1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
// Copyright 2015-2016 Brian Smith.
//
// Permission to use, copy, modify, and/or 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 AUTHORS DISCLAIM ALL WARRANTIES
// WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
// MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHORS 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.

use super::{
    bigint::{self, Prime},
    verification, RsaEncoding, N,
};
/// RSA PKCS#1 1.5 signatures.
use crate::{
    arithmetic::montgomery::R,
    bits, digest,
    error::{self, KeyRejected},
    io::{self, der, der_writer},
    pkcs8, rand, signature,
};
use alloc::boxed::Box;

/// An RSA key pair, used for signing.
pub struct RsaKeyPair {
    p: PrivatePrime<P>,
    q: PrivatePrime<Q>,
    qInv: bigint::Elem<P, R>,
    qq: bigint::Modulus<QQ>,
    q_mod_n: bigint::Elem<N, R>,
    public: verification::Key,
    public_key: RsaSubjectPublicKey,
}

derive_debug_via_field!(RsaKeyPair, stringify!(RsaKeyPair), public_key);

impl RsaKeyPair {
    /// Parses an unencrypted PKCS#8-encoded RSA private key.
    ///
    /// Only two-prime (not multi-prime) keys are supported. The public modulus
    /// (n) must be at least 2047 bits. The public modulus must be no larger
    /// than 4096 bits. It is recommended that the public modulus be exactly
    /// 2048 or 3072 bits. The public exponent must be at least 65537.
    ///
    /// This will generate a 2048-bit RSA private key of the correct form using
    /// OpenSSL's command line tool:
    ///
    /// ```sh
    ///    openssl genpkey -algorithm RSA \
    ///        -pkeyopt rsa_keygen_bits:2048 \
    ///        -pkeyopt rsa_keygen_pubexp:65537 | \
    ///      openssl pkcs8 -topk8 -nocrypt -outform der > rsa-2048-private-key.pk8
    /// ```
    ///
    /// This will generate a 3072-bit RSA private key of the correct form:
    ///
    /// ```sh
    ///    openssl genpkey -algorithm RSA \
    ///        -pkeyopt rsa_keygen_bits:3072 \
    ///        -pkeyopt rsa_keygen_pubexp:65537 | \
    ///      openssl pkcs8 -topk8 -nocrypt -outform der > rsa-3072-private-key.pk8
    /// ```
    ///
    /// Often, keys generated for use in OpenSSL-based software are stored in
    /// the Base64 “PEM” format without the PKCS#8 wrapper. Such keys can be
    /// converted to binary PKCS#8 form using the OpenSSL command line tool like
    /// this:
    ///
    /// ```sh
    /// openssl pkcs8 -topk8 -nocrypt -outform der \
    ///     -in rsa-2048-private-key.pem > rsa-2048-private-key.pk8
    /// ```
    ///
    /// Base64 (“PEM”) PKCS#8-encoded keys can be converted to the binary PKCS#8
    /// form like this:
    ///
    /// ```sh
    /// openssl pkcs8 -nocrypt -outform der \
    ///     -in rsa-2048-private-key.pem > rsa-2048-private-key.pk8
    /// ```
    ///
    /// The private key is validated according to [NIST SP-800-56B rev. 1]
    /// section 6.4.1.4.3, crt_pkv (Intended Exponent-Creation Method Unknown),
    /// with the following exceptions:
    ///
    /// * Section 6.4.1.2.1, Step 1: Neither a target security level nor an
    ///   expected modulus length is provided as a parameter, so checks
    ///   regarding these expectations are not done.
    /// * Section 6.4.1.2.1, Step 3: Since neither the public key nor the
    ///   expected modulus length is provided as a parameter, the consistency
    ///   check between these values and the private key's value of n isn't
    ///   done.
    /// * Section 6.4.1.2.1, Step 5: No primality tests are done, both for
    ///   performance reasons and to avoid any side channels that such tests
    ///   would provide.
    /// * Section 6.4.1.2.1, Step 6, and 6.4.1.4.3, Step 7:
    ///     * *ring* has a slightly looser lower bound for the values of `p`
    ///     and `q` than what the NIST document specifies. This looser lower
    ///     bound matches what most other crypto libraries do. The check might
    ///     be tightened to meet NIST's requirements in the future. Similarly,
    ///     the check that `p` and `q` are not too close together is skipped
    ///     currently, but may be added in the future.
    ///     - The validity of the mathematical relationship of `dP`, `dQ`, `e`
    ///     and `n` is verified only during signing. Some size checks of `d`,
    ///     `dP` and `dQ` are performed at construction, but some NIST checks
    ///     are skipped because they would be expensive and/or they would leak
    ///     information through side channels. If a preemptive check of the
    ///     consistency of `dP`, `dQ`, `e` and `n` with each other is
    ///     necessary, that can be done by signing any message with the key
    ///     pair.
    ///
    ///     * `d` is not fully validated, neither at construction nor during
    ///     signing. This is OK as far as *ring*'s usage of the key is
    ///     concerned because *ring* never uses the value of `d` (*ring* always
    ///     uses `p`, `q`, `dP` and `dQ` via the Chinese Remainder Theorem,
    ///     instead). However, *ring*'s checks would not be sufficient for
    ///     validating a key pair for use by some other system; that other
    ///     system must check the value of `d` itself if `d` is to be used.
    ///
    /// In addition to the NIST requirements, *ring* requires that `p > q` and
    /// that `e` must be no more than 33 bits.
    ///
    /// See [RFC 5958] and [RFC 3447 Appendix A.1.2] for more details of the
    /// encoding of the key.
    ///
    /// [NIST SP-800-56B rev. 1]:
    ///     http://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-56Br1.pdf
    ///
    /// [RFC 3447 Appendix A.1.2]:
    ///     https://tools.ietf.org/html/rfc3447#appendix-A.1.2
    ///
    /// [RFC 5958]:
    ///     https://tools.ietf.org/html/rfc5958
    pub fn from_pkcs8(pkcs8: &[u8]) -> Result<Self, KeyRejected> {
        const RSA_ENCRYPTION: &[u8] = include_bytes!("../data/alg-rsa-encryption.der");
        let (der, _) = pkcs8::unwrap_key_(
            untrusted::Input::from(&RSA_ENCRYPTION),
            pkcs8::Version::V1Only,
            untrusted::Input::from(pkcs8),
        )?;
        Self::from_der(der.as_slice_less_safe())
    }

