1use crate::{Array, BatchVerifier, PrivateKeyExt};
2use bytes::{Buf, BufMut};
3use commonware_codec::{Error as CodecError, FixedSize, Read, ReadExt, Write};
4use commonware_utils::{hex, union_unique, Span};
5use ed25519_consensus::{self, VerificationKey};
6use rand::{CryptoRng, Rng, RngCore};
7use std::{
8 borrow::Cow,
9 fmt::{Debug, Display},
10 hash::{Hash, Hasher},
11 ops::Deref,
12};
13use zeroize::{Zeroize, ZeroizeOnDrop};
14
15const CURVE_NAME: &str = "ed25519";
16const PRIVATE_KEY_LENGTH: usize = 32;
17const PUBLIC_KEY_LENGTH: usize = 32;
18const SIGNATURE_LENGTH: usize = 64;
19
20#[derive(Clone, Zeroize, ZeroizeOnDrop)]
22pub struct PrivateKey {
23 raw: [u8; PRIVATE_KEY_LENGTH],
24 key: ed25519_consensus::SigningKey,
25}
26
27impl crate::PrivateKey for PrivateKey {}
28
29impl crate::Signer for PrivateKey {
30 type Signature = Signature;
31 type PublicKey = PublicKey;
32
33 fn sign(&self, namespace: Option<&[u8]>, msg: &[u8]) -> Self::Signature {
34 let sig = match namespace {
35 Some(namespace) => self.key.sign(&union_unique(namespace, msg)),
36 None => self.key.sign(msg),
37 };
38 Signature::from(sig)
39 }
40
41 fn public_key(&self) -> Self::PublicKey {
42 let raw = self.key.verification_key().to_bytes();
43 Self::PublicKey {
44 raw,
45 key: self.key.verification_key().to_owned(),
46 }
47 }
48}
49
50impl PrivateKeyExt for PrivateKey {
51 fn from_rng<R: Rng + CryptoRng>(rng: &mut R) -> Self {
52 let key = ed25519_consensus::SigningKey::new(rng);
53 let raw = key.to_bytes();
54 Self { raw, key }
55 }
56}
57
58impl Write for PrivateKey {
59 fn write(&self, buf: &mut impl BufMut) {
60 self.raw.write(buf);
61 }
62}
63
64impl Read for PrivateKey {
65 type Cfg = ();
66
67 fn read_cfg(buf: &mut impl Buf, _: &()) -> Result<Self, CodecError> {
68 let raw = <[u8; Self::SIZE]>::read(buf)?;
69 let key = ed25519_consensus::SigningKey::from(raw);
70 Ok(Self { raw, key })
71 }
72}
73
74impl FixedSize for PrivateKey {
75 const SIZE: usize = PRIVATE_KEY_LENGTH;
76}
77
78impl Span for PrivateKey {}
79
80impl Array for PrivateKey {}
81
82impl Eq for PrivateKey {}
83
84impl Hash for PrivateKey {
85 fn hash<H: Hasher>(&self, state: &mut H) {
86 self.raw.hash(state);
87 }
88}
89
90impl PartialEq for PrivateKey {
91 fn eq(&self, other: &Self) -> bool {
92 self.raw == other.raw
93 }
94}
95
96impl Ord for PrivateKey {
97 fn cmp(&self, other: &Self) -> std::cmp::Ordering {
98 self.raw.cmp(&other.raw)
99 }
100}
101
102impl PartialOrd for PrivateKey {
103 fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
104 Some(self.cmp(other))
105 }
106}
107
108impl AsRef<[u8]> for PrivateKey {
109 fn as_ref(&self) -> &[u8] {
110 &self.raw
111 }
112}
113
114impl Deref for PrivateKey {
115 type Target = [u8];
116 fn deref(&self) -> &[u8] {
117 &self.raw
118 }
119}
120
121impl From<ed25519_consensus::SigningKey> for PrivateKey {
122 fn from(key: ed25519_consensus::SigningKey) -> Self {
123 let raw = key.to_bytes();
124 Self { raw, key }
125 }
126}
127
128impl Debug for PrivateKey {
129 fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
130 write!(f, "{}", hex(&self.raw))
131 }
132}
133
134impl Display for PrivateKey {
135 fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
136 write!(f, "{}", hex(&self.raw))
137 }
138}
139
140#[derive(Clone, Eq, PartialEq, Ord, PartialOrd, Hash)]
142pub struct PublicKey {
143 raw: [u8; PUBLIC_KEY_LENGTH],
144 key: ed25519_consensus::VerificationKey,
145}
146
147impl From<PrivateKey> for PublicKey {
148 fn from(value: PrivateKey) -> Self {
