kernel / lib / crypto / mldsa.c

// SPDX-License-Identifier: GPL-2.0-or-later
/*
 * Support for verifying ML-DSA signatures
 *
 * Copyright 2025 Google LLC
 */

#include <crypto/mldsa.h>
#include <crypto/sha3.h>
#include <kunit/visibility.h>
#include <linux/export.h>
#include <linux/module.h>
#include <linux/slab.h>
#include <linux/string.h>
#include <linux/unaligned.h>
#include "fips-mldsa.h"

#define Q 8380417 /* The prime q = 2^23 - 2^13 + 1 */
#define QINV_MOD_2_32 58728449 /* Multiplicative inverse of q mod 2^32 */
#define N 256 /* Number of components per ring element */
#define D 13 /* Number of bits dropped from the public key vector t */
#define RHO_LEN 32 /* Length of the public random seed in bytes */
#define MAX_W1_ENCODED_LEN 192 /* Max encoded length of one element of w'_1 */

/*
 * The zetas array in Montgomery form, i.e. with extra factor of 2^32.
 * Reference: FIPS 204 Section 7.5 "NTT and NTT^-1"
 * Generated by the following Python code:
 * q=8380417; [a%q - q*(a%q > q//2) for a in [1753**(int(f'{i:08b}'[::-1], 2)) << 32 for i in range(256)]]
 */
static const s32 zetas_times_2_32[N] = {
	-4186625, 25847,    -2608894, -518909,	237124,	  -777960,  -876248,
	466468,	  1826347,  2353451,  -359251,	-2091905, 3119733,  -2884855,
	3111497,  2680103,  2725464,  1024112,	-1079900, 3585928,  -549488,
	-1119584, 2619752,  -2108549, -2118186, -3859737, -1399561, -3277672,
	1757237,  -19422,   4010497,  280005,	2706023,  95776,    3077325,
	3530437,  -1661693, -3592148, -2537516, 3915439,  -3861115, -3043716,
	3574422,  -2867647, 3539968,  -300467,	2348700,  -539299,  -1699267,
	-1643818, 3505694,  -3821735, 3507263,	-2140649, -1600420, 3699596,
	811944,	  531354,   954230,   3881043,	3900724,  -2556880, 2071892,
	-2797779, -3930395, -1528703, -3677745, -3041255, -1452451, 3475950,
	2176455,  -1585221, -1257611, 1939314,	-4083598, -1000202, -3190144,
	-3157330, -3632928, 126922,   3412210,	-983419,  2147896,  2715295,
	-2967645, -3693493, -411027,  -2477047, -671102,  -1228525, -22981,
	-1308169, -381987,  1349076,  1852771,	-1430430, -3343383, 264944,
	508951,	  3097992,  44288,    -1100098, 904516,	  3958618,  -3724342,
	-8578,	  1653064,  -3249728, 2389356,	-210977,  759969,   -1316856,
	189548,	  -3553272, 3159746,  -1851402, -2409325, -177440,  1315589,
	1341330,  1285669,  -1584928, -812732,	-1439742, -3019102, -3881060,
	-3628969, 3839961,  2091667,  3407706,	2316500,  3817976,  -3342478,
	2244091,  -2446433, -3562462, 266997,	2434439,  -1235728, 3513181,
	-3520352, -3759364, -1197226, -3193378, 900702,	  1859098,  909542,
	819034,	  495491,   -1613174, -43260,	-522500,  -655327,  -3122442,
	2031748,  3207046,  -3556995, -525098,	-768622,  -3595838, 342297,
	286988,	  -2437823, 4108315,  3437287,	-3342277, 1735879,  203044,
	2842341,  2691481,  -2590150, 1265009,	4055324,  1247620,  2486353,
	1595974,  -3767016, 1250494,  2635921,	-3548272, -2994039, 1869119,
	1903435,  -1050970, -1333058, 1237275,	-3318210, -1430225, -451100,
	1312455,  3306115,  -1962642, -1279661, 1917081,  -2546312, -1374803,
	1500165,  777191,   2235880,  3406031,	-542412,  -2831860, -1671176,
	-1846953, -2584293, -3724270, 594136,	-3776993, -2013608, 2432395,
	2454455,  -164721,  1957272,  3369112,	185531,	  -1207385, -3183426,
	162844,	  1616392,  3014001,  810149,	1652634,  -3694233, -1799107,
	-3038916, 3523897,  3866901,  269760,	2213111,  -975884,  1717735,
	472078,	  -426683,  1723600,  -1803090, 1910376,  -1667432, -1104333,
	-260646,  -3833893, -2939036, -2235985, -420899,  -2286327, 183443,
	-976891,  1612842,  -3545687, -554416,	3919660,  -48306,   -1362209,
	3937738,  1400424,  -846154,  1976782
};