    /// Parses an RSA private key that is not inside a PKCS#8 wrapper.
    ///
    /// The private key must be encoded as a binary DER-encoded ASN.1
    /// `RSAPrivateKey` as described in [RFC 3447 Appendix A.1.2]). In all other
    /// respects, this is just like `from_pkcs8()`. See the documentation for
    /// `from_pkcs8()` for more details.
    ///
    /// It is recommended to use `from_pkcs8()` (with a PKCS#8-encoded key)
    /// instead.
    ///
    /// [RFC 3447 Appendix A.1.2]:
    ///     https://tools.ietf.org/html/rfc3447#appendix-A.1.2
    ///
    /// [NIST SP-800-56B rev. 1]:
    ///     http://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-56Br1.pdf
    pub fn from_der(input: &[u8]) -> Result<Self, KeyRejected> {
        untrusted::Input::from(input).read_all(KeyRejected::invalid_encoding(), |input| {
            der::nested(
                input,
                der::Tag::Sequence,
                error::KeyRejected::invalid_encoding(),
                Self::from_der_reader,
            )
        })
    }

    fn from_der_reader(input: &mut untrusted::Reader) -> Result<Self, KeyRejected> {
        let version = der::small_nonnegative_integer(input)
            .map_err(|error::Unspecified| KeyRejected::invalid_encoding())?;
        if version != 0 {
            return Err(KeyRejected::version_not_supported());
        }

