form but another signed binary sparse representation. The seeds in the SageMath script are copied from the preprint.

The estimated security in F_p^k is from eprint 2019/1371 Tables 4 and 10 in the version of December 6, 2019 (Table 10 was later removed in the version of February 2020 to fit in the 30 pages of the published version).

Some seeds for KSS do not provide integer values of p and r. Some seeds for k=27 and D=-3 are wrong. There are two families with D=-3 for k=27, construction 6.6 from Freeman, Scott, Teske (DOI 10.1007/s00145-009-9048-z, eprint 2006/372) and a BLS-like construction, with trace t(x) = x+1. It seems that the polynomials p(x) and r(x) from these two families were mixed when generating seeds, though the polynomials are correct in the paper.

Table 10 construction 6.2 k=1 mod 2

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.2	9	`0x400637`	2²²+2¹⁰+2⁹+2⁵+2⁴+2²+2+1	2²²+2¹¹-2⁹+2⁶-2³-1	483	265	4339	116
6.2	11	`0x40ff`		2¹⁴+2⁸-1	363	281	3989	122
6.2	13	`0x10451b`	2²⁰+2¹⁴+2¹⁰+2⁸+2⁴+2³+2+1	2²⁰+2¹⁴+2¹⁰+2⁸+2⁵-2²-1	599	481	7784	162

Table 10 construction 6.3 k=2 mod 4

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.3	10	`0x18000000119`	2⁴⁰+2³⁹+2⁸+2⁴+2³+1	2⁴¹-2³⁹+2⁸+2⁵-2³+1	567	325	5662	Error p is not prime Error r is not prime
6.3	14	`0x37723d`		2²²-2¹⁹-2¹⁵-2¹²+2⁹+2⁶-2²+1	391	262	5464	131
6.3	18	`0x4035ab`	2²²+2¹³+2¹²+2¹⁰+2⁸+2⁷+2⁵+2³+2+1	2²²+2¹⁴-2¹¹-2⁹-2⁶-2⁴-2²-1	483	265	8678
6.3	22	`0xc013`	2¹⁵+2¹⁴+2⁴+2+1	2¹⁶-2¹⁴+2⁴+2²-1	404	312	8871
6.3	26	`0x1101`	2¹²+2⁸+1	2¹²+2⁸+1	361	291	9377
6.3	30	`0x13c0d`	2¹⁶+2¹³+2¹²+2¹¹+2¹⁰+2³+2²+1	2¹⁶+2¹⁴-2¹⁰+2⁴-2²+1	553	261	16571
6.3	34	`0x43f1`		2¹⁴+2¹⁰-2⁴+1	534	451	18132
6.3	38	`0x20a09`	2¹⁷+2¹¹+2⁹+2³+1	2¹⁷+2¹¹+2⁹+2³+1	714	614	27101
6.3	42	`0xd91`	2¹¹+2¹⁰+2⁸+2⁷+2⁴+1	2¹²-2⁹-2⁷+2⁴+1	540	283	22641
6.3	46	`0x2603`	2¹³+2¹⁰+2⁹+2+1	2¹³+2¹¹-2⁹+2²-1	661	583	30380
6.3	50	`0x4b91`	2¹⁴+2¹¹+2⁹+2⁸+2⁷+2⁴+1	2¹⁴+2¹²-2¹⁰-2⁷+2⁴+1	767	570	38348
6.3	54	`0xb2b`	2¹¹+2⁹+2⁸+2⁵+2³+2+1	2¹²-2¹⁰-2⁸+2⁶-2⁴-2²-1	664	414	35852