/* Reference: FIPS 204 Section 4 "Parameter Sets" */
static const struct mldsa_parameter_set {
	u8 k; /* num rows in the matrix A */
	u8 l; /* num columns in the matrix A */
	u8 ctilde_len; /* length of commitment hash ctilde in bytes; lambda/4 */
	u8 omega; /* max num of 1's in the hint vector h */
	u8 tau; /* num of +-1's in challenge c */
	u8 beta; /* tau times eta */
	u16 pk_len; /* length of public keys in bytes */
	u16 sig_len; /* length of signatures in bytes */
	s32 gamma1; /* coefficient range of y */
} mldsa_parameter_sets[] = {
	[MLDSA44] = {
		.k = 4,
		.l = 4,
		.ctilde_len = 32,
		.omega = 80,
		.tau = 39,
		.beta = 78,
		.pk_len = MLDSA44_PUBLIC_KEY_SIZE,
		.sig_len = MLDSA44_SIGNATURE_SIZE,
		.gamma1 = 1 << 17,
	},
	[MLDSA65] = {
		.k = 6,
		.l = 5,
		.ctilde_len = 48,
		.omega = 55,
		.tau = 49,
		.beta = 196,
		.pk_len = MLDSA65_PUBLIC_KEY_SIZE,
		.sig_len = MLDSA65_SIGNATURE_SIZE,
		.gamma1 = 1 << 19,
	},
	[MLDSA87] = {
		.k = 8,
		.l = 7,
		.ctilde_len = 64,
		.omega = 75,
		.tau = 60,
		.beta = 120,
		.pk_len = MLDSA87_PUBLIC_KEY_SIZE,
		.sig_len = MLDSA87_SIGNATURE_SIZE,
		.gamma1 = 1 << 19,
	},
};

/*
 * An element of the ring R_q (normal form) or the ring T_q (NTT form).  It
 * consists of N integers mod q: either the polynomial coefficients of the R_q
 * element or the components of the T_q element.  In either case, whether they
 * are fully reduced to [0, q - 1] varies in the different parts of the code.
 */
struct mldsa_ring_elem {
	s32 x[N];
};

struct mldsa_verification_workspace {
	/* SHAKE context for computing c, mu, and ctildeprime */
	struct shake_ctx shake;
	/* The fields in this union are used in their order of declaration. */
	union {
		/* The hash of the public key */
		u8 tr[64];
		/* The message representative mu */
		u8 mu[64];
		/* Temporary space for rej_ntt_poly() */
		u8 block[SHAKE128_BLOCK_SIZE + 1];
		/* Encoded element of w'_1 */
		u8 w1_encoded[MAX_W1_ENCODED_LEN];
		/* The commitment hash.  Real length is params->ctilde_len */
		u8 ctildeprime[64];
	};
	/* SHAKE context for generating elements of the matrix A */
	struct shake_ctx a_shake;
	/*
	 * An element of the matrix A generated from the public seed, or an
	 * element of the vector t_1 decoded from the public key and pre-scaled
	 * by 2^d.  Both are in NTT form.  To reduce memory usage, we generate
	 * or decode these elements only as needed.
	 */
	union {
		struct mldsa_ring_elem a;
		struct mldsa_ring_elem t1_scaled;
	};
	/* The challenge c, generated from ctilde */
	struct mldsa_ring_elem c;
	/* A temporary element used during calculations */
	struct mldsa_ring_elem tmp;

	/* The following fields are variable-length: */

	/* The signer's response vector */
	struct mldsa_ring_elem z[/* l */];

	/* The signer's hint vector */
	/* u8 h[k * N]; */
};