        fn positive_integer<'a>(
            input: &mut untrusted::Reader<'a>,
        ) -> Result<io::Positive<'a>, KeyRejected> {
            der::positive_integer(input)
                .map_err(|error::Unspecified| KeyRejected::invalid_encoding())
        }

        let n = positive_integer(input)?;
        let e = positive_integer(input)?;
        let d = positive_integer(input)?.big_endian_without_leading_zero_as_input();
        let p = positive_integer(input)?.big_endian_without_leading_zero_as_input();
        let q = positive_integer(input)?.big_endian_without_leading_zero_as_input();
        let dP = positive_integer(input)?.big_endian_without_leading_zero_as_input();
        let dQ = positive_integer(input)?.big_endian_without_leading_zero_as_input();
        let qInv = positive_integer(input)?.big_endian_without_leading_zero_as_input();

        let (p, p_bits) = bigint::Nonnegative::from_be_bytes_with_bit_length(p)
            .map_err(|error::Unspecified| KeyRejected::invalid_encoding())?;
        let (q, q_bits) = bigint::Nonnegative::from_be_bytes_with_bit_length(q)
            .map_err(|error::Unspecified| KeyRejected::invalid_encoding())?;

        // Our implementation of CRT-based modular exponentiation used requires
        // that `p > q` so swap them if `p < q`. If swapped, `qInv` is
        // recalculated below. `p != q` is verified implicitly below, e.g. when
        // `q_mod_p` is constructed.
        let ((p, p_bits, dP), (q, q_bits, dQ, qInv)) = match q.verify_less_than(&p) {
            Ok(_) => ((p, p_bits, dP), (q, q_bits, dQ, Some(qInv))),
            Err(error::Unspecified) => {
                // TODO: verify `q` and `qInv` are inverses (mod p).
                ((q, q_bits, dQ), (p, p_bits, dP, None))
            }
        };

        // XXX: Some steps are done out of order, but the NIST steps are worded
        // in such a way that it is clear that NIST intends for them to be done
        // in order. TODO: Does this matter at all?

        // 6.4.1.4.3/6.4.1.2.1 - Step 1.

        // Step 1.a is omitted, as explained above.

        // Step 1.b is omitted per above. Instead, we check that the public
        // modulus is 2048 to `PRIVATE_KEY_PUBLIC_MODULUS_MAX_BITS` bits.
        // XXX: The maximum limit of 4096 bits is primarily due to lack of
        // testing of larger key sizes; see, in particular,
        // https://www.mail-archive.com/openssl-dev@openssl.org/msg44586.html
        // and
        // https://www.mail-archive.com/openssl-dev@openssl.org/msg44759.html.
        // Also, this limit might help with memory management decisions later.

        // Step 1.c. We validate e >= 65537.
        let public_key = verification::Key::from_modulus_and_exponent(
            n.big_endian_without_leading_zero_as_input(),
            e.big_endian_without_leading_zero_as_input(),
            bits::BitLength::from_usize_bits(2048),
            super::PRIVATE_KEY_PUBLIC_MODULUS_MAX_BITS,
            65537,
        )?;

        // 6.4.1.4.3 says to skip 6.4.1.2.1 Step 2.

        // 6.4.1.4.3 Step 3.

        // Step 3.a is done below, out of order.
        // Step 3.b is unneeded since `n_bits` is derived here from `n`.

        // 6.4.1.4.3 says to skip 6.4.1.2.1 Step 4. (We don't need to recover
        // the prime factors since they are already given.)

        // 6.4.1.4.3 - Step 5.

        // Steps 5.a and 5.b are omitted, as explained above.

        // Step 5.c.
        //
        // TODO: First, stop if `p < (√2) * 2**((nBits/2) - 1)`.
        //
        // Second, stop if `p > 2**(nBits/2) - 1`.
        let half_n_bits = public_key.n_bits.half_rounded_up();
        if p_bits != half_n_bits {
            return Err(KeyRejected::inconsistent_components());
        }

        // TODO: Step 5.d: Verify GCD(p - 1, e) == 1.

