curve25519 for wireguard.

via Matt Dunwoodie and Jason A. Donenfeld

ok deraadt@

1 files changed, 890 insertions, 0 deletions

diff --git a/sys/crypto/curve25519.c b/sys/crypto/curve25519.c
new file mode 100644
index 00000000000..3e3aa1b9c4a
--- /dev/null
+++ b/sys/crypto/curve25519.c
@@ -0,0 +1,890 @@
+/*
+ * Copyright (C) 2018-2020 Jason A. Donenfeld <Jason@zx2c4.com>. All Rights Reserved.
+ * Copyright (C) 2015-2016 The fiat-crypto Authors.
+ *
+ * Permission to use, copy, modify, and distribute this software for any
+ * purpose with or without fee is hereby granted, provided that the above
+ * copyright notice and this permission notice appear in all copies.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+ * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
+ * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+ * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
+ * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
+ * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
+ *
+ * This contains two implementation: a machine-generated formally verified
+ * implementation of Curve25519 ECDH from:
+ * <https://github.com/mit-plv/fiat-crypto>. Though originally machine
+ * generated, it has been tweaked to be suitable for use in the kernel.  It is
+ * optimized for 32-bit machines and machines that cannot work efficiently with
+ * 128-bit integer types.
+ */
+
+#include <sys/types.h>
+#include <sys/systm.h>
+#include <crypto/curve25519.h>
+
+#define __always_inline __inline __attribute__((__always_inline__))
+static const uint8_t null_point[CURVE25519_KEY_SIZE];
+static const uint8_t base_point[CURVE25519_KEY_SIZE] = { 9 };
+
+int curve25519_generate_public(uint8_t pub[CURVE25519_KEY_SIZE],
+			       const uint8_t secret[CURVE25519_KEY_SIZE])
+{
+	if (timingsafe_bcmp(secret, null_point, CURVE25519_KEY_SIZE) == 0)
+		return 0;
+	return curve25519(pub, secret, base_point);
+}
+
+static __always_inline uint32_t get_unaligned_le32(const uint8_t *a)
+{
+	uint32_t l;
+	__builtin_memcpy(&l, a, sizeof(l));
+	return letoh32(l);
+}
+
+/* fe means field element. Here the field is \Z/(2^255-19). An element t,
+ * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
+ * t[3]+2^102 t[4]+...+2^230 t[9].
+ * fe limbs are bounded by 1.125*2^26,1.125*2^25,1.125*2^26,1.125*2^25,etc.
+ * Multiplication and carrying produce fe from fe_loose.
+ */
+typedef struct fe { uint32_t v[10]; } fe;
+
+/* fe_loose limbs are bounded by 3.375*2^26,3.375*2^25,3.375*2^26,3.375*2^25,etc
+ * Addition and subtraction produce fe_loose from (fe, fe).
+ */
+typedef struct fe_loose { uint32_t v[10]; } fe_loose;
+
+static __always_inline void fe_frombytes_impl(uint32_t h[10], const uint8_t *s)
+{
+	/* Ignores top bit of s. */
+	uint32_t a0 = get_unaligned_le32(s);
+	uint32_t a1 = get_unaligned_le32(s+4);
+	uint32_t a2 = get_unaligned_le32(s+8);
+	uint32_t a3 = get_unaligned_le32(s+12);
+	uint32_t a4 = get_unaligned_le32(s+16);
+	uint32_t a5 = get_unaligned_le32(s+20);
+	uint32_t a6 = get_unaligned_le32(s+24);
+	uint32_t a7 = get_unaligned_le32(s+28);
+	h[0] = a0&((1<<26)-1);                    /* 26 used, 32-26 left.   26 */
+	h[1] = (a0>>26) | ((a1&((1<<19)-1))<< 6); /* (32-26) + 19 =  6+19 = 25 */
+	h[2] = (a1>>19) | ((a2&((1<<13)-1))<<13); /* (32-19) + 13 = 13+13 = 26 */
+	h[3] = (a2>>13) | ((a3&((1<< 6)-1))<<19); /* (32-13) +  6 = 19+ 6 = 25 */
+	h[4] = (a3>> 6);                          /* (32- 6)              = 26 */
+	h[5] = a4&((1<<25)-1);                    /*                        25 */
+	h[6] = (a4>>25) | ((a5&((1<<19)-1))<< 7); /* (32-25) + 19 =  7+19 = 26 */
+	h[7] = (a5>>19) | ((a6&((1<<12)-1))<<13); /* (32-19) + 12 = 13+12 = 25 */
+	h[8] = (a6>>12) | ((a7&((1<< 6)-1))<<20); /* (32-12) +  6 = 20+ 6 = 26 */
+	h[9] = (a7>> 6)&((1<<25)-1); /*                                     25 */
+}
+
+static __always_inline void fe_frombytes(fe *h, const uint8_t *s)
+{
+	fe_frombytes_impl(h->v, s);
+}
+
+static __always_inline uint8_t /*bool*/
+addcarryx_u25(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low)
+{
+	/* This function extracts 25 bits of result and 1 bit of carry
+	 * (26 total), so a 32-bit intermediate is sufficient.
+	 */
+	uint32_t x = a + b + c;
+	*low = x & ((1 << 25) - 1);
+	return (x >> 25) & 1;
+}
+
+static __always_inline uint8_t /*bool*/
+addcarryx_u26(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low)
+{
+	/* This function extracts 26 bits of result and 1 bit of carry
+	 * (27 total), so a 32-bit intermediate is sufficient.
+	 */
+	uint32_t x = a + b + c;
+	*low = x & ((1 << 26) - 1);
+	return (x >> 26) & 1;
+}
+
+static __always_inline uint8_t /*bool*/
+subborrow_u25(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low)
+{
+	/* This function extracts 25 bits of result and 1 bit of borrow
+	 * (26 total), so a 32-bit intermediate is sufficient.
+	 */
+	uint32_t x = a - b - c;
+	*low = x & ((1 << 25) - 1);
+	return x >> 31;
+}
+
+static __always_inline uint8_t /*bool*/
+subborrow_u26(uint8_t /*bool*/ c, uint32_t a, uint32_t b, uint32_t *low)
+{
+	/* This function extracts 26 bits of result and 1 bit of borrow
+	 *(27 total), so a 32-bit intermediate is sufficient.