Table 10 construction 6.4 k=4 mod 8

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.4	12	`0x10000000000000b0b`	2⁶⁴+2¹¹+2⁹+2⁸+2³+2+1	2⁶⁴+2¹²-2¹⁰-2⁸+2⁴-2²-1	511	257	6121	138
6.4	20	`0x100010011`	2³²+2¹⁶+2⁴+1	2³²+2¹⁶+2⁴+1	383	257	7641
6.4	28	`0x40031b`	2²²+2⁹+2⁸+2⁴+2³+2+1	2²²+2¹⁰-2⁸+2⁵-2²-1	351	265	9801
6.4	36	`0x414405`	2²²+2¹⁶+2¹⁴+2¹⁰+2²+1	2²²+2¹⁶+2¹⁴+2¹⁰+2²+1	439	265	15789
6.4	44	`0x5181`	2¹⁴+2¹²+2⁸+2⁷+1	2¹⁴+2¹²+2⁹-2⁷+1	343	287	15065
6.4	52	`0x31c1`	2¹³+2¹²+2⁸+2⁷+2⁶+1	2¹⁴-2¹²+2⁹-2⁶+1	380	328	19752

Table 10 construction 6.7 even k

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.7	12	`0x100024001`	2³²+2¹⁷+2¹⁴+1	2³²+2¹⁷+2¹⁴+1	446	257	5341	134
6.7	24	`0x100000305`	2³²+2⁹+2⁸+2²+1	2³²+2¹⁰-2⁸+2²+1	382	257	9145
6.7	30	`0xfa3`	2¹¹+2¹⁰+2⁹+2⁸+2⁷+2⁵+2+1	2¹²-2⁷+2⁵+2²-1	692	383	20733
6.7	36	`0x12729`	2¹⁶+2¹³+2¹⁰+2⁹+2⁸+2⁵+2³+1	2¹⁶+2¹³+2¹¹-2⁸+2⁵+2³+1	548	389	19728
6.7	48	`0x1004569`	2²⁴+2¹⁴+2¹⁰+2⁸+2⁶+2⁵+2³+1	2²⁴+2¹⁴+2¹¹-2⁹-2⁷-2⁵+2³+1	526	385	25202

Table 10 construction 6.7 odd k

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.7	9	`0xa2f`	2¹¹+2⁹+2⁵+2³+2²+2+1	2¹¹+2⁹+2⁶-2⁴-1	520	273	4672	140
6.7	15	`0x5695`	2¹⁴+2¹²+2¹⁰+2⁹+2⁷+2⁴+2²+1	2¹⁵-2¹³-2¹¹-2⁹+2⁷+2⁴+2²+1	950	462	14247
6.7	21	`0x1c93`	2¹²+2¹¹+2¹⁰+2⁷+2⁴+2+1	2¹³-2¹⁰+2⁷+2⁴+2²-1	1101	617	23120
6.7	27	`0xb7d`		2¹²-2¹⁰-2⁷-2²+1	1219	830	32896

Table 11 construction 6.6 k=0 mod 6

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.6	12	`-0x1ffffffbfffe00000000`		-(-2⁷⁷+2⁵⁰+2³³)	461	308	5525	134
6.6	24	`-0xeffff000`		-(-2³²+2²⁸+2¹²)	318	256	7620	162
6.6	30	`0x100006009`	2³²+2¹⁴+2¹³+2³+1	2³²+2¹⁵-2¹³+2³+1	383	257	11473
6.6	42	`-0x3bffc0`		-(-2²²+2¹⁸+2⁶)	349	263	14655
6.6	48	`0x16840`	2¹⁶+2¹⁴+2¹³+2¹¹+2⁶	2¹⁷-2¹⁵-2¹³+2¹¹+2⁶	296	264	14174