        // Steps 5.e and 5.f are omitted as explained above.

        // Step 5.g.
        //
        // TODO: First, stop if `q < (√2) * 2**((nBits/2) - 1)`.
        //
        // Second, stop if `q > 2**(nBits/2) - 1`.
        if p_bits != q_bits {
            return Err(KeyRejected::inconsistent_components());
        }

        // TODO: Step 5.h: Verify GCD(p - 1, e) == 1.

        let q_mod_n_decoded = q
            .to_elem(&public_key.n)
            .map_err(|error::Unspecified| KeyRejected::inconsistent_components())?;

        // TODO: Step 5.i
        //
        // 3.b is unneeded since `n_bits` is derived here from `n`.

        // 6.4.1.4.3 - Step 3.a (out of order).
        //
        // Verify that p * q == n. We restrict ourselves to modular
        // multiplication. We rely on the fact that we've verified
        // 0 < q < p < n. We check that q and p are close to sqrt(n) and then
        // assume that these preconditions are enough to let us assume that
        // checking p * q == 0 (mod n) is equivalent to checking p * q == n.
        let q_mod_n = bigint::elem_mul(
            public_key.n.oneRR().as_ref(),
            q_mod_n_decoded.clone(),
            &public_key.n,
        );
        let p_mod_n = p
            .to_elem(&public_key.n)
            .map_err(|error::Unspecified| KeyRejected::inconsistent_components())?;
        let pq_mod_n = bigint::elem_mul(&q_mod_n, p_mod_n, &public_key.n);
        if !pq_mod_n.is_zero() {
            return Err(KeyRejected::inconsistent_components());
        }

        // 6.4.1.4.3/6.4.1.2.1 - Step 6.

        // Step 6.a, partial.
        //
        // First, validate `2**half_n_bits < d`. Since 2**half_n_bits has a bit
        // length of half_n_bits + 1, this check gives us 2**half_n_bits <= d,
        // and knowing d is odd makes the inequality strict.
        let (d, d_bits) = bigint::Nonnegative::from_be_bytes_with_bit_length(d)
            .map_err(|_| error::KeyRejected::invalid_encoding())?;
        if !(half_n_bits < d_bits) {
            return Err(KeyRejected::inconsistent_components());
        }
        // XXX: This check should be `d < LCM(p - 1, q - 1)`, but we don't have
        // a good way of calculating LCM, so it is omitted, as explained above.
        d.verify_less_than_modulus(&public_key.n)
            .map_err(|error::Unspecified| KeyRejected::inconsistent_components())?;
        if !d.is_odd() {
            return Err(KeyRejected::invalid_component());
        }

        // Step 6.b is omitted as explained above.

        // 6.4.1.4.3 - Step 7.

        // Step 7.a.
        let p = PrivatePrime::new(p, dP)?;

        // Step 7.b.
        let q = PrivatePrime::new(q, dQ)?;

        let q_mod_p = q.modulus.to_elem(&p.modulus);

        // Step 7.c.
        let qInv = if let Some(qInv) = qInv {
            bigint::Elem::from_be_bytes_padded(qInv, &p.modulus)
                .map_err(|error::Unspecified| KeyRejected::invalid_component())?
        } else {
            // We swapped `p` and `q` above, so we need to calculate `qInv`.
            // Step 7.f below will verify `qInv` is correct.
            let q_mod_p = bigint::elem_mul(p.modulus.oneRR().as_ref(), q_mod_p.clone(), &p.modulus);
            bigint::elem_inverse_consttime(q_mod_p, &p.modulus)
                .map_err(|error::Unspecified| KeyRejected::unexpected_error())?
        };

        // Steps 7.d and 7.e are omitted per the documentation above, and
        // because we don't (in the long term) have a good way to do modulo
        // with an even modulus.