+	 */
+	uint32_t x = a - b - c;
+	*low = x & ((1 << 26) - 1);
+	return x >> 31;
+}
+
+static __always_inline uint32_t cmovznz32(uint32_t t, uint32_t z, uint32_t nz)
+{
+	t = -!!t; /* all set if nonzero, 0 if 0 */
+	return (t&nz) | ((~t)&z);
+}
+
+static __always_inline void fe_freeze(uint32_t out[10], const uint32_t in1[10])
+{
+	const uint32_t x17 = in1[9];
+	const uint32_t x18 = in1[8];
+	const uint32_t x16 = in1[7];
+	const uint32_t x14 = in1[6];
+	const uint32_t x12 = in1[5];
+	const uint32_t x10 = in1[4];
+	const uint32_t x8 = in1[3];
+	const uint32_t x6 = in1[2];
+	const uint32_t x4 = in1[1];
+	const uint32_t x2 = in1[0];
+	uint32_t x20; uint8_t/*bool*/ x21 = subborrow_u26(0x0, x2, 0x3ffffed, &x20);
+	uint32_t x23; uint8_t/*bool*/ x24 = subborrow_u25(x21, x4, 0x1ffffff, &x23);
+	uint32_t x26; uint8_t/*bool*/ x27 = subborrow_u26(x24, x6, 0x3ffffff, &x26);
+	uint32_t x29; uint8_t/*bool*/ x30 = subborrow_u25(x27, x8, 0x1ffffff, &x29);
+	uint32_t x32; uint8_t/*bool*/ x33 = subborrow_u26(x30, x10, 0x3ffffff, &x32);
+	uint32_t x35; uint8_t/*bool*/ x36 = subborrow_u25(x33, x12, 0x1ffffff, &x35);
+	uint32_t x38; uint8_t/*bool*/ x39 = subborrow_u26(x36, x14, 0x3ffffff, &x38);
+	uint32_t x41; uint8_t/*bool*/ x42 = subborrow_u25(x39, x16, 0x1ffffff, &x41);
+	uint32_t x44; uint8_t/*bool*/ x45 = subborrow_u26(x42, x18, 0x3ffffff, &x44);
+	uint32_t x47; uint8_t/*bool*/ x48 = subborrow_u25(x45, x17, 0x1ffffff, &x47);
+	uint32_t x49 = cmovznz32(x48, 0x0, 0xffffffff);
+	uint32_t x50 = (x49 & 0x3ffffed);
+	uint32_t x52; uint8_t/*bool*/ x53 = addcarryx_u26(0x0, x20, x50, &x52);
+	uint32_t x54 = (x49 & 0x1ffffff);
+	uint32_t x56; uint8_t/*bool*/ x57 = addcarryx_u25(x53, x23, x54, &x56);
+	uint32_t x58 = (x49 & 0x3ffffff);
+	uint32_t x60; uint8_t/*bool*/ x61 = addcarryx_u26(x57, x26, x58, &x60);
+	uint32_t x62 = (x49 & 0x1ffffff);
+	uint32_t x64; uint8_t/*bool*/ x65 = addcarryx_u25(x61, x29, x62, &x64);
+	uint32_t x66 = (x49 & 0x3ffffff);
+	uint32_t x68; uint8_t/*bool*/ x69 = addcarryx_u26(x65, x32, x66, &x68);
+	uint32_t x70 = (x49 & 0x1ffffff);
+	uint32_t x72; uint8_t/*bool*/ x73 = addcarryx_u25(x69, x35, x70, &x72);
+	uint32_t x74 = (x49 & 0x3ffffff);
+	uint32_t x76; uint8_t/*bool*/ x77 = addcarryx_u26(x73, x38, x74, &x76);
+	uint32_t x78 = (x49 & 0x1ffffff);
+	uint32_t x80; uint8_t/*bool*/ x81 = addcarryx_u25(x77, x41, x78, &x80);
+	uint32_t x82 = (x49 & 0x3ffffff);
+	uint32_t x84; uint8_t/*bool*/ x85 = addcarryx_u26(x81, x44, x82, &x84);
+	uint32_t x86 = (x49 & 0x1ffffff);
+	uint32_t x88; addcarryx_u25(x85, x47, x86, &x88);
+	out[0] = x52;
+	out[1] = x56;
+	out[2] = x60;
+	out[3] = x64;
+	out[4] = x68;
+	out[5] = x72;
+	out[6] = x76;
+	out[7] = x80;
+	out[8] = x84;
+	out[9] = x88;
+}
+
+static __always_inline void fe_tobytes(uint8_t s[32], const fe *f)
+{
+	uint32_t h[10];
+	fe_freeze(h, f->v);
+	s[0] = h[0] >> 0;
+	s[1] = h[0] >> 8;
+	s[2] = h[0] >> 16;
+	s[3] = (h[0] >> 24) | (h[1] << 2);
+	s[4] = h[1] >> 6;
+	s[5] = h[1] >> 14;
+	s[6] = (h[1] >> 22) | (h[2] << 3);
+	s[7] = h[2] >> 5;
+	s[8] = h[2] >> 13;
+	s[9] = (h[2] >> 21) | (h[3] << 5);
+	s[10] = h[3] >> 3;
+	s[11] = h[3] >> 11;
+	s[12] = (h[3] >> 19) | (h[4] << 6);
+	s[13] = h[4] >> 2;
+	s[14] = h[4] >> 10;
+	s[15] = h[4] >> 18;
+	s[16] = h[5] >> 0;
+	s[17] = h[5] >> 8;
+	s[18] = h[5] >> 16;
+	s[19] = (h[5] >> 24) | (h[6] << 1);
+	s[20] = h[6] >> 7;
+	s[21] = h[6] >> 15;
+	s[22] = (h[6] >> 23) | (h[7] << 3);
+	s[23] = h[7] >> 5;
+	s[24] = h[7] >> 13;
+	s[25] = (h[7] >> 21) | (h[8] << 4);
+	s[26] = h[8] >> 4;
+	s[27] = h[8] >> 12;
+	s[28] = (h[8] >> 20) | (h[9] << 6);
+	s[29] = h[9] >> 2;
+	s[30] = h[9] >> 10;
+	s[31] = h[9] >> 18;
+}
+
+/* h = f */
+static __always_inline void fe_copy(fe *h, const fe *f)
+{
+	memmove(h, f, sizeof(uint32_t) * 10);
+}
+
+static __always_inline void fe_copy_lt(fe_loose *h, const fe *f)
+{
+	memmove(h, f, sizeof(uint32_t) * 10);
+}
+
+/* h = 0 */
+static __always_inline void fe_0(fe *h)
+{
+	memset(h, 0, sizeof(uint32_t) * 10);
+}
+
+/* h = 1 */
+static __always_inline void fe_1(fe *h)
+{
+	memset(h, 0, sizeof(uint32_t) * 10);
+	h->v[0] = 1;
+}
+
+static void fe_add_impl(uint32_t out[10], const uint32_t in1[10], const uint32_t in2[10])
+{
+	const uint32_t x20 = in1[9];
+	const uint32_t x21 = in1[8];
+	const uint32_t x19 = in1[7];
+	const uint32_t x17 = in1[6];
+	const uint32_t x15 = in1[5];
+	const uint32_t x13 = in1[4];
+	const uint32_t x11 = in1[3];
+	const uint32_t x9 = in1[2];
+	const uint32_t x7 = in1[1];
+	const uint32_t x5 = in1[0];
+	const uint32_t x38 = in2[9];
+	const uint32_t x39 = in2[8];
+	const uint32_t x37 = in2[7];
+	const uint32_t x35 = in2[6];
+	const uint32_t x33 = in2[5];
+	const uint32_t x31 = in2[4];
+	const uint32_t x29 = in2[3];
+	const uint32_t x27 = in2[2];
+	const uint32_t x25 = in2[1];
+	const uint32_t x23 = in2[0];
+	out[0] = (x5 + x23);
+	out[1] = (x7 + x25);
+	out[2] = (x9 + x27);
+	out[3] = (x11 + x29);
+	out[4] = (x13 + x31);
+	out[5] = (x15 + x33);
+	out[6] = (x17 + x35);
+	out[7] = (x19 + x37);
+	out[8] = (x21 + x39);
+	out[9] = (x20 + x38);
+}
+
+/* h = f + g
+ * Can overlap h with f or g.
