Merge Fotiadis-Martindale curves with embedding degree 18.

include/relic_pc.h

@@ -62,7 +62,7 @@
 #elif FP_PRIME == 315 || FP_PRIME == 317 || FP_PRIME == 330 || FP_PRIME == 509 || FP_PRIME == 765 || FP_PRIME == 766
 #define RLC_G2_LOWER			ep4_
 #define RLC_G2_BASEF(A)			A[0][0]
-#elif FP_PRIME == 508 || FP_PRIME == 638 && !defined(FP_QNRES)
+#elif FP_PRIME == 508 || FP_PRIME == 768 || FP_PRIME == 638 && !defined(FP_QNRES)
 #define RLC_G2_LOWER			ep3_
 #define RLC_G2_BASEF(A)			A[0]
 #else
@@ -78,7 +78,7 @@
 #elif FP_PRIME == 315 || FP_PRIME == 317 || FP_PRIME == 509
 #define RLC_GT_LOWER			fp24_
 #define RLC_GT_EMBED      		24
-#elif FP_PRIME == 508 || FP_PRIME == 638 && !defined(FP_QNRES)
+#elif FP_PRIME == 508 || FP_PRIME == 768 || FP_PRIME == 638 && !defined(FP_QNRES)
 #define RLC_GT_LOWER			fp18_
 #define RLC_GT_EMBED      		18
 #elif FP_PRIME == 330 || FP_PRIME == 765 || FP_PRIME == 766

src/ep/relic_ep_param.c

@@ -1756,6 +1756,9 @@ void ep_param_print(void) {
 		case K16_P766:
 			util_banner("Curve K16-P766:", 0);
 			break;
+		case FM18_P768:
+			util_banner("Curve FM18-P768:", 0);
+			break;
 		case SS_P1536:
 			util_banner("Curve SS-P1536:", 0);
 			break;
@@ -1818,6 +1821,7 @@ int ep_param_level(void) {
 		case N16_P765:
 		case FM16_P765:
 		case K16_P766:
+		case FM18_P768:
 		case K18_P638:
 		case B24_P509:
 			return 192;