Table 11 construction 6.6 k=2,4 mod 6

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.6	14	`0x4226bf`		2²²+2¹⁷+2¹³+2¹¹-2⁸-2⁶-1	352	265	4917	150
6.6	16	`0x1c768`	2¹⁶+2¹⁵+2¹⁴+2¹⁰+2⁹+2⁸+2⁶+2⁵+2³	2¹⁷-2¹⁴+2¹¹-2⁷-2⁵+2³	369	270	5900
6.6	20	`0x20041`	2¹⁷+2⁶+1	2¹⁷+2⁶+1	373	273	7449
6.6	22	`0x20020`	2¹⁷+2⁵	2¹⁷+2⁵	475	341	10438
6.6	26	`0x60ac`	2¹⁴+2¹³+2⁷+2⁵+2³+2²	2¹⁵-2¹³+2⁸-2⁶-2⁴-2²	408	351	10584
6.6	28	`0x457c`	2¹⁴+2¹⁰+2⁸+2⁶+2⁵+2⁴+2³+2²	2¹⁴+2¹¹-2⁹-2⁷-2²	479	339	13397
6.6	32	`0x232`	2⁹+2⁵+2⁴+2	2⁹+2⁶-2⁴+2	309	293	9888
6.6	34	`0x42a`	2¹⁰+2⁵+2³+2	2¹⁰+2⁵+2³+2	401	322	13625
6.6	38	`0x4d5`	2¹⁰+2⁷+2⁶+2⁴+2²+1	2¹⁰+2⁸-2⁶+2⁴+2²+1	410	370	15555
6.6	40	`0x478`	2¹⁰+2⁶+2⁵+2⁴+2³	2¹⁰+2⁷-2³	466	326	18631

Table 11 construction 6.6 k=3 mod 6

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.6	15	`0x100011005`	2³²+2¹⁶+2¹²+2²+1	2³²+2¹⁶+2¹²+2²+1	383	257	5737	138
6.6	21	`0x41037e`		2²²+2¹⁶+2¹⁰-2⁷-2	351	265	7367	Error p is not prime
BLS3	27	`0x7c09`	2¹⁴+2¹³+2¹²+2¹¹+2¹⁰+2³+1	2¹⁵-2¹⁰+2³+1	298	268	8033
6.6	33	`0x4283`	2¹⁴+2⁹+2⁷+2+1	2¹⁴+2⁹+2⁷+2²-1	336	282	11080
6.6	39	`0x2c90`	2¹³+2¹¹+2¹⁰+2⁷+2⁴	2¹⁴-2¹²-2¹⁰+2⁷+2⁴	376	324	14656
6.6	45	`0xcfb`		2¹²-2¹⁰+2⁸-2²-1	373	281	16775

Table 11 construction 6.6 k=1,5 mod 6

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.6	11	`0x46d0`	2¹⁴+2¹⁰+2⁹+2⁷+2⁶+2⁴	2¹⁴+2¹¹-2⁸-2⁶+2⁴	338	283	3718	118
6.6	13	`0x2c90`	2¹³+2¹¹+2¹⁰+2⁷+2⁴	2¹⁴-2¹²-2¹⁰+2⁷+2⁴	376	324	4886	152
6.6	29	`0xb90`	2¹¹+2⁹+2⁸+2⁷+2⁴	2¹²-2¹⁰-2⁷+2⁴	691	646	20019

Table 14 KSS

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
KSS	16	`-0x3f87007ff`		-(-2³⁴+2²⁷-2²³+2²⁰-2¹¹+1)	330	257	5280	140
KSS	18	`0x1000003ffe02`		2⁴⁴+2²²-2⁹+2	348	256	6257
KSS	32	`0x180c20`	2²⁰+2¹⁹+2¹¹+2¹⁰+2⁵	2²¹-2¹⁹+2¹²-2¹⁰+2⁵	350	283	11171	Error p(u) is not in Z Error r(u) is not in Z
KSS	36	`0x1824213`	2²⁴+2²³+2¹⁷+2¹⁴+2⁹+2⁴+2+1	2²⁵-2²³+2¹⁷+2¹⁴+2⁹+2⁴+2²-1	330	268	11862	Error p(u) is not in Z Error r(u) is not in Z
KSS	40	`0x42191`	2¹⁸+2¹³+2⁸+2⁷+2⁴+1	2¹⁸+2¹³+2⁹-2⁷+2⁴+1	377	258	15077	Error p(u) is not in Z Error r(u) is not in Z
KSS	54	`0x18888`	2¹⁶+2¹⁵+2¹¹+2⁷+2³	2¹⁷-2¹⁵+2¹¹+2⁷+2³	349	314	18802	Error r is not prime