+ */
+static __always_inline void fe_add(fe_loose *h, const fe *f, const fe *g)
+{
+	fe_add_impl(h->v, f->v, g->v);
+}
+
+static void fe_sub_impl(uint32_t out[10], const uint32_t in1[10], const uint32_t in2[10])
+{
+	const uint32_t x20 = in1[9];
+	const uint32_t x21 = in1[8];
+	const uint32_t x19 = in1[7];
+	const uint32_t x17 = in1[6];
+	const uint32_t x15 = in1[5];
+	const uint32_t x13 = in1[4];
+	const uint32_t x11 = in1[3];
+	const uint32_t x9 = in1[2];
+	const uint32_t x7 = in1[1];
+	const uint32_t x5 = in1[0];
+	const uint32_t x38 = in2[9];
+	const uint32_t x39 = in2[8];
+	const uint32_t x37 = in2[7];
+	const uint32_t x35 = in2[6];
+	const uint32_t x33 = in2[5];
+	const uint32_t x31 = in2[4];
+	const uint32_t x29 = in2[3];
+	const uint32_t x27 = in2[2];
+	const uint32_t x25 = in2[1];
+	const uint32_t x23 = in2[0];
+	out[0] = ((0x7ffffda + x5) - x23);
+	out[1] = ((0x3fffffe + x7) - x25);
+	out[2] = ((0x7fffffe + x9) - x27);
+	out[3] = ((0x3fffffe + x11) - x29);
+	out[4] = ((0x7fffffe + x13) - x31);
+	out[5] = ((0x3fffffe + x15) - x33);
+	out[6] = ((0x7fffffe + x17) - x35);
+	out[7] = ((0x3fffffe + x19) - x37);
+	out[8] = ((0x7fffffe + x21) - x39);
+	out[9] = ((0x3fffffe + x20) - x38);
+}
+
+/* h = f - g
+ * Can overlap h with f or g.
+ */
+static __always_inline void fe_sub(fe_loose *h, const fe *f, const fe *g)
+{
+	fe_sub_impl(h->v, f->v, g->v);
+}
+
+static void fe_mul_impl(uint32_t out[10], const uint32_t in1[10], const uint32_t in2[10])
+{
+	const uint32_t x20 = in1[9];
+	const uint32_t x21 = in1[8];
+	const uint32_t x19 = in1[7];
+	const uint32_t x17 = in1[6];
+	const uint32_t x15 = in1[5];
+	const uint32_t x13 = in1[4];
+	const uint32_t x11 = in1[3];
+	const uint32_t x9 = in1[2];
+	const uint32_t x7 = in1[1];
+	const uint32_t x5 = in1[0];
+	const uint32_t x38 = in2[9];
+	const uint32_t x39 = in2[8];
+	const uint32_t x37 = in2[7];
+	const uint32_t x35 = in2[6];
+	const uint32_t x33 = in2[5];
+	const uint32_t x31 = in2[4];
+	const uint32_t x29 = in2[3];
+	const uint32_t x27 = in2[2];
+	const uint32_t x25 = in2[1];
+	const uint32_t x23 = in2[0];
+	uint64_t x40 = ((uint64_t)x23 * x5);
+	uint64_t x41 = (((uint64_t)x23 * x7) + ((uint64_t)x25 * x5));
+	uint64_t x42 = ((((uint64_t)(0x2 * x25) * x7) + ((uint64_t)x23 * x9)) + ((uint64_t)x27 * x5));
+	uint64_t x43 = (((((uint64_t)x25 * x9) + ((uint64_t)x27 * x7)) + ((uint64_t)x23 * x11)) + ((uint64_t)x29 * x5));
+	uint64_t x44 = (((((uint64_t)x27 * x9) + (0x2 * (((uint64_t)x25 * x11) + ((uint64_t)x29 * x7)))) + ((uint64_t)x23 * x13)) + ((uint64_t)x31 * x5));
+	uint64_t x45 = (((((((uint64_t)x27 * x11) + ((uint64_t)x29 * x9)) + ((uint64_t)x25 * x13)) + ((uint64_t)x31 * x7)) + ((uint64_t)x23 * x15)) + ((uint64_t)x33 * x5));
+	uint64_t x46 = (((((0x2 * ((((uint64_t)x29 * x11) + ((uint64_t)x25 * x15)) + ((uint64_t)x33 * x7))) + ((uint64_t)x27 * x13)) + ((uint64_t)x31 * x9)) + ((uint64_t)x23 * x17)) + ((uint64_t)x35 * x5));
+	uint64_t x47 = (((((((((uint64_t)x29 * x13) + ((uint64_t)x31 * x11)) + ((uint64_t)x27 * x15)) + ((uint64_t)x33 * x9)) + ((uint64_t)x25 * x17)) + ((uint64_t)x35 * x7)) + ((uint64_t)x23 * x19)) + ((uint64_t)x37 * x5));
+	uint64_t x48 = (((((((uint64_t)x31 * x13) + (0x2 * (((((uint64_t)x29 * x15) + ((uint64_t)x33 * x11)) + ((uint64_t)x25 * x19)) + ((uint64_t)x37 * x7)))) + ((uint64_t)x27 * x17)) + ((uint64_t)x35 * x9)) + ((uint64_t)x23 * x21)) + ((uint64_t)x39 * x5));
+	uint64_t x49 = (((((((((((uint64_t)x31 * x15) + ((uint64_t)x33 * x13)) + ((uint64_t)x29 * x17)) + ((uint64_t)x35 * x11)) + ((uint64_t)x27 * x19)) + ((uint64_t)x37 * x9)) + ((uint64_t)x25 * x21)) + ((uint64_t)x39 * x7)) + ((uint64_t)x23 * x20)) + ((uint64_t)x38 * x5));
+	uint64_t x50 = (((((0x2 * ((((((uint64_t)x33 * x15) + ((uint64_t)x29 * x19)) + ((uint64_t)x37 * x11)) + ((uint64_t)x25 * x20)) + ((uint64_t)x38 * x7))) + ((uint64_t)x31 * x17)) + ((uint64_t)x35 * x13)) + ((uint64_t)x27 * x21)) + ((uint64_t)x39 * x9));
+	uint64_t x51 = (((((((((uint64_t)x33 * x17) + ((uint64_t)x35 * x15)) + ((uint64_t)x31 * x19)) + ((uint64_t)x37 * x13)) + ((uint64_t)x29 * x21)) + ((uint64_t)x39 * x11)) + ((uint64_t)x27 * x20)) + ((uint64_t)x38 * x9));