src/epx/relic_ep3_curve.c

@@ -99,17 +99,17 @@
 #define FM18_P768_A0		"0"
 #define FM18_P768_A1		"0"
 #define FM18_P768_A2		"0"
-#define FM18_P768_B0		"0"
+#define FM18_P768_B0		"-14"
 #define FM18_P768_B1		"5"
 #define FM18_P768_B2		"0"
-#define FM18_P768_X0		"FE31E13463CB8897FE1708543815C50C0FFBC6AAE7360ACF2B36B64FC0D806FDF4B80FCFCD28CA748F6964B9C7848882AA1DEC9640D8CF09344B95F78E4E85F790959CE583A201FCAD217D3E39CADBE3B7B3F9BC0AE6E7753FA806880660B538"
-#define FM18_P768_X1		"E69ECCB9020052A03987AFA34E9F30D7298AA8E02B157866C37A84B444775BF7A439E5A3BD611738444BF7DF8237B478D6A284A7D46736BC29528A88C6CE23E997AABA613211A42D027D83AFEC558BD69FDCCBEEE7273F335E449FAD8E4B1F92"
-#define FM18_P768_X2		"F21BA095672FE95B02E5C5CD933F403D9C48FC980D378FC5184893E8D3E34834B32BA9D1B7A540A102EF345CA0C7E4B134F925948FA4BB3EB00184FD829FA8553EE43938E783687A553F3E75851E6B2C039698C77899D849AB8016BF9041AFC2"
-#define FM18_P768_Y0		"D16A99C5ED5D491EA7B51110063533ABCC26EC4C77232489F82F04A862F9C7FCF9672EFE6960F5A2D11FABFCBBE1499F79373324699E557280FE3A588B8DEC2FB6A5376172179DA428BBECAC45F245747DD2D47D0A014119FA441326A50765F4"
-#define FM18_P768_Y1		"D009FDA9F7B1666F00A9DDBAA872AA0FFD60F9436E9DFED25AE3FEF0D346C232E8B25624106D30CA7A22986423D922A1C1291D370369826DC1E189A489581191C66F1E983521C3318A66FFCA874EB12A07CAF39048C9E985C6068C7571FF6790"
-#define FM18_P768_Y2		"6FAD5BE9CBC44355D5745ABB148D5132DDBBF1BEAB092D03954FD7DB16D0EC25C1EDA664057E758481AF124666DD668478B452C43B9C1B78602193262DD9AE8DD749F40F58FC458D6CF1CD3442C2822A5C976D4195276A5FC19E16759DAD1FE3"
-#define FM18_P768_R		"FFFFFFF27FA00045F4380E5F9EE3795E88D88C72E7B408B61E4CA1FB2558E7C336F40FAAEC98807AF3600C06C0300001"
-#define FM18_P768_H		"FFFFFFBC7E20087CAA5905F0B82ABB93AF7E81A53882226042648999C855F369B72CECF33CAFCEE57D8E28C84A08805B59DE451E30AC535A4DD982BCD28F9915B40B200C183FEEB08DC7E199BD1C0BC98FB2653657500B21B5876AA05AA4870EF344801BFE0329F91259BB407D680660A83C30FD7B152124C532AE9C1834BB5967AA84FD428CC94EDE0BBF89981839AC28D48E78F22A4695C2C0CC996FFAF415B9B684B1B2427CAA3EE14D0B8463A431ACC81B81F156C21F6B157D8BE3DCB1E0459DE473AC6EFAF4BD9EBBC861C799A4F9EF7D445FB243587315E11A04E4BF85CBE901BA1273F2C229293D5102400004"
+#define FM18_P768_X0		"1D930B0B04D7465AB78CA64B2B5DC31F76CA89CABAF298299A7A9F372BD3FF506AC99CB9BE1915BA02C7ED843B32E5B7117F4FF177B9F8EA8E917EA292C020ABF01B8EC7702797F46657A4AC50E0B1BC0E3EFAB4668822FC41DAA3E3E4B54361"
+#define FM18_P768_X1		"EC814ED9C93EB3787C1D58406920E3941BF9B39200431A5DD3114F3E9A8202CCCA742B4D0A96D083D11883E4CD4A5AA2FF3D10CAFF789D754C4AF11C09BD453FF379F07C778CFDE0156B87B8A731DD5998715486EBFA0A0F40A7591AC19AF918"
+#define FM18_P768_X2		"FCDA64C5662F0E0F0A1FCDCC80BBDCC4B99BAEB06917E0BE604828C636927F9A08A40C184490CE3E5982EDECD2E9D0F93A97BCE43E3C05EC502E781C47866BE70B51F542027C96A2090E96B6428177465C4645F5808ECCDA5A0CB34F689F0E7E"
+#define FM18_P768_Y0		"51CC2E24BFC22B4D46B33FF63B91D0B6EF6CDAF5AEFA9B54588F08787362B29489CD18D3EDEEF975B3E07090149EFEEC95455CBD7FADAC43284D4836A2581F81214FF8D7273AA46DF440628B1B4AEBE2BD30809C675EC62B365DE91484BAA6D8"
+#define FM18_P768_Y1		"87A1EEC340AB26B14EBE6F2056CE339553090C95A0653432428B62FE34BCCA238F0D067B8E8F2345BC2F46DE68493C75C2ED07C5F2D9B29A6BF9FA730D1C78582819735D7FED671B03C359DECC81765A8A8E1DD16C87801FC3FC40176FB55C5C"
+#define FM18_P768_Y2		"DFE9CABA0A20C5FB26091329C60716EC027B828A173120F1AC7BAA5336E97348B26071A022DC0352074F0EA581F63632B6BBBBE9F06CE2762F66087D50D7DB9C96B0192E30F4406104F5D53DCC922A40AEA7600FD79AFB7E2A83350A4D0AD74A"
+#define FM18_P768_R			"FFFFFFF27FA00045F4380E5F9EE3795E88D88C72E7B408B61E4CA1FB2558E7C336F40FAAEC98807AF3600C06C0300001"
+#define FM18_P768_H			"FFFFFFBC7E20087CAA5905F0B82ABB93AF7E81A53882226042648999C855F369B72CECF33CAFCEE57D8E28C84A08805B59DE451E30AC535A4DD982BCD28F9915B40B200C183FEEB08DC7E199BD1C0BC98FB2653657500B21B5876AA05AA4870EF344801BFE0329F91259BB407D680660A83C30FD7B152124C532AE9C1834BB5967AA84FD428CC94EDE0BBF89981839AB28D48E93F2EA45539030933B28949C5EA287BBE39E92B7641ABED52EC449C1F8561BF845845AF977FFF07688C87D2320CF7D4DDBA602142190F32D2FCD0DFB975E2FF266BD2130ABC252AC90AFF19B742F92924F46054CE84DB1865E82A00003"
 /** @} */
 #endif
 