/*
 * Compute a * b * 2^-32 mod q.  a * b must be in the range [-2^31 * q, 2^31 * q
 * - 1] before reduction.  The return value is in the range [-q + 1, q - 1].
 *
 * To reduce mod q efficiently, this uses Montgomery reduction with R=2^32.
 * That's where the factor of 2^-32 comes from.  The caller must include a
 * factor of 2^32 at some point to compensate for that.
 *
 * To keep the input and output ranges very close to symmetric, this
 * specifically does a "signed" Montgomery reduction.  That is, when computing
 * d = c * q^-1 mod 2^32, this chooses a representative in [S32_MIN, S32_MAX]
 * rather than [0, U32_MAX], i.e. s32 rather than u32.  This matters in the
 * wider multiplication d * Q when d keeps its value via sign extension.
 *
 * Reference: FIPS 204 Appendix A "Montgomery Multiplication".  But, it doesn't
 * explain it properly: it has an off-by-one error in the upper end of the input
 * range, it doesn't clarify that the signed version should be used, and it
 * gives an unnecessarily large output range.  A better citation is perhaps the
 * Dilithium reference code, which functionally matches the below code and
 * merely has the (benign) off-by-one error in its documentation.
 */
static inline s32 Zq_mult(s32 a, s32 b)
{
	/* Compute the unreduced product c. */
	s64 c = (s64)a * b;

	/*
	 * Compute d = c * q^-1 mod 2^32.  Generate a signed result, as
	 * explained above, but do the actual multiplication using an unsigned
	 * type to avoid signed integer overflow which is undefined behavior.
	 */
	s32 d = (u32)c * QINV_MOD_2_32;

	/*
	 * Compute e = c - d * q.  This makes the low 32 bits zero, since
	 *   c - (c * q^-1) * q mod 2^32
	 * = c - c * (q^-1 * q) mod 2^32
	 * = c - c * 1 mod 2^32
	 * = c - c mod 2^32
	 * = 0 mod 2^32
	 */
	s64 e = c - (s64)d * Q;

	/* Finally, return e * 2^-32. */
	return e >> 32;
}

/*
 * Convert @w to its number-theoretically-transformed representation in-place.
 * Reference: FIPS 204 Algorithm 41, NTT
 *
 * To prevent intermediate overflows, all input coefficients must have absolute
 * value < q.  All output components have absolute value < 9*q.
 */
static void ntt(struct mldsa_ring_elem *w)
{
	int m = 0; /* index in zetas_times_2_32 */

	for (int len = 128; len >= 1; len /= 2) {
		for (int start = 0; start < 256; start += 2 * len) {
			const s32 z = zetas_times_2_32[++m];

			for (int j = start; j < start + len; j++) {
				s32 t = Zq_mult(z, w->x[j + len]);

				w->x[j + len] = w->x[j] - t;
				w->x[j] += t;
			}
		}
	}
}

/*
 * Convert @w from its number-theoretically-transformed representation in-place.
 * Reference: FIPS 204 Algorithm 42, NTT^-1
 *
 * This also multiplies the coefficients by 2^32, undoing an extra factor of
 * 2^-32 introduced earlier, and reduces the coefficients to [0, q - 1].
 */
static void invntt_and_mul_2_32(struct mldsa_ring_elem *w)
{
	int m = 256; /* index in zetas_times_2_32 */

	/* Prevent intermediate overflows. */
	for (int j = 0; j < 256; j++)
		w->x[j] %= Q;

	for (int len = 1; len < 256; len *= 2) {
		for (int start = 0; start < 256; start += 2 * len) {
			const s32 z = -zetas_times_2_32[--m];

			for (int j = start; j < start + len; j++) {
				s32 t = w->x[j];

				w->x[j] = t + w->x[j + len];
				w->x[j + len] = Zq_mult(z, t - w->x[j + len]);
			}
		}
	}
	/*
	 * Multiply by 2^32 * 256^-1.  2^32 cancels the factor of 2^-32 from
	 * earlier Montgomery multiplications.  256^-1 is for NTT^-1.  This
	 * itself uses Montgomery multiplication, so *another* 2^32 is needed.
	 * Thus the actual multiplicand is 2^32 * 2^32 * 256^-1 mod q = 41978.
	 *
	 * Finally, also reduce from [-q + 1, q - 1] to [0, q - 1].
	 */
	for (int j = 0; j < 256; j++) {
		w->x[j] = Zq_mult(w->x[j], 41978);
		w->x[j] += (w->x[j] >> 31) & Q;
	}
}