        // Step 7.f.
        let qInv = bigint::elem_mul(p.modulus.oneRR().as_ref(), qInv, &p.modulus);
        bigint::verify_inverses_consttime(&qInv, q_mod_p, &p.modulus)
            .map_err(|error::Unspecified| KeyRejected::inconsistent_components())?;

        let qq = bigint::elem_mul(&q_mod_n, q_mod_n_decoded, &public_key.n).into_modulus::<QQ>()?;

        let public_key_serialized = RsaSubjectPublicKey::from_n_and_e(n, e);

        Ok(Self {
            p,
            q,
            qInv,
            q_mod_n,
            qq,
            public: public_key,
            public_key: public_key_serialized,
        })
    }

    /// Returns the length in bytes of the key pair's public modulus.
    ///
    /// A signature has the same length as the public modulus.
    pub fn public_modulus_len(&self) -> usize {
        self.public_key
            .modulus()
            .big_endian_without_leading_zero_as_input()
            .as_slice_less_safe()
            .len()
    }
}

impl signature::KeyPair for RsaKeyPair {
    type PublicKey = RsaSubjectPublicKey;

    fn public_key(&self) -> &Self::PublicKey {
        &self.public_key
    }
}

/// A serialized RSA public key.
#[derive(Clone)]
pub struct RsaSubjectPublicKey(Box<[u8]>);

impl AsRef<[u8]> for RsaSubjectPublicKey {
    fn as_ref(&self) -> &[u8] {
        self.0.as_ref()
    }
}

derive_debug_self_as_ref_hex_bytes!(RsaSubjectPublicKey);

impl RsaSubjectPublicKey {
    fn from_n_and_e(n: io::Positive, e: io::Positive) -> Self {
        let bytes = der_writer::write_all(der::Tag::Sequence, &|output| {
            der_writer::write_positive_integer(output, &n);
            der_writer::write_positive_integer(output, &e);
        });
        RsaSubjectPublicKey(bytes)
    }

    /// The public modulus (n).
    pub fn modulus(&self) -> io::Positive {
        // Parsing won't fail because we serialized it ourselves.
        let (public_key, _exponent) =
            super::parse_public_key(untrusted::Input::from(self.as_ref())).unwrap();
        public_key
    }

    /// The public exponent (e).
    pub fn exponent(&self) -> io::Positive {
        // Parsing won't fail because we serialized it ourselves.
        let (_public_key, exponent) =
            super::parse_public_key(untrusted::Input::from(self.as_ref())).unwrap();
        exponent
    }
}

struct PrivatePrime<M: Prime> {
    modulus: bigint::Modulus<M>,
    exponent: bigint::PrivateExponent<M>,
}

impl<M: Prime + Clone> PrivatePrime<M> {
    /// Constructs a `PrivatePrime` from the private prime `p` and `dP` where
    /// dP == d % (p - 1).
    fn new(p: bigint::Nonnegative, dP: untrusted::Input) -> Result<Self, KeyRejected> {
        let (p, p_bits) = bigint::Modulus::from_nonnegative_with_bit_length(p)?;
        if p_bits.as_usize_bits() % 512 != 0 {
            return Err(error::KeyRejected::private_modulus_len_not_multiple_of_512_bits());
        }

        // [NIST SP-800-56B rev. 1] 6.4.1.4.3 - Steps 7.a & 7.b.
        let dP = bigint::PrivateExponent::from_be_bytes_padded(dP, &p)
            .map_err(|error::Unspecified| KeyRejected::inconsistent_components())?;

        // XXX: Steps 7.d and 7.e are omitted. We don't check that
        // `dP == d % (p - 1)` because we don't (in the long term) have a good
        // way to do modulo with an even modulus. Instead we just check that
        // `1 <= dP < p - 1`. We'll check it, to some unknown extent, when we
        // do the private key operation, since we verify that the result of the
        // private key operation using the CRT parameters is consistent with `n`
        // and `e`. TODO: Either prove that what we do is sufficient, or make
        // it so.