src/epx/relic_ep3_mul.c

@@ -70,9 +70,9 @@ static void ep3_psi(ep3_t r, const ep3_t p) {
 				ep3_frb(r, r, 1);
 				break;
 			case EP_FM18:
-				/* For FM18, we have that -u = (p-p^4) mod r. */
+				/* For FM18, we have that u = (p^4-p) mod r. */
 				ep3_frb(q, p, 3);
-				ep3_sub(r, p, q);
+				ep3_sub(r, q, p);
 				ep3_frb(r, r, 1);
 				break;
 		}

src/epx/relic_ep3_mul_cof.c

@@ -134,6 +134,7 @@
 		ep3_dbl(t4, t4);
 		/* r = [5u+18, u^3+3u^2+1, -3u^2-8u, -3u+1, -u^2-2, u^2+5u]. */
 		ep3_add(r, t4, t3);
+		ep3_norm(r, r);
 	} RLC_CATCH_ANY {
 		RLC_THROW(ERR_CAUGHT);
 	} RLC_FINALLY {
@@ -202,6 +203,86 @@
 		ep3_dbl(t4, t0);
 		ep3_frb(t4, t4, 3);
 		ep3_sub(r, t3, t4);
+		ep3_norm(r, r);
+	} RLC_CATCH_ANY {
+		RLC_THROW(ERR_CAUGHT);
+	} RLC_FINALLY {
+		ep3_free(t0);
+		ep3_free(t1);
+		ep3_free(t2);
+		ep3_free(t3);
+		ep3_free(t4);
+		bn_free(x);
+	}
+}
+
+/**
+ * Multiplies a point by the cofactor in a Fotiadis-Mardindale curve.
+ *
+ * @param[out] r			- the result.
+ * @param[in] p				- the point to multiply.
+ */
+static void ep3_mul_cof_fm18(ep3_t r, const ep3_t p) {
+	ep3_t t0, t1, t2, t3, t4;
+	bn_t x;
+
+	ep3_null(t0);
+	ep3_null(t1);
+	ep3_null(t2);
+	ep3_null(t3);
+	ep3_null(t4);
+	bn_null(x);
+
+	RLC_TRY {
+		ep3_new(t0);
+		ep3_new(t1);
+		ep3_new(t2);
+		ep3_new(t3);
+		ep3_new(t4);
+		bn_new(x);
+
+		/* Vector computed by Guillevic's MAGMA script:
+		[2*x*(x+2)/3, x^3-(x+2)/3, -2*x^2*(x+2)/3, -x*(x^3+(x+2)/3), 2*(x+2)/3, x^2*(x^3+(x+2)/3)-1] */
+		fp_prime_get_par(x);
+
+		/* t0 = [(x+2)/3]P, t1 = [x]P. */
+		bn_add_dig(x, x, 2);
+		bn_div_dig(x, x, 3);
+		ep3_mul_basic(t0, p, x);
+		ep3_dbl(t1, t0);
+		ep3_add(t1, t1, t0);
+		ep3_dbl(t2, p);
+		ep3_sub(t1, t1, t2);
+
+		/* Compute t2 = [x*(x+2)/3]P, t1 = [3*x*(x+2)/3-2x]P = [x^2]P. */
+		fp_prime_get_par(x);
+		ep3_frb(t3, t0, 4);
+		ep3_mul_basic(t2, t0, x);
+		ep3_add(t3, t3, t2);
+		ep3_dbl(t4, t2);
+		ep3_add(t4, t4, t2);
+		ep3_dbl(t1, t1);
+		ep3_sub(t1, t4, t1);
+		ep3_norm(t1, t1);
+		/* Compute t2 = [x^2*(x+2)/3]P, */
+		ep3_mul_basic(t2, t2, x);
+		ep3_frb(t4, t2, 2);
+		ep3_sub(t3, t3, t4);
+		ep3_dbl(t3, t3);
+		ep3_mul_basic(t2, t1, x);
+		ep3_sub(t4, t2, t0);
+		ep3_frb(t4, t4, 1);
+		ep3_add(t3, t3, t4);
+		ep3_add(t4, t2, t0);
+		ep3_norm(t4, t4);
+		ep3_mul_basic(t2, t4, x);
+		ep3_frb(t4, t2, 3);
+		ep3_sub(t3, t3, t4);
+		ep3_mul_basic(t2, t2, x);
+		ep3_sub(t2, t2, p);
+		ep3_frb(t2, t2, 5);
+		ep3_add(t3, t3, t2);
+		ep3_norm(r, t3);
 	} RLC_CATCH_ANY {
 		RLC_THROW(ERR_CAUGHT);
 	} RLC_FINALLY {
@@ -231,6 +312,9 @@
 			case EP_SG18:
 				ep3_mul_cof_sg18(r, p);
 				break;
+			case EP_FM18:
+				ep3_mul_cof_fm18(r, p);
+				break;
 			default:
 				/* Now, multiply by cofactor to get the correct group. */
 				ep3_curve_get_cof(k);