/*
 * Decode an element of t_1, i.e. the high d bits of t = A*s_1 + s_2.
 * Reference: FIPS 204 Algorithm 23, pkDecode.
 * Also multiply it by 2^d and convert it to NTT form.
 */
static const u8 *decode_t1_elem(struct mldsa_ring_elem *out,
				const u8 *t1_encoded)
{
	for (int j = 0; j < N; j += 4, t1_encoded += 5) {
		u32 v = get_unaligned_le32(t1_encoded);

		out->x[j + 0] = ((v >> 0) & 0x3ff) << D;
		out->x[j + 1] = ((v >> 10) & 0x3ff) << D;
		out->x[j + 2] = ((v >> 20) & 0x3ff) << D;
		out->x[j + 3] = ((v >> 30) | (t1_encoded[4] << 2)) << D;
		static_assert(0x3ff << D < Q); /* All coefficients < q. */
	}
	ntt(out);
	return t1_encoded; /* Return updated pointer. */
}

/*
 * Decode the signer's response vector 'z' from the signature.
 * Reference: FIPS 204 Algorithm 27, sigDecode.
 *
 * This also validates that the coefficients of z are in range, corresponding
 * the infinity norm check at the end of Algorithm 8, ML-DSA.Verify_internal.
 *
 * Finally, this also converts z to NTT form.
 */
static bool decode_z(struct mldsa_ring_elem z[/* l */], int l, s32 gamma1,
		     int beta, const u8 **sig_ptr)
{
	const u8 *sig = *sig_ptr;

	for (int i = 0; i < l; i++) {
		if (l == 4) { /* ML-DSA-44? */
			/* 18-bit coefficients: decode 4 from 9 bytes. */
			for (int j = 0; j < N; j += 4, sig += 9) {
				u64 v = get_unaligned_le64(sig);

				z[i].x[j + 0] = (v >> 0) & 0x3ffff;
				z[i].x[j + 1] = (v >> 18) & 0x3ffff;
				z[i].x[j + 2] = (v >> 36) & 0x3ffff;
				z[i].x[j + 3] = (v >> 54) | (sig[8] << 10);
			}
		} else {
			/* 20-bit coefficients: decode 4 from 10 bytes. */
			for (int j = 0; j < N; j += 4, sig += 10) {
				u64 v = get_unaligned_le64(sig);

				z[i].x[j + 0] = (v >> 0) & 0xfffff;
				z[i].x[j + 1] = (v >> 20) & 0xfffff;
				z[i].x[j + 2] = (v >> 40) & 0xfffff;
				z[i].x[j + 3] =
					(v >> 60) |
					(get_unaligned_le16(&sig[8]) << 4);
			}
		}
		for (int j = 0; j < N; j++) {
			z[i].x[j] = gamma1 - z[i].x[j];
			if (z[i].x[j] <= -(gamma1 - beta) ||
			    z[i].x[j] >= gamma1 - beta)
				return false;
		}
		ntt(&z[i]);
	}
	*sig_ptr = sig; /* Return updated pointer. */
	return true;
}

/*
 * Decode the signer's hint vector 'h' from the signature.
 * Reference: FIPS 204 Algorithm 21, HintBitUnpack
 *
 * Note that there are several ways in which the hint vector can be malformed.
 */
static bool decode_hint_vector(u8 h[/* k * N */], int k, int omega, const u8 *y)
{
	int index = 0;

	memset(h, 0, k * N);
	for (int i = 0; i < k; i++) {
		int count = y[omega + i]; /* num 1's in elems 0 through i */
		int prev = -1;

		/* Cumulative count mustn't decrease or exceed omega. */
		if (count < index || count > omega)
			return false;
		for (; index < count; index++) {
			if (prev >= y[index]) /* Coefficients out of order? */
				return false;
			prev = y[index];
			h[i * N + y[index]] = 1;
		}
	}
	return mem_is_zero(&y[index], omega - index);
}