+	uint64_t x52 = (((((uint64_t)x35 * x17) + (0x2 * (((((uint64_t)x33 * x19) + ((uint64_t)x37 * x15)) + ((uint64_t)x29 * x20)) + ((uint64_t)x38 * x11)))) + ((uint64_t)x31 * x21)) + ((uint64_t)x39 * x13));
+	uint64_t x53 = (((((((uint64_t)x35 * x19) + ((uint64_t)x37 * x17)) + ((uint64_t)x33 * x21)) + ((uint64_t)x39 * x15)) + ((uint64_t)x31 * x20)) + ((uint64_t)x38 * x13));
+	uint64_t x54 = (((0x2 * ((((uint64_t)x37 * x19) + ((uint64_t)x33 * x20)) + ((uint64_t)x38 * x15))) + ((uint64_t)x35 * x21)) + ((uint64_t)x39 * x17));
+	uint64_t x55 = (((((uint64_t)x37 * x21) + ((uint64_t)x39 * x19)) + ((uint64_t)x35 * x20)) + ((uint64_t)x38 * x17));
+	uint64_t x56 = (((uint64_t)x39 * x21) + (0x2 * (((uint64_t)x37 * x20) + ((uint64_t)x38 * x19))));
+	uint64_t x57 = (((uint64_t)x39 * x20) + ((uint64_t)x38 * x21));
+	uint64_t x58 = ((uint64_t)(0x2 * x38) * x20);
+	uint64_t x59 = (x48 + (x58 << 0x4));
+	uint64_t x60 = (x59 + (x58 << 0x1));
+	uint64_t x61 = (x60 + x58);
+	uint64_t x62 = (x47 + (x57 << 0x4));
+	uint64_t x63 = (x62 + (x57 << 0x1));
+	uint64_t x64 = (x63 + x57);
+	uint64_t x65 = (x46 + (x56 << 0x4));
+	uint64_t x66 = (x65 + (x56 << 0x1));
+	uint64_t x67 = (x66 + x56);
+	uint64_t x68 = (x45 + (x55 << 0x4));
+	uint64_t x69 = (x68 + (x55 << 0x1));
+	uint64_t x70 = (x69 + x55);
+	uint64_t x71 = (x44 + (x54 << 0x4));
+	uint64_t x72 = (x71 + (x54 << 0x1));
+	uint64_t x73 = (x72 + x54);
+	uint64_t x74 = (x43 + (x53 << 0x4));
+	uint64_t x75 = (x74 + (x53 << 0x1));
+	uint64_t x76 = (x75 + x53);
+	uint64_t x77 = (x42 + (x52 << 0x4));
+	uint64_t x78 = (x77 + (x52 << 0x1));
+	uint64_t x79 = (x78 + x52);
+	uint64_t x80 = (x41 + (x51 << 0x4));
+	uint64_t x81 = (x80 + (x51 << 0x1));
+	uint64_t x82 = (x81 + x51);
+	uint64_t x83 = (x40 + (x50 << 0x4));
+	uint64_t x84 = (x83 + (x50 << 0x1));
+	uint64_t x85 = (x84 + x50);
+	uint64_t x86 = (x85 >> 0x1a);
+	uint32_t x87 = ((uint32_t)x85 & 0x3ffffff);
+	uint64_t x88 = (x86 + x82);
+	uint64_t x89 = (x88 >> 0x19);
+	uint32_t x90 = ((uint32_t)x88 & 0x1ffffff);
+	uint64_t x91 = (x89 + x79);
+	uint64_t x92 = (x91 >> 0x1a);
+	uint32_t x93 = ((uint32_t)x91 & 0x3ffffff);
+	uint64_t x94 = (x92 + x76);
+	uint64_t x95 = (x94 >> 0x19);
+	uint32_t x96 = ((uint32_t)x94 & 0x1ffffff);
+	uint64_t x97 = (x95 + x73);
+	uint64_t x98 = (x97 >> 0x1a);
+	uint32_t x99 = ((uint32_t)x97 & 0x3ffffff);
+	uint64_t x100 = (x98 + x70);
+	uint64_t x101 = (x100 >> 0x19);
+	uint32_t x102 = ((uint32_t)x100 & 0x1ffffff);
+	uint64_t x103 = (x101 + x67);
+	uint64_t x104 = (x103 >> 0x1a);
+	uint32_t x105 = ((uint32_t)x103 & 0x3ffffff);
+	uint64_t x106 = (x104 + x64);
+	uint64_t x107 = (x106 >> 0x19);
+	uint32_t x108 = ((uint32_t)x106 & 0x1ffffff);
+	uint64_t x109 = (x107 + x61);
+	uint64_t x110 = (x109 >> 0x1a);
+	uint32_t x111 = ((uint32_t)x109 & 0x3ffffff);
+	uint64_t x112 = (x110 + x49);
+	uint64_t x113 = (x112 >> 0x19);
+	uint32_t x114 = ((uint32_t)x112 & 0x1ffffff);
+	uint64_t x115 = (x87 + (0x13 * x113));
+	uint32_t x116 = (uint32_t) (x115 >> 0x1a);
+	uint32_t x117 = ((uint32_t)x115 & 0x3ffffff);
+	uint32_t x118 = (x116 + x90);
+	uint32_t x119 = (x118 >> 0x19);
+	uint32_t x120 = (x118 & 0x1ffffff);
+	out[0] = x117;
+	out[1] = x120;
+	out[2] = (x119 + x93);
+	out[3] = x96;
+	out[4] = x99;
+	out[5] = x102;
+	out[6] = x105;
+	out[7] = x108;
+	out[8] = x111;
+	out[9] = x114;
+}
+
+static __always_inline void fe_mul_ttt(fe *h, const fe *f, const fe *g)
+{
+	fe_mul_impl(h->v, f->v, g->v);
+}
+
+static __always_inline void fe_mul_tlt(fe *h, const fe_loose *f, const fe *g)
+{
+	fe_mul_impl(h->v, f->v, g->v);
+}
+
+static __always_inline void
+fe_mul_tll(fe *h, const fe_loose *f, const fe_loose *g)
+{
+	fe_mul_impl(h->v, f->v, g->v);
+}
+
+static void fe_sqr_impl(uint32_t out[10], const uint32_t in1[10])
+{
+	const uint32_t x17 = in1[9];
+	const uint32_t x18 = in1[8];
+	const uint32_t x16 = in1[7];
+	const uint32_t x14 = in1[6];
+	const uint32_t x12 = in1[5];
+	const uint32_t x10 = in1[4];
+	const uint32_t x8 = in1[3];
+	const uint32_t x6 = in1[2];
+	const uint32_t x4 = in1[1];