        Ok(PrivatePrime {
            modulus: p,
            exponent: dP,
        })
    }
}

fn elem_exp_consttime<M, MM>(
    c: &bigint::Elem<MM>,
    p: &PrivatePrime<M>,
) -> Result<bigint::Elem<M>, error::Unspecified>
where
    M: bigint::NotMuchSmallerModulus<MM>,
    M: Prime,
{
    let c_mod_m = bigint::elem_reduced(c, &p.modulus);
    // We could precompute `oneRRR = elem_squared(&p.oneRR`) as mentioned
    // in the Smooth CRT-RSA paper.
    let c_mod_m = bigint::elem_mul(p.modulus.oneRR().as_ref(), c_mod_m, &p.modulus);
    let c_mod_m = bigint::elem_mul(p.modulus.oneRR().as_ref(), c_mod_m, &p.modulus);
    bigint::elem_exp_consttime(c_mod_m, &p.exponent, &p.modulus)
}

// Type-level representations of the different moduli used in RSA signing, in
// addition to `super::N`. See `super::bigint`'s modulue-level documentation.

#[derive(Copy, Clone)]
enum P {}
unsafe impl Prime for P {}
unsafe impl bigint::SmallerModulus<N> for P {}
unsafe impl bigint::NotMuchSmallerModulus<N> for P {}

#[derive(Copy, Clone)]
enum QQ {}
unsafe impl bigint::SmallerModulus<N> for QQ {}
unsafe impl bigint::NotMuchSmallerModulus<N> for QQ {}

// `q < p < 2*q` since `q` is slightly smaller than `p` (see below). Thus:
//
//                         q <  p  < 2*q
//                       q*q < p*q < 2*q*q.
//                      q**2 <  n  < 2*(q**2).
unsafe impl bigint::SlightlySmallerModulus<N> for QQ {}

#[derive(Copy, Clone)]
enum Q {}
unsafe impl Prime for Q {}
unsafe impl bigint::SmallerModulus<N> for Q {}
unsafe impl bigint::SmallerModulus<P> for Q {}

// q < p && `p.bit_length() == q.bit_length()` implies `q < p < 2*q`.
unsafe impl bigint::SlightlySmallerModulus<P> for Q {}

unsafe impl bigint::SmallerModulus<QQ> for Q {}
unsafe impl bigint::NotMuchSmallerModulus<QQ> for Q {}

impl RsaKeyPair {
    /// Sign `msg`. `msg` is digested using the digest algorithm from
    /// `padding_alg` and the digest is then padded using the padding algorithm
    /// from `padding_alg`. The signature it written into `signature`;
    /// `signature`'s length must be exactly the length returned by
    /// `public_modulus_len()`. `rng` may be used to randomize the padding
    /// (e.g. for PSS).
    ///
    /// Many other crypto libraries have signing functions that takes a
    /// precomputed digest as input, instead of the message to digest. This
    /// function does *not* take a precomputed digest; instead, `sign`
    /// calculates the digest itself.
    ///
    /// Lots of effort has been made to make the signing operations close to
    /// constant time to protect the private key from side channel attacks. On
    /// x86-64, this is done pretty well, but not perfectly. On other
    /// platforms, it is done less perfectly.
    pub fn sign(
        &self,
        padding_alg: &'static dyn RsaEncoding,
        rng: &dyn rand::SecureRandom,
        msg: &[u8],
        signature: &mut [u8],
    ) -> Result<(), error::Unspecified> {
        let mod_bits = self.public.n_bits;
        if signature.len() != mod_bits.as_usize_bytes_rounded_up() {
            return Err(error::Unspecified);
        }

        let m_hash = digest::digest(padding_alg.digest_alg(), msg);
        padding_alg.encode(&m_hash, signature, mod_bits, rng)?;

        // RFC 8017 Section 5.1.2: RSADP, using the Chinese Remainder Theorem
        // with Garner's algorithm.