/*
 * Expand @seed into an element of R_q @c with coefficients in {-1, 0, 1},
 * exactly @tau of them nonzero.  Reference: FIPS 204 Algorithm 29, SampleInBall
 */
static void sample_in_ball(struct mldsa_ring_elem *c, const u8 *seed,
			   size_t seed_len, int tau, struct shake_ctx *shake)
{
	u64 signs;
	u8 j;

	shake256_init(shake);
	shake_update(shake, seed, seed_len);
	shake_squeeze(shake, (u8 *)&signs, sizeof(signs));
	le64_to_cpus(&signs);
	*c = (struct mldsa_ring_elem){};
	for (int i = N - tau; i < N; i++, signs >>= 1) {
		do {
			shake_squeeze(shake, &j, 1);
		} while (j > i);
		c->x[i] = c->x[j];
		c->x[j] = 1 - 2 * (s32)(signs & 1);
	}
}

/*
 * Expand the public seed @rho and @row_and_column into an element of T_q @out.
 * Reference: FIPS 204 Algorithm 30, RejNTTPoly
 *
 * @shake and @block are temporary space used by the expansion.  @block has
 * space for one SHAKE128 block, plus an extra byte to allow reading a u32 from
 * the final 3-byte group without reading out-of-bounds.
 */
static void rej_ntt_poly(struct mldsa_ring_elem *out, const u8 rho[RHO_LEN],
			 __le16 row_and_column, struct shake_ctx *shake,
			 u8 block[SHAKE128_BLOCK_SIZE + 1])
{
	shake128_init(shake);
	shake_update(shake, rho, RHO_LEN);
	shake_update(shake, (u8 *)&row_and_column, sizeof(row_and_column));
	for (int i = 0; i < N;) {
		shake_squeeze(shake, block, SHAKE128_BLOCK_SIZE);
		block[SHAKE128_BLOCK_SIZE] = 0; /* for KMSAN */
		static_assert(SHAKE128_BLOCK_SIZE % 3 == 0);
		for (int j = 0; j < SHAKE128_BLOCK_SIZE && i < N; j += 3) {
			u32 x = get_unaligned_le32(&block[j]) & 0x7fffff;

			if (x < Q) /* Ignore values >= q. */
				out->x[i++] = x;
		}
	}
}

/*
 * Return the HighBits of r adjusted according to hint h
 * Reference: FIPS 204 Algorithm 40, UseHint
 *
 * This is needed because of the public key compression in ML-DSA.
 *
 * h is either 0 or 1, r is in [0, q - 1], and gamma2 is either (q - 1) / 88 or
 * (q - 1) / 32.  Except when invoked via the unit test interface, gamma2 is a
 * compile-time constant, so compilers will optimize the code accordingly.
 */
static __always_inline s32 use_hint(u8 h, s32 r, const s32 gamma2)
{
	const s32 m = (Q - 1) / (2 * gamma2); /* 44 or 16, compile-time const */
	s32 r1;

	/*
	 * Handle the special case where r - (r mod+- (2 * gamma2)) == q - 1,
	 * i.e. r >= q - gamma2.  This is also exactly where the computation of
	 * r1 below would produce 'm' and would need a correction.
	 */
	if (r >= Q - gamma2)
		return h == 0 ? 0 : m - 1;

	/*
	 * Compute the (non-hint-adjusted) HighBits r1 as:
	 *
	 *  r1 = (r - (r mod+- (2 * gamma2))) / (2 * gamma2)
	 *     = floor((r + gamma2 - 1) / (2 * gamma2))
	 *
	 * Note that when '2 * gamma2' is a compile-time constant, compilers
	 * optimize the division to a reciprocal multiplication and shift.
	 */
	r1 = (u32)(r + gamma2 - 1) / (2 * gamma2);

	/*
	 * Return the HighBits r1:
	 *	+ 0 if the hint is 0;
	 *	+ 1 (mod m) if the hint is 1 and the LowBits are positive;
	 *	- 1 (mod m) if the hint is 1 and the LowBits are negative or 0.
	 *
	 * r1 is in (and remains in) [0, m - 1].  Note that when 'm' is a
	 * compile-time constant, compilers optimize the '% m' accordingly.
	 */
	if (h == 0)
		return r1;
	if (r > r1 * (2 * gamma2))
		return (u32)(r1 + 1) % m;
	return (u32)(r1 + m - 1) % m;
}

static __always_inline void use_hint_elem(struct mldsa_ring_elem *w,
					  const u8 h[N], const s32 gamma2)
{
	for (int j = 0; j < N; j++)
		w->x[j] = use_hint(h[j], w->x[j], gamma2);
}