Table 14 other

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
LZZW	9	`0x40000000007ffc00002`		2⁷⁴+2³⁵-2²²+2	591	443	5314	128
BN	12	`0x4001fffffffffffffffffffffbfff`		2¹¹⁴+2¹⁰¹-2¹⁴-1	462	462	5535	135
DCC	15	`0x100090402`	2³²+2¹⁹+2¹⁶+2¹⁰+2	2³²+2¹⁹+2¹⁶+2¹⁰+2	383	257	5737	138

Table 15 128 bits

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.3	14	`0x37723d`		2²²-2¹⁹-2¹⁵-2¹²+2⁹+2⁶-2²+1	391	262	5464	131
6.4	20	`0x100010011`	2³²+2¹⁶+2⁴+1	2³²+2¹⁶+2⁴+1	383	257	7641
6.4	28	`0x40031b`	2²²+2⁹+2⁸+2⁴+2³+2+1	2²²+2¹⁰-2⁸+2⁵-2²-1	351	265	9801
6.6	12	`-0x1ffffffbfffe00000000`		-(-2⁷⁷+2⁵⁰+2³³)	461	308	5525	134
6.6	15	`0x100011005`	2³²+2¹⁶+2¹²+2²+1	2³²+2¹⁶+2¹²+2²+1	383	257	5737	138
6.6	24	`-0xeffff000`		-(-2³²+2²⁸+2¹²)	318	256	7620	162
BLS3	27	`0x7c09`	2¹⁴+2¹³+2¹²+2¹¹+2¹⁰+2³+1	2¹⁵-2¹⁰+2³+1	298	268	8033
6.7	12	`0x100024001`	2³²+2¹⁷+2¹⁴+1	2³²+2¹⁷+2¹⁴+1	446	257	5341	134
KSS	16	`-0x3f87007ff`		-(-2³⁴+2²⁷-2²³+2²⁰-2¹¹+1)	330	257	5280	140
DCC	15	`0x100090402`	2³²+2¹⁹+2¹⁶+2¹⁰+2	2³²+2¹⁹+2¹⁶+2¹⁰+2	383	257	5737	138

Table 15 192 bits

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)	DL sec. in F_p^k
6.3	14	`0x10000000979`	2⁴⁰+2¹¹+2⁸+2⁶+2⁵+2⁴+2³+1	2⁴⁰+2¹¹+2⁹-2⁷-2³+1	719	481	10053	172 Error p is not prime
6.4	20	`0x2000000000011a41`	2⁶¹+2¹⁶+2¹²+2¹¹+2⁹+2⁶+1	2⁶¹+2¹⁶+2¹³-2¹¹+2⁹+2⁶+1	731	489	14601
6.4	28	`-0x80002003`	-(2³¹+2¹³+2+1)	-(-2³¹-2¹³-2²+1)	495	373	13833	200
6.6	12	`0xc0000000000000000000000002 000000000000000000000008c6`	2²⁰⁷+2²⁰⁶+2¹⁰⁵+2¹¹+2⁷+2⁶+2²+2	2²⁰⁸-2²⁰⁶+2¹⁰⁵+2¹¹+2⁸-2⁶+2³-2	1244	831	14928
6.6	15	`0x7000000000000018f01`	2⁷⁴+2⁷³+2⁷²+2¹⁶+2¹⁵+2¹¹+2¹⁰+2⁹+2⁸+1	2⁷⁵-2⁷²+2¹⁷-2¹⁵+2¹²-2⁸+1	897	599	13442
BLS3	27	`0x6110e0`	2²²+2²¹+2¹⁶+2¹²+2⁷+2⁶+2⁵	2²³-2²¹+2¹⁶+2¹²+2⁸-2⁵	451	406	12162
6.6	24	`-0x10007fffffffe40`		-(-2⁵⁶-2⁴³+2⁹-2⁶)	559	449	13403	202
6.7	24	`-0xfbffffffefff`		-(-2⁴⁸+2⁴²+2¹²+1)	573	384	13746	205
KSS	16	`0xa0000000000000201afd0`		2⁸³+2⁸¹+2²⁵+2¹⁷-2¹⁴-2¹²-2⁶+2⁴	824	651	13173	Error p(u) is not in Z Error r(u) is not in Z
KSS	18	`0x4000000000000000049e2`	2⁸²+2¹⁴+2¹¹+2⁸+2⁷+2⁶+2⁵+2	2⁸²+2¹⁴+2¹¹+2⁹-2⁵+2	652	484	11729	Error p(u) is not in Z Error r(u) is not in Z
DCC	15	`0x200000000000001d0108`	2⁷⁷+2²⁰+2¹⁹+2¹⁸+2¹⁶+2⁸+2³	2⁷⁷+2²¹-2¹⁸+2¹⁶+2⁸+2³	923	617	13837