        let n = &self.public.n;

        // Step 1. The value zero is also rejected.
        let base = bigint::Elem::from_be_bytes_padded(untrusted::Input::from(signature), n)?;

        // Step 2
        let c = base;

        // Step 2.b.i.
        let m_1 = elem_exp_consttime(&c, &self.p)?;
        let c_mod_qq = bigint::elem_reduced_once(&c, &self.qq);
        let m_2 = elem_exp_consttime(&c_mod_qq, &self.q)?;

        // Step 2.b.ii isn't needed since there are only two primes.

        // Step 2.b.iii.
        let p = &self.p.modulus;
        let m_2 = bigint::elem_widen(m_2, p);
        let m_1_minus_m_2 = bigint::elem_sub(m_1, &m_2, p);
        let h = bigint::elem_mul(&self.qInv, m_1_minus_m_2, p);

        // Step 2.b.iv. The reduction in the modular multiplication isn't
        // necessary because `h < p` and `p * q == n` implies `h * q < n`.
        // Modular arithmetic is used simply to avoid implementing
        // non-modular arithmetic.
        let h = bigint::elem_widen(h, n);
        let q_times_h = bigint::elem_mul(&self.q_mod_n, h, n);
        let m_2 = bigint::elem_widen(m_2, n);
        let m = bigint::elem_add(m_2, q_times_h, n);

        // Step 2.b.v isn't needed since there are only two primes.

        // Verify the result to protect against fault attacks as described
        // in "On the Importance of Checking Cryptographic Protocols for
        // Faults" by Dan Boneh, Richard A. DeMillo, and Richard J. Lipton.
        // This check is cheap assuming `e` is small, which is ensured during
        // `KeyPair` construction. Note that this is the only validation of `e`
        // that is done other than basic checks on its size, oddness, and
        // minimum value, since the relationship of `e` to `d`, `p`, and `q` is
        // not verified during `KeyPair` construction.
        {
            let verify = bigint::elem_exp_vartime(m.clone(), self.public.e, n);
            let verify = verify.into_unencoded(n);
            bigint::elem_verify_equal_consttime(&verify, &c)?;
        }

        // Step 3.
        //
        // See Falko Strenzke, "Manger's Attack revisited", ICICS 2010.
        m.fill_be_bytes(signature);

        Ok(())
    }
}

#[cfg(test)]
mod tests {
    // We intentionally avoid `use super::*` so that we are sure to use only
    // the public API; this ensures that enough of the API is public.
    use crate::{rand, signature};
    use alloc::vec;

    // `KeyPair::sign` requires that the output buffer is the same length as
    // the public key modulus. Test what happens when it isn't the same length.
    #[test]
    fn test_signature_rsa_pkcs1_sign_output_buffer_len() {
        // Sign the message "hello, world", using PKCS#1 v1.5 padding and the
        // SHA256 digest algorithm.
        const MESSAGE: &[u8] = b"hello, world";
        let rng = rand::SystemRandom::new();

        const PRIVATE_KEY_DER: &[u8] = include_bytes!("signature_rsa_example_private_key.der");
        let key_pair = signature::RsaKeyPair::from_der(PRIVATE_KEY_DER).unwrap();

        // The output buffer is one byte too short.
        let mut signature = vec![0; key_pair.public_modulus_len() - 1];

        assert!(key_pair
            .sign(&signature::RSA_PKCS1_SHA256, &rng, MESSAGE, &mut signature)
            .is_err());

        // The output buffer is the right length.
        signature.push(0);
        assert!(key_pair
            .sign(&signature::RSA_PKCS1_SHA256, &rng, MESSAGE, &mut signature)
            .is_ok());

        // The output buffer is one byte too long.
        signature.push(0);
        assert!(key_pair
            .sign(&signature::RSA_PKCS1_SHA256, &rng, MESSAGE, &mut signature)
            .is_err());
    }
}