#if IS_ENABLED(CONFIG_CRYPTO_LIB_MLDSA_KUNIT_TEST)
/* Allow the __always_inline function use_hint() to be unit-tested. */
s32 mldsa_use_hint(u8 h, s32 r, s32 gamma2)
{
	return use_hint(h, r, gamma2);
}
EXPORT_SYMBOL_IF_KUNIT(mldsa_use_hint);
#endif

/*
 * Encode one element of the commitment vector w'_1 into a byte string.
 * Reference: FIPS 204 Algorithm 28, w1Encode.
 * Return the number of bytes used: 192 for ML-DSA-44 and 128 for the others.
 */
static size_t encode_w1(u8 out[MAX_W1_ENCODED_LEN],
			const struct mldsa_ring_elem *w1, int k)
{
	size_t pos = 0;

	static_assert(N * 6 / 8 == MAX_W1_ENCODED_LEN);
	if (k == 4) { /* ML-DSA-44? */
		/* 6 bits per coefficient.  Pack 4 at a time. */
		for (int j = 0; j < N; j += 4) {
			u32 v = (w1->x[j + 0] << 0) | (w1->x[j + 1] << 6) |
				(w1->x[j + 2] << 12) | (w1->x[j + 3] << 18);
			out[pos++] = v >> 0;
			out[pos++] = v >> 8;
			out[pos++] = v >> 16;
		}
	} else {
		/* 4 bits per coefficient.  Pack 2 at a time. */
		for (int j = 0; j < N; j += 2)
			out[pos++] = w1->x[j] | (w1->x[j + 1] << 4);
	}
	return pos;
}

int mldsa_verify(enum mldsa_alg alg, const u8 *sig, size_t sig_len,
		 const u8 *msg, size_t msg_len, const u8 *pk, size_t pk_len)
{
	const struct mldsa_parameter_set *params = &mldsa_parameter_sets[alg];
	const int k = params->k, l = params->l;
	/* For now this just does pure ML-DSA with an empty context string. */
	static const u8 msg_prefix[2] = { /* dom_sep= */ 0, /* ctx_len= */ 0 };
	const u8 *ctilde; /* The signer's commitment hash */
	const u8 *t1_encoded = &pk[RHO_LEN]; /* Next encoded element of t_1 */
	u8 *h; /* The signer's hint vector, length k * N */
	size_t w1_enc_len;

	/* Validate the public key and signature lengths. */
	if (pk_len != params->pk_len || sig_len != params->sig_len)
		return -EBADMSG;

	/*
	 * Allocate the workspace, including variable-length fields.  Its size
	 * depends only on the ML-DSA parameter set, not the other inputs.
	 *
	 * For freeing it, use kfree_sensitive() rather than kfree().  This is
	 * mainly to comply with FIPS 204 Section 3.6.3 "Intermediate Values".
	 * In reality it's a bit gratuitous, as this is a public key operation.
	 */
	struct mldsa_verification_workspace *ws __free(kfree_sensitive) =
		kmalloc(sizeof(*ws) + (l * sizeof(ws->z[0])) + (k * N),
			GFP_KERNEL);
	if (!ws)
		return -ENOMEM;
	h = (u8 *)&ws->z[l];

	/* Decode the signature.  Reference: FIPS 204 Algorithm 27, sigDecode */
	ctilde = sig;
	sig += params->ctilde_len;
	if (!decode_z(ws->z, l, params->gamma1, params->beta, &sig))
		return -EBADMSG;
	if (!decode_hint_vector(h, k, params->omega, sig))
		return -EBADMSG;

	/* Recreate the challenge c from the signer's commitment hash. */
	sample_in_ball(&ws->c, ctilde, params->ctilde_len, params->tau,
		       &ws->shake);
	ntt(&ws->c);

	/* Compute the message representative mu. */
	shake256(pk, pk_len, ws->tr, sizeof(ws->tr));
	shake256_init(&ws->shake);
	shake_update(&ws->shake, ws->tr, sizeof(ws->tr));
	shake_update(&ws->shake, msg_prefix, sizeof(msg_prefix));
	shake_update(&ws->shake, msg, msg_len);
	shake_squeeze(&ws->shake, ws->mu, sizeof(ws->mu));