+	const uint32_t x2 = in1[0];
+	uint64_t x19 = ((uint64_t)x2 * x2);
+	uint64_t x20 = ((uint64_t)(0x2 * x2) * x4);
+	uint64_t x21 = (0x2 * (((uint64_t)x4 * x4) + ((uint64_t)x2 * x6)));
+	uint64_t x22 = (0x2 * (((uint64_t)x4 * x6) + ((uint64_t)x2 * x8)));
+	uint64_t x23 = ((((uint64_t)x6 * x6) + ((uint64_t)(0x4 * x4) * x8)) + ((uint64_t)(0x2 * x2) * x10));
+	uint64_t x24 = (0x2 * ((((uint64_t)x6 * x8) + ((uint64_t)x4 * x10)) + ((uint64_t)x2 * x12)));
+	uint64_t x25 = (0x2 * (((((uint64_t)x8 * x8) + ((uint64_t)x6 * x10)) + ((uint64_t)x2 * x14)) + ((uint64_t)(0x2 * x4) * x12)));
+	uint64_t x26 = (0x2 * (((((uint64_t)x8 * x10) + ((uint64_t)x6 * x12)) + ((uint64_t)x4 * x14)) + ((uint64_t)x2 * x16)));
+	uint64_t x27 = (((uint64_t)x10 * x10) + (0x2 * ((((uint64_t)x6 * x14) + ((uint64_t)x2 * x18)) + (0x2 * (((uint64_t)x4 * x16) + ((uint64_t)x8 * x12))))));
+	uint64_t x28 = (0x2 * ((((((uint64_t)x10 * x12) + ((uint64_t)x8 * x14)) + ((uint64_t)x6 * x16)) + ((uint64_t)x4 * x18)) + ((uint64_t)x2 * x17)));
+	uint64_t x29 = (0x2 * (((((uint64_t)x12 * x12) + ((uint64_t)x10 * x14)) + ((uint64_t)x6 * x18)) + (0x2 * (((uint64_t)x8 * x16) + ((uint64_t)x4 * x17)))));
+	uint64_t x30 = (0x2 * (((((uint64_t)x12 * x14) + ((uint64_t)x10 * x16)) + ((uint64_t)x8 * x18)) + ((uint64_t)x6 * x17)));
+	uint64_t x31 = (((uint64_t)x14 * x14) + (0x2 * (((uint64_t)x10 * x18) + (0x2 * (((uint64_t)x12 * x16) + ((uint64_t)x8 * x17))))));
+	uint64_t x32 = (0x2 * ((((uint64_t)x14 * x16) + ((uint64_t)x12 * x18)) + ((uint64_t)x10 * x17)));
+	uint64_t x33 = (0x2 * ((((uint64_t)x16 * x16) + ((uint64_t)x14 * x18)) + ((uint64_t)(0x2 * x12) * x17)));
+	uint64_t x34 = (0x2 * (((uint64_t)x16 * x18) + ((uint64_t)x14 * x17)));
+	uint64_t x35 = (((uint64_t)x18 * x18) + ((uint64_t)(0x4 * x16) * x17));
+	uint64_t x36 = ((uint64_t)(0x2 * x18) * x17);
+	uint64_t x37 = ((uint64_t)(0x2 * x17) * x17);
+	uint64_t x38 = (x27 + (x37 << 0x4));
+	uint64_t x39 = (x38 + (x37 << 0x1));
+	uint64_t x40 = (x39 + x37);
+	uint64_t x41 = (x26 + (x36 << 0x4));
+	uint64_t x42 = (x41 + (x36 << 0x1));
+	uint64_t x43 = (x42 + x36);
+	uint64_t x44 = (x25 + (x35 << 0x4));
+	uint64_t x45 = (x44 + (x35 << 0x1));
+	uint64_t x46 = (x45 + x35);
+	uint64_t x47 = (x24 + (x34 << 0x4));
+	uint64_t x48 = (x47 + (x34 << 0x1));
+	uint64_t x49 = (x48 + x34);
+	uint64_t x50 = (x23 + (x33 << 0x4));
+	uint64_t x51 = (x50 + (x33 << 0x1));
+	uint64_t x52 = (x51 + x33);
+	uint64_t x53 = (x22 + (x32 << 0x4));
+	uint64_t x54 = (x53 + (x32 << 0x1));
+	uint64_t x55 = (x54 + x32);
+	uint64_t x56 = (x21 + (x31 << 0x4));
+	uint64_t x57 = (x56 + (x31 << 0x1));
+	uint64_t x58 = (x57 + x31);
+	uint64_t x59 = (x20 + (x30 << 0x4));
+	uint64_t x60 = (x59 + (x30 << 0x1));
+	uint64_t x61 = (x60 + x30);
+	uint64_t x62 = (x19 + (x29 << 0x4));
+	uint64_t x63 = (x62 + (x29 << 0x1));
+	uint64_t x64 = (x63 + x29);
+	uint64_t x65 = (x64 >> 0x1a);
+	uint32_t x66 = ((uint32_t)x64 & 0x3ffffff);
+	uint64_t x67 = (x65 + x61);
+	uint64_t x68 = (x67 >> 0x19);
+	uint32_t x69 = ((uint32_t)x67 & 0x1ffffff);
+	uint64_t x70 = (x68 + x58);
+	uint64_t x71 = (x70 >> 0x1a);
+	uint32_t x72 = ((uint32_t)x70 & 0x3ffffff);
+	uint64_t x73 = (x71 + x55);
+	uint64_t x74 = (x73 >> 0x19);
+	uint32_t x75 = ((uint32_t)x73 & 0x1ffffff);
+	uint64_t x76 = (x74 + x52);
+	uint64_t x77 = (x76 >> 0x1a);
+	uint32_t x78 = ((uint32_t)x76 & 0x3ffffff);
+	uint64_t x79 = (x77 + x49);
+	uint64_t x80 = (x79 >> 0x19);
+	uint32_t x81 = ((uint32_t)x79 & 0x1ffffff);
+	uint64_t x82 = (x80 + x46);
+	uint64_t x83 = (x82 >> 0x1a);
+	uint32_t x84 = ((uint32_t)x82 & 0x3ffffff);
+	uint64_t x85 = (x83 + x43);
+	uint64_t x86 = (x85 >> 0x19);
+	uint32_t x87 = ((uint32_t)x85 & 0x1ffffff);
+	uint64_t x88 = (x86 + x40);
+	uint64_t x89 = (x88 >> 0x1a);
+	uint32_t x90 = ((uint32_t)x88 & 0x3ffffff);
+	uint64_t x91 = (x89 + x28);
+	uint64_t x92 = (x91 >> 0x19);
+	uint32_t x93 = ((uint32_t)x91 & 0x1ffffff);
+	uint64_t x94 = (x66 + (0x13 * x92));
+	uint32_t x95 = (uint32_t) (x94 >> 0x1a);
+	uint32_t x96 = ((uint32_t)x94 & 0x3ffffff);
+	uint32_t x97 = (x95 + x69);