src/fpx/relic_fp3_mul.c

@@ -190,6 +190,7 @@ void fp3_mul_nor(fp3_t c, const fp3_t a) {
 		fp3_mul_art(t, a);
 
 		int cnr = fp3_field_get_cnr();
+		cnr = (cnr < 0 ? -cnr : cnr);
 		switch (fp_prime_get_mod18()) {
 			case 1:
 			case 7:
@@ -202,7 +203,11 @@ void fp3_mul_nor(fp3_t c, const fp3_t a) {
 						}
 						cnr = cnr >> 1;
 					}
-					fp3_add(t, t, u);
+					if (fp3_field_get_cnr() > 0) {
+						fp3_add(t, t, u);
+					} else {
+						fp3_sub(t, t, u);
+					}
 				}
 				break;
 		}

src/fpx/relic_fpx_cyc.c

@@ -1557,21 +1557,32 @@ static void fp18_gls(fp18_t c, const fp18_t a) {
 	RLC_TRY {
 		fp18_new(b);
 
-		if (ep_curve_is_pairf() == EP_SG18) {
-			/* -3*u = (2*p^2 - p^5) mod r */
-			fp18_frb(b, a, 5);
-			fp18_inv_cyc(b, b);
-			fp18_frb(c, a, 2);
-			fp18_sqr_cyc(c, c);
-			fp18_mul(c, c, b);
-		} else {
-			/* For KSS18, we have that x = p^4 - 3*p = (p^3 - 3)p mod n. */
-			fp18_sqr_cyc(b, a);
-			fp18_mul(b, b, a);
-			fp18_frb(c, a, 3);
-			fp18_inv_cyc(b, b);
-			fp18_mul(c, c, b);
-			fp18_frb(c, c, 1);
+		switch (ep_curve_is_pairf()) {
+			case EP_SG18:
+				/* -3*u = (2*p^2 - p^5) mod r */
+				fp18_frb(b, a, 5);
+				fp18_inv_cyc(b, b);
+				fp18_frb(c, a, 2);
+				fp18_sqr_cyc(c, c);
+				fp18_mul(c, c, b);
+				break;
+			case EP_K18:
+				/* For KSS18, we have that x = p^4 - 3*p = (p^3 - 3)p mod n. */
+				fp18_sqr_cyc(b, a);
+				fp18_mul(b, b, a);
+				fp18_frb(c, a, 3);
+				fp18_inv_cyc(b, b);
+				fp18_mul(c, c, b);
+				fp18_frb(c, c, 1);
+				break;
+			case EP_FM18:
+				/* For FM18, we have that u = (p^4-p) mod r. */
+				fp18_frb(b, a, 3);
+				fp18_inv_cyc(b, b);
+				fp18_mul(c, a, b);
+				fp18_frb(c, c, 1);
+				fp18_inv_cyc(c, c);
+				break;
 		}
 	}
 	RLC_CATCH_ANY {

src/fpx/relic_fpx_field.c

@@ -54,8 +54,9 @@ int fp3_field_get_cnr() {
 	} else {
 		return 3;
 	}
+#elif FP_PRIME == 768
+	return -4;
 #endif
-
 	return core_get()->cnr3;
 }
 
@@ -188,7 +189,7 @@ void fp3_field_init(void) {
 		fp_zero(t0[2]);
 		/* If it does not work, attempt (u + 1), otherwise double. */
 		/* This code will fail if p \neq 1 mod 8 because square root in Fp^3
-		 * relic on Frobenius. Must implement explicit test for those cases. */
+		 * relies on Frobenius. Must implement explicit test for those cases. */
 		if (fp3_srt(t1, t0)) {
 			ctx->cnr3 = 1;
 			fp_set_dig(t0[0], ctx->cnr3);

src/low/easy/relic_fpx_add_low.c

@@ -306,6 +306,7 @@ void fp3_nord_low(dv3_t c, dv3_t a) {
 		dv_copy(t[1], a[0], 2 * RLC_FP_DIGS);
 
 		int cnr = fp3_field_get_cnr();
+		cnr = (cnr < 0 ? -cnr : cnr);
 		switch (fp_prime_get_mod18()) {
 			case 1:
 			case 7:
@@ -320,7 +321,11 @@ void fp3_nord_low(dv3_t c, dv3_t a) {
 						}
 						cnr = cnr >> 1;
 					}
-					fp3_addc_low(t, t, u);
+					if (fp3_field_get_cnr() > 0) {
+						fp3_addc_low(t, t, u);
+					} else {
+						fp3_subc_low(t, t, u);
+					}
 				}
 				break;
 		}