	/* Start computing ctildeprime = H(mu || w1Encode(w'_1)). */
	shake256_init(&ws->shake);
	shake_update(&ws->shake, ws->mu, sizeof(ws->mu));

	/*
	 * Compute the commitment w'_1 from A, z, c, t_1, and h.
	 *
	 * The computation is the same for each of the k rows.  Just do each row
	 * before moving on to the next, resulting in only one loop over k.
	 */
	for (int i = 0; i < k; i++) {
		/*
		 * tmp = NTT(A) * NTT(z) * 2^-32
		 * To reduce memory use, generate each element of NTT(A)
		 * on-demand.  Note that each element is used only once.
		 */
		ws->tmp = (struct mldsa_ring_elem){};
		for (int j = 0; j < l; j++) {
			rej_ntt_poly(&ws->a, pk /* rho is first field of pk */,
				     cpu_to_le16((i << 8) | j), &ws->a_shake,
				     ws->block);
			for (int n = 0; n < N; n++)
				ws->tmp.x[n] +=
					Zq_mult(ws->a.x[n], ws->z[j].x[n]);
		}
		/* All components of tmp now have abs value < l*q. */

		/* Decode the next element of t_1. */
		t1_encoded = decode_t1_elem(&ws->t1_scaled, t1_encoded);

		/*
		 * tmp -= NTT(c) * NTT(t_1 * 2^d) * 2^-32
		 *
		 * Taking a conservative bound for the output of ntt(), the
		 * multiplicands can have absolute value up to 9*q.  That
		 * corresponds to a product with absolute value 81*q^2.  That is
		 * within the limits of Zq_mult() which needs < ~256*q^2.
		 */
		for (int j = 0; j < N; j++)
			ws->tmp.x[j] -= Zq_mult(ws->c.x[j], ws->t1_scaled.x[j]);
		/* All components of tmp now have abs value < (l+1)*q. */

		/* tmp = w'_Approx = NTT^-1(tmp) * 2^32 */
		invntt_and_mul_2_32(&ws->tmp);
		/* All coefficients of tmp are now in [0, q - 1]. */

		/*
		 * tmp = w'_1 = UseHint(h, w'_Approx)
		 * For efficiency, set gamma2 to a compile-time constant.
		 */
		if (k == 4)
			use_hint_elem(&ws->tmp, &h[i * N], (Q - 1) / 88);
		else
			use_hint_elem(&ws->tmp, &h[i * N], (Q - 1) / 32);

		/* Encode and hash the next element of w'_1. */
		w1_enc_len = encode_w1(ws->w1_encoded, &ws->tmp, k);
		shake_update(&ws->shake, ws->w1_encoded, w1_enc_len);
	}

	/* Finish computing ctildeprime. */
	shake_squeeze(&ws->shake, ws->ctildeprime, params->ctilde_len);

	/* Verify that ctilde == ctildeprime. */
	if (memcmp(ws->ctildeprime, ctilde, params->ctilde_len) != 0)
		return -EKEYREJECTED;
	/* ||z||_infinity < gamma1 - beta was already checked in decode_z(). */
	return 0;
}
EXPORT_SYMBOL_GPL(mldsa_verify);

#ifdef CONFIG_CRYPTO_FIPS
static int __init mldsa_mod_init(void)
{
	if (fips_enabled) {
		/*
		 * FIPS cryptographic algorithm self-test.  As per the FIPS
		 * Implementation Guidance, testing any ML-DSA parameter set
		 * satisfies the test requirement for all of them, and only a
		 * positive test is required.
		 */
		int err = mldsa_verify(MLDSA65, fips_test_mldsa65_signature,
				       sizeof(fips_test_mldsa65_signature),
				       fips_test_mldsa65_message,
				       sizeof(fips_test_mldsa65_message),
				       fips_test_mldsa65_public_key,
				       sizeof(fips_test_mldsa65_public_key));
		if (err)
			panic("mldsa: FIPS self-test failed; err=%pe\n",
			      ERR_PTR(err));
	}
	return 0;
}
subsys_initcall(mldsa_mod_init);

static void __exit mldsa_mod_exit(void)
{
}
module_exit(mldsa_mod_exit);
#endif /* CONFIG_CRYPTO_FIPS */

MODULE_DESCRIPTION("ML-DSA signature verification");
MODULE_LICENSE("GPL");