+	uint32_t x98 = (x97 >> 0x19);
+	uint32_t x99 = (x97 & 0x1ffffff);
+	out[0] = x96;
+	out[1] = x99;
+	out[2] = (x98 + x72);
+	out[3] = x75;
+	out[4] = x78;
+	out[5] = x81;
+	out[6] = x84;
+	out[7] = x87;
+	out[8] = x90;
+	out[9] = x93;
+}
+
+static __always_inline void fe_sq_tl(fe *h, const fe_loose *f)
+{
+	fe_sqr_impl(h->v, f->v);
+}
+
+static __always_inline void fe_sq_tt(fe *h, const fe *f)
+{
+	fe_sqr_impl(h->v, f->v);
+}
+
+static __always_inline void fe_loose_invert(fe *out, const fe_loose *z)
+{
+	fe t0;
+	fe t1;
+	fe t2;
+	fe t3;
+	int i;
+
+	fe_sq_tl(&t0, z);
+	fe_sq_tt(&t1, &t0);
+	for (i = 1; i < 2; ++i)
+		fe_sq_tt(&t1, &t1);
+	fe_mul_tlt(&t1, z, &t1);
+	fe_mul_ttt(&t0, &t0, &t1);
+	fe_sq_tt(&t2, &t0);
+	fe_mul_ttt(&t1, &t1, &t2);
+	fe_sq_tt(&t2, &t1);
+	for (i = 1; i < 5; ++i)
+		fe_sq_tt(&t2, &t2);
+	fe_mul_ttt(&t1, &t2, &t1);
+	fe_sq_tt(&t2, &t1);
+	for (i = 1; i < 10; ++i)
+		fe_sq_tt(&t2, &t2);
+	fe_mul_ttt(&t2, &t2, &t1);
+	fe_sq_tt(&t3, &t2);
+	for (i = 1; i < 20; ++i)
+		fe_sq_tt(&t3, &t3);
+	fe_mul_ttt(&t2, &t3, &t2);
+	fe_sq_tt(&t2, &t2);
+	for (i = 1; i < 10; ++i)
+		fe_sq_tt(&t2, &t2);
+	fe_mul_ttt(&t1, &t2, &t1);
+	fe_sq_tt(&t2, &t1);
+	for (i = 1; i < 50; ++i)
+		fe_sq_tt(&t2, &t2);
+	fe_mul_ttt(&t2, &t2, &t1);
+	fe_sq_tt(&t3, &t2);
+	for (i = 1; i < 100; ++i)
+		fe_sq_tt(&t3, &t3);
+	fe_mul_ttt(&t2, &t3, &t2);
+	fe_sq_tt(&t2, &t2);
+	for (i = 1; i < 50; ++i)
+		fe_sq_tt(&t2, &t2);
+	fe_mul_ttt(&t1, &t2, &t1);
+	fe_sq_tt(&t1, &t1);
+	for (i = 1; i < 5; ++i)
+		fe_sq_tt(&t1, &t1);
+	fe_mul_ttt(out, &t1, &t0);
+}
+
+static __always_inline void fe_invert(fe *out, const fe *z)
+{
+	fe_loose l;
+	fe_copy_lt(&l, z);
+	fe_loose_invert(out, &l);
+}
+
+/* Replace (f,g) with (g,f) if b == 1;
+ * replace (f,g) with (f,g) if b == 0.
+ *
+ * Preconditions: b in {0,1}
+ */
+static __always_inline void fe_cswap(fe *f, fe *g, unsigned int b)
+{
+	unsigned i;
+	b = 0 - b;
+	for (i = 0; i < 10; i++) {
+		uint32_t x = f->v[i] ^ g->v[i];
+		x &= b;
+		f->v[i] ^= x;
+		g->v[i] ^= x;
+	}
+}
+
+/* NOTE: based on fiat-crypto fe_mul, edited for in2=121666, 0, 0.*/
+static __always_inline void fe_mul_121666_impl(uint32_t out[10], const uint32_t in1[10])
+{
+	const uint32_t x20 = in1[9];
+	const uint32_t x21 = in1[8];
+	const uint32_t x19 = in1[7];
+	const uint32_t x17 = in1[6];
+	const uint32_t x15 = in1[5];
+	const uint32_t x13 = in1[4];
+	const uint32_t x11 = in1[3];
+	const uint32_t x9 = in1[2];
+	const uint32_t x7 = in1[1];
+	const uint32_t x5 = in1[0];
+	const uint32_t x38 = 0;
+	const uint32_t x39 = 0;
+	const uint32_t x37 = 0;
+	const uint32_t x35 = 0;
+	const uint32_t x33 = 0;
+	const uint32_t x31 = 0;
+	const uint32_t x29 = 0;
+	const uint32_t x27 = 0;
+	const uint32_t x25 = 0;
+	const uint32_t x23 = 121666;
+	uint64_t x40 = ((uint64_t)x23 * x5);
+	uint64_t x41 = (((uint64_t)x23 * x7) + ((uint64_t)x25 * x5));
+	uint64_t x42 = ((((uint64_t)(0x2 * x25) * x7) + ((uint64_t)x23 * x9)) + ((uint64_t)x27 * x5));
+	uint64_t x43 = (((((uint64_t)x25 * x9) + ((uint64_t)x27 * x7)) + ((uint64_t)x23 * x11)) + ((uint64_t)x29 * x5));
+	uint64_t x44 = (((((uint64_t)x27 * x9) + (0x2 * (((uint64_t)x25 * x11) + ((uint64_t)x29 * x7)))) + ((uint64_t)x23 * x13)) + ((uint64_t)x31 * x5));
+	uint64_t x45 = (((((((uint64_t)x27 * x11) + ((uint64_t)x29 * x9)) + ((uint64_t)x25 * x13)) + ((uint64_t)x31 * x7)) + ((uint64_t)x23 * x15)) + ((uint64_t)x33 * x5));
+	uint64_t x46 = (((((0x2 * ((((uint64_t)x29 * x11) + ((uint64_t)x25 * x15)) + ((uint64_t)x33 * x7))) + ((uint64_t)x27 * x13)) + ((uint64_t)x31 * x9)) + ((uint64_t)x23 * x17)) + ((uint64_t)x35 * x5));
+	uint64_t x47 = (((((((((uint64_t)x29 * x13) + ((uint64_t)x31 * x11)) + ((uint64_t)x27 * x15)) + ((uint64_t)x33 * x9)) + ((uint64_t)x25 * x17)) + ((uint64_t)x35 * x7)) + ((uint64_t)x23 * x19)) + ((uint64_t)x37 * x5));
+	uint64_t x48 = (((((((uint64_t)x31 * x13) + (0x2 * (((((uint64_t)x29 * x15) + ((uint64_t)x33 * x11)) + ((uint64_t)x25 * x19)) + ((uint64_t)x37 * x7)))) + ((uint64_t)x27 * x17)) + ((uint64_t)x35 * x9)) + ((uint64_t)x23 * x21)) + ((uint64_t)x39 * x5));