Table 15 256 bits exotic k

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)
6.2	17	`0x1000067bf`		2³²+2¹⁵-2¹³+2¹¹-2⁶-1	1215	1025	20639
6.3	18	`0x8000000553f`		2⁴³+2¹⁴+2¹²+2¹⁰+2⁸+2⁶-1	945	517	16993
6.6	19	`0x9bf0`		2¹⁵+2¹³-2¹⁰-2⁴	610	551	11587
6.6	20	`0x400000003df2`		2⁴⁶+2¹⁴-2⁹-2⁴+2	1011	737	20209
6.6	22	`0x200008bd`	2²⁹+2¹¹+2⁷+2⁵+2⁴+2³+2²+1	2²⁹+2¹¹+2⁸-2⁶-2²+1	811	581	17830
6.6	26	`0x821104`	2²³+2¹⁷+2¹²+2⁸+2²	2²³+2¹⁷+2¹²+2⁸+2²	644	553	16720
6.6	28	`0x41f956`		2²²+2¹⁷-2¹¹+2⁹-2⁷-2⁵-2³-2	748	530	20942
6.7	9	`0x305be9`		2²²-2²⁰+2¹⁵-2¹³-2¹⁰-2⁵+2³+1	991	519	8914

Table 15 256 bits

method	k	seed u	u sparse	u 2-NAF	p (bits)	r (bits)	p^k (bits)
6.4	20	`0xe00000000000000073c9`		2⁸⁰-2⁷⁷+2¹⁵-2¹²+2¹⁰-2⁶+2³+1	956	639	19114
6.4	28	`0x4000000000007d83`		2⁶²+2¹⁵-2⁹-2⁷+2²-1	991	745	27721
6.6	15	`0x40000000000000000000000000000040000c00`	2¹⁵⁰+2³⁰+2¹¹+2¹⁰	2¹⁵⁰+2³⁰+2¹²-2¹⁰	1799	1201	26977
6.6	24	`0x60000000100004000000000000`	2¹⁰²+2¹⁰¹+2⁶⁸+2⁵⁰	2¹⁰³-2¹⁰¹+2⁶⁸+2⁵⁰	1025	821	24583	Error p(u) is not in Z Error r is not prime
BLS3	27	`-0xffffc14`		-(-2²⁸+2¹⁰-2⁴-2²)	559	503	15078
6.7	12	`0x80000000000000000000000000720f`	2¹¹⁹+2¹⁴+2¹³+2¹²+2⁹+2³+2²+2+1	2¹¹⁹+2¹⁵-2¹²+2⁹+2⁴-1	1664	953	19957
KSS	16	`0x600000000000000000000000000000000016e1`	2¹⁵⁰+2¹⁴⁹+2¹²+2¹⁰+2⁹+2⁷+2⁶+2⁵+1	2¹⁵¹-2¹⁴⁹+2¹³-2¹¹-2⁸-2⁵+1	1496	1189	23935	Error p(u) is not in Z Error r(u) is not in Z
LZZW	9	`0x80000000000000000000000000000000000 000000000000000000000000000000005f090`	2²⁸⁷+2¹⁸+2¹⁶+2¹⁵+2¹⁴+2¹³+2¹²+2⁷+2⁴	2²⁸⁷+2¹⁹-2¹⁷-2¹²+2⁷+2⁴	2295	1721	20650
DCC	15	`0x10000000000000000000000007f6b`		2¹¹²+2¹⁵-2⁷-2⁴-2²-1	1343	897	20137

Remarks on the version of September 13, 2020 of eprint 2019/485 (Barbulescu, El Mrabet and Ghammam)