src/pc/relic_pc_util.c

@@ -56,7 +56,7 @@ void gt_rand(gt_t a) {
 	pp_exp_k24(a, a);
 #elif FP_PRIME == 330 || FP_PRIME == 765 || FP_PRIME == 766
 	pp_exp_k16(a, a);
-#elif FP_PRIME == 508 || FP_PRIME == 638 && !defined(FP_QNRES)
+#elif FP_PRIME == 508 || FP_PRIME == 768 || FP_PRIME == 638 && !defined(FP_QNRES)
 	pp_exp_k18(a, a);
 #else
 	pp_exp_k12(a, a);
@@ -292,6 +292,7 @@ int g2_is_valid(const g2_t a) {
 		g2_new(v);
 		g2_new(w);
 
+		fp_prime_get_par(n);
 		switch (ep_curve_is_pairf()) {
 #if defined(EP_ENDOM) && !defined(STRIP)
 			/* Formulas from "Co-factor clearing and subgroup membership
@@ -319,7 +320,6 @@ int g2_is_valid(const g2_t a) {
 			 * https://eprint.iacr.org/2022/348.pdf */
 			case EP_BN:
 				/*Check that [z+1]P+[z]\psi(P)+[z]\psi^2(P)=[2z]\psi^3(P)*/
-				fp_prime_get_par(n);
 				g2_mul_any(u, a, n);
 				g2_frb(v, u, 1);
 				g2_add(u, u, a);
@@ -333,7 +333,6 @@ int g2_is_valid(const g2_t a) {
 			/* If u is even, check that [u*p^3]P = P
 			 * else check [p^5]P = [u]P. */
 			case EP_N16:
-				fp_prime_get_par(n);
 				g2_mul_any(u, a, n);
 				if (bn_is_even(n)) {
 					g2_frb(v, u, 3);
@@ -348,7 +347,6 @@ int g2_is_valid(const g2_t a) {
 			 * https://eprint.iacr.org/2022/348.pdf
 			 * Paper has u = 45 mod 70, we ran their code for u = 25 mod 70. */
 			case EP_K16:
-				fp_prime_get_par(n);
 				bn_mod_dig(&rem, n, 70);
 				if (rem == 45) {
 					bn_neg(n, n);
@@ -400,14 +398,12 @@ int g2_is_valid(const g2_t a) {
 				break;
 			case EP_FM16:
 				/* Check that u*Q == psi(Q). */
-				fp_prime_get_par(n);
 				g2_mul_any(u, a, n);
 				g2_frb(v, a, 1);
 				r = g2_on_curve(a) && (g2_cmp(u, v) == RLC_EQ);
 				break;
 			case EP_K18:
 				/* Check that P + u*psi2P + 2*psi3P == \mathcal{O}. */
-				fp_prime_get_par(n);
 				g2_frb(u, a, 2);
 				g2_frb(v, u, 1);
 				g2_dbl(v, v);
@@ -416,9 +412,15 @@ int g2_is_valid(const g2_t a) {
 				g2_neg(u, v);
 				r = g2_on_curve(a) && (g2_cmp(u, a) == RLC_EQ);
 				break;
+			case EP_FM18:
+				/* Check that Q == -u*\psi^2(Q). */
+				bn_neg(n, n);
+				g2_mul_any(u, a, n);
+				g2_frb(u, u, 2);
+				r = g2_on_curve(a) && (g2_cmp(u, a) == RLC_EQ);
+				break;
 			case EP_SG18:
 				/* Check that 3u*P + 2\psi^2(P) == \psi^5P] and [3]P \eq O. */
-				fp_prime_get_par(n);
 				bn_mul_dig(n, n, 3);
 				g2_mul_any(u, a, n);
 				r = g2_is_infty(a) == 0;
@@ -611,6 +613,14 @@ int gt_is_valid(const gt_t a) {
 				r = (gt_cmp(u, a) == RLC_EQ);
 				r &= fp18_test_cyc((void *)a);
 				break;
+			case EP_FM18:
+				/* Check that Q == -u*\psi^2(Q). */
+				bn_neg(n, n);
+				gt_exp(u, a, n);
+				gt_frb(u, u, 2);
+				r = (gt_cmp(u, a) == RLC_EQ);
+				r &= fp18_test_cyc((void *)a);
+				break;
 			case EP_SG18:
 				/* Check that 3u*P + 2\psi^2(P) == \psi^5P] and [3]P \eq O. */
 				fp_prime_get_par(n);