+	uint64_t x49 = (((((((((((uint64_t)x31 * x15) + ((uint64_t)x33 * x13)) + ((uint64_t)x29 * x17)) + ((uint64_t)x35 * x11)) + ((uint64_t)x27 * x19)) + ((uint64_t)x37 * x9)) + ((uint64_t)x25 * x21)) + ((uint64_t)x39 * x7)) + ((uint64_t)x23 * x20)) + ((uint64_t)x38 * x5));
+	uint64_t x50 = (((((0x2 * ((((((uint64_t)x33 * x15) + ((uint64_t)x29 * x19)) + ((uint64_t)x37 * x11)) + ((uint64_t)x25 * x20)) + ((uint64_t)x38 * x7))) + ((uint64_t)x31 * x17)) + ((uint64_t)x35 * x13)) + ((uint64_t)x27 * x21)) + ((uint64_t)x39 * x9));
+	uint64_t x51 = (((((((((uint64_t)x33 * x17) + ((uint64_t)x35 * x15)) + ((uint64_t)x31 * x19)) + ((uint64_t)x37 * x13)) + ((uint64_t)x29 * x21)) + ((uint64_t)x39 * x11)) + ((uint64_t)x27 * x20)) + ((uint64_t)x38 * x9));
+	uint64_t x52 = (((((uint64_t)x35 * x17) + (0x2 * (((((uint64_t)x33 * x19) + ((uint64_t)x37 * x15)) + ((uint64_t)x29 * x20)) + ((uint64_t)x38 * x11)))) + ((uint64_t)x31 * x21)) + ((uint64_t)x39 * x13));
+	uint64_t x53 = (((((((uint64_t)x35 * x19) + ((uint64_t)x37 * x17)) + ((uint64_t)x33 * x21)) + ((uint64_t)x39 * x15)) + ((uint64_t)x31 * x20)) + ((uint64_t)x38 * x13));
+	uint64_t x54 = (((0x2 * ((((uint64_t)x37 * x19) + ((uint64_t)x33 * x20)) + ((uint64_t)x38 * x15))) + ((uint64_t)x35 * x21)) + ((uint64_t)x39 * x17));
+	uint64_t x55 = (((((uint64_t)x37 * x21) + ((uint64_t)x39 * x19)) + ((uint64_t)x35 * x20)) + ((uint64_t)x38 * x17));
+	uint64_t x56 = (((uint64_t)x39 * x21) + (0x2 * (((uint64_t)x37 * x20) + ((uint64_t)x38 * x19))));
+	uint64_t x57 = (((uint64_t)x39 * x20) + ((uint64_t)x38 * x21));
+	uint64_t x58 = ((uint64_t)(0x2 * x38) * x20);
+	uint64_t x59 = (x48 + (x58 << 0x4));
+	uint64_t x60 = (x59 + (x58 << 0x1));
+	uint64_t x61 = (x60 + x58);
+	uint64_t x62 = (x47 + (x57 << 0x4));
+	uint64_t x63 = (x62 + (x57 << 0x1));
+	uint64_t x64 = (x63 + x57);
+	uint64_t x65 = (x46 + (x56 << 0x4));
+	uint64_t x66 = (x65 + (x56 << 0x1));
+	uint64_t x67 = (x66 + x56);
+	uint64_t x68 = (x45 + (x55 << 0x4));
+	uint64_t x69 = (x68 + (x55 << 0x1));
+	uint64_t x70 = (x69 + x55);
+	uint64_t x71 = (x44 + (x54 << 0x4));
+	uint64_t x72 = (x71 + (x54 << 0x1));
+	uint64_t x73 = (x72 + x54);
+	uint64_t x74 = (x43 + (x53 << 0x4));
+	uint64_t x75 = (x74 + (x53 << 0x1));
+	uint64_t x76 = (x75 + x53);
+	uint64_t x77 = (x42 + (x52 << 0x4));
+	uint64_t x78 = (x77 + (x52 << 0x1));
+	uint64_t x79 = (x78 + x52);
+	uint64_t x80 = (x41 + (x51 << 0x4));
+	uint64_t x81 = (x80 + (x51 << 0x1));
+	uint64_t x82 = (x81 + x51);
+	uint64_t x83 = (x40 + (x50 << 0x4));
+	uint64_t x84 = (x83 + (x50 << 0x1));
+	uint64_t x85 = (x84 + x50);
+	uint64_t x86 = (x85 >> 0x1a);
+	uint32_t x87 = ((uint32_t)x85 & 0x3ffffff);
+	uint64_t x88 = (x86 + x82);
+	uint64_t x89 = (x88 >> 0x19);
+	uint32_t x90 = ((uint32_t)x88 & 0x1ffffff);
+	uint64_t x91 = (x89 + x79);
+	uint64_t x92 = (x91 >> 0x1a);
+	uint32_t x93 = ((uint32_t)x91 & 0x3ffffff);
+	uint64_t x94 = (x92 + x76);
+	uint64_t x95 = (x94 >> 0x19);
+	uint32_t x96 = ((uint32_t)x94 & 0x1ffffff);
+	uint64_t x97 = (x95 + x73);
+	uint64_t x98 = (x97 >> 0x1a);
+	uint32_t x99 = ((uint32_t)x97 & 0x3ffffff);
+	uint64_t x100 = (x98 + x70);
+	uint64_t x101 = (x100 >> 0x19);
+	uint32_t x102 = ((uint32_t)x100 & 0x1ffffff);
+	uint64_t x103 = (x101 + x67);
+	uint64_t x104 = (x103 >> 0x1a);
+	uint32_t x105 = ((uint32_t)x103 & 0x3ffffff);
+	uint64_t x106 = (x104 + x64);
+	uint64_t x107 = (x106 >> 0x19);
+	uint32_t x108 = ((uint32_t)x106 & 0x1ffffff);
+	uint64_t x109 = (x107 + x61);
+	uint64_t x110 = (x109 >> 0x1a);
+	uint32_t x111 = ((uint32_t)x109 & 0x3ffffff);
+	uint64_t x112 = (x110 + x49);
+	uint64_t x113 = (x112 >> 0x19);
+	uint32_t x114 = ((uint32_t)x112 & 0x1ffffff);
+	uint64_t x115 = (x87 + (0x13 * x113));
+	uint32_t x116 = (uint32_t) (x115 >> 0x1a);
+	uint32_t x117 = ((uint32_t)x115 & 0x3ffffff);
+	uint32_t x118 = (x116 + x90);
+	uint32_t x119 = (x118 >> 0x19);
+	uint32_t x120 = (x118 & 0x1ffffff);
+	out[0] = x117;
+	out[1] = x120;
+	out[2] = (x119 + x93);
+	out[3] = x96;
+	out[4] = x99;
+	out[5] = x102;
+	out[6] = x105;
+	out[7] = x108;
+	out[8] = x111;
+	out[9] = x114;
+}
+
+static __always_inline void fe_mul121666(fe *h, const fe_loose *f)
+{
+	fe_mul_121666_impl(h->v, f->v);
+}
+
+int curve25519(uint8_t out[CURVE25519_KEY_SIZE],
+	       const uint8_t scalar[CURVE25519_KEY_SIZE],
+	       const uint8_t point[CURVE25519_KEY_SIZE])
+{
+	fe x1, x2, z2, x3, z3;
+	fe_loose x2l, z2l, x3l;
+	unsigned swap = 0;
+	int pos;
+	uint8_t e[32];
+
+	memcpy(e, scalar, 32);
+	curve25519_clamp_secret(e);
+
+	/* The following implementation was transcribed to Coq and proven to
+	 * correspond to unary scalar multiplication in affine coordinates given
+	 * that x1 != 0 is the x coordinate of some point on the curve. It was
+	 * also checked in Coq that doing a ladderstep with x1 = x3 = 0 gives
+	 * z2' = z3' = 0, and z2 = z3 = 0 gives z2' = z3' = 0. The statement was
+	 * quantified over the underlying field, so it applies to Curve25519
+	 * itself and the quadratic twist of Curve25519. It was not proven in
+	 * Coq that prime-field arithmetic correctly simulates extension-field
+	 * arithmetic on prime-field values. The decoding of the byte array
+	 * representation of e was not considered.
+	 *
+	 * Specification of Montgomery curves in affine coordinates:
+	 * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Spec/MontgomeryCurve.v#L27>
+	 *
+	 * Proof that these form a group that is isomorphic to a Weierstrass
+	 * curve:
+	 * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/AffineProofs.v#L35>
+	 *
+	 * Coq transcription and correctness proof of the loop
+	 * (where scalarbits=255):
+	 * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZ.v#L118>
+	 * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L278>
+	 * preconditions: 0 <= e < 2^255 (not necessarily e < order),
+	 * fe_invert(0) = 0
+	 */
+	fe_frombytes(&x1, point);
+	fe_1(&x2);
+	fe_0(&z2);
+	fe_copy(&x3, &x1);
+	fe_1(&z3);
+
+	for (pos = 254; pos >= 0; --pos) {
+		fe tmp0, tmp1;
+		fe_loose tmp0l, tmp1l;
+		/* loop invariant as of right before the test, for the case
+		 * where x1 != 0:
+		 *   pos >= -1; if z2 = 0 then x2 is nonzero; if z3 = 0 then x3
+		 *   is nonzero
+		 *   let r := e >> (pos+1) in the following equalities of
+		 *   projective points:
+		 *   to_xz (r*P)     === if swap then (x3, z3) else (x2, z2)
+		 *   to_xz ((r+1)*P) === if swap then (x2, z2) else (x3, z3)
+		 *   x1 is the nonzero x coordinate of the nonzero
+		 *   point (r*P-(r+1)*P)
+		 */
+		unsigned b = 1 & (e[pos / 8] >> (pos & 7));
+		swap ^= b;
+		fe_cswap(&x2, &x3, swap);
+		fe_cswap(&z2, &z3, swap);
+		swap = b;
+		/* Coq transcription of ladderstep formula (called from
+		 * transcribed loop):
+		 * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZ.v#L89>
+		 * <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L131>
+		 * x1 != 0 <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L217>
+		 * x1  = 0 <https://github.com/mit-plv/fiat-crypto/blob/2456d821825521f7e03e65882cc3521795b0320f/src/Curves/Montgomery/XZProofs.v#L147>
+		 */
+		fe_sub(&tmp0l, &x3, &z3);
+		fe_sub(&tmp1l, &x2, &z2);
+		fe_add(&x2l, &x2, &z2);
+		fe_add(&z2l, &x3, &z3);
+		fe_mul_tll(&z3, &tmp0l, &x2l);
+		fe_mul_tll(&z2, &z2l, &tmp1l);
+		fe_sq_tl(&tmp0, &tmp1l);
+		fe_sq_tl(&tmp1, &x2l);
+		fe_add(&x3l, &z3, &z2);
+		fe_sub(&z2l, &z3, &z2);
+		fe_mul_ttt(&x2, &tmp1, &tmp0);
+		fe_sub(&tmp1l, &tmp1, &tmp0);
+		fe_sq_tl(&z2, &z2l);
+		fe_mul121666(&z3, &tmp1l);
+		fe_sq_tl(&x3, &x3l);
+		fe_add(&tmp0l, &tmp0, &z3);
+		fe_mul_ttt(&z3, &x1, &z2);
+		fe_mul_tll(&z2, &tmp1l, &tmp0l);
+	}
+	/* here pos=-1, so r=e, so to_xz (e*P) === if swap then (x3, z3)
+	 * else (x2, z2)
+	 */
+	fe_cswap(&x2, &x3, swap);
+	fe_cswap(&z2, &z3, swap);
+
+	fe_invert(&z2, &z2);
+	fe_mul_ttt(&x2, &x2, &z2);
+	fe_tobytes(out, &x2);
+
+	explicit_bzero(&x1, sizeof(x1));
+	explicit_bzero(&x2, sizeof(x2));
+	explicit_bzero(&z2, sizeof(z2));
+	explicit_bzero(&x3, sizeof(x3));
+	explicit_bzero(&z3, sizeof(z3));
+	explicit_bzero(&x2l, sizeof(x2l));
+	explicit_bzero(&z2l, sizeof(z2l));
+	explicit_bzero(&x3l, sizeof(x3l));
+	explicit_bzero(&e, sizeof(e));
+	return timingsafe_bcmp(out, null_point, CURVE25519_KEY_SIZE);
+}