149 let raw = value.key.verification_key().to_bytes();
150 Self {
151 raw,
152 key: value.key.verification_key(),
153 }
154 }
155}
156
157impl crate::PublicKey for PublicKey {}
158
159impl crate::Verifier for PublicKey {
160 type Signature = Signature;
161
162 fn verify(&self, namespace: Option<&[u8]>, msg: &[u8], sig: &Self::Signature) -> bool {
163 match namespace {
164 Some(namespace) => {
165 let payload = union_unique(namespace, msg);
166 self.key.verify(&sig.signature, &payload).is_ok()
167 }
168 None => self.key.verify(&sig.signature, msg).is_ok(),
169 }
170 }
171}
172
173impl Write for PublicKey {
174 fn write(&self, buf: &mut impl BufMut) {
175 self.raw.write(buf);
176 }
177}
178
179impl Read for PublicKey {
180 type Cfg = ();
181
182 fn read_cfg(buf: &mut impl Buf, _: &()) -> Result<Self, CodecError> {
183 let raw = <[u8; Self::SIZE]>::read(buf)?;
184 let key = VerificationKey::try_from(raw)
185 .map_err(|e| CodecError::Wrapped(CURVE_NAME, e.into()))?;
186 Ok(Self { raw, key })
187 }
188}
189
190impl FixedSize for PublicKey {
191 const SIZE: usize = PUBLIC_KEY_LENGTH;
192}
193
194impl Span for PublicKey {}
195
196impl Array for PublicKey {}
197
198impl AsRef<[u8]> for PublicKey {
199 fn as_ref(&self) -> &[u8] {
200 &self.raw
201 }
202}
203
204impl Deref for PublicKey {
205 type Target = [u8];
206 fn deref(&self) -> &[u8] {
207 &self.raw
208 }
209}
210
211impl From<VerificationKey> for PublicKey {
212 fn from(key: VerificationKey) -> Self {
213 let raw = key.to_bytes();
214 Self { raw, key }
215 }
216}
217
218impl Debug for PublicKey {
219 fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
220 write!(f, "{}", hex(&self.raw))
221 }
222}
223
224impl Display for PublicKey {
225 fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
226 write!(f, "{}", hex(&self.raw))
227 }
228}
229
230#[derive(Clone, Eq, PartialEq)]
232pub struct Signature {
233 raw: [u8; SIGNATURE_LENGTH],
234 signature: ed25519_consensus::Signature,
235}
236
237impl crate::Signature for Signature {}
238
239impl Write for Signature {
240 fn write(&self, buf: &mut impl BufMut) {
241 self.raw.write(buf);
242 }
243}
244
245impl Read for Signature {
246 type Cfg = ();
247
248 fn read_cfg(buf: &mut impl Buf, _: &()) -> Result<Self, CodecError> {
249 let raw = <[u8; Self::SIZE]>::read(buf)?;
250 let signature = ed25519_consensus::Signature::from(raw);
251 Ok(Self { raw, signature })
252 }
253}
254
255impl FixedSize for Signature {
256 const SIZE: usize = SIGNATURE_LENGTH;
257}
258
259impl Span for Signature {}
260
261impl Array for Signature {}
262
263impl Hash for Signature {
264 fn hash<H: Hasher>(&self, state: &mut H) {
265 self.raw.hash(state);
266 }
267}
268
269impl Ord for Signature {
270 fn cmp(&self, other: &Self) -> std::cmp::Ordering {
271 self.raw.cmp(&other.raw)
272 }
273}
274
275impl PartialOrd for Signature {
276 fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
277 Some(self.cmp(other))
278 }
279}
280
281impl AsRef<[u8]> for Signature {
282 fn as_ref(&self) -> &[u8] {
283 &self.raw
284 }
285}
286
287impl Deref for Signature {
288 type Target = [u8];
289 fn deref(&self) -> &[u8] {
290 &self.raw
291 }
292}
293
294impl From<ed25519_consensus::Signature> for Signature {
295 fn from(value: ed25519_consensus::Signature) -> Self {
296 let raw = value.to_bytes();
297 Self {
298 raw,
299 signature: value,
300 }
301 }
302}
303
304impl Debug for Signature {
305 fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
306 write!(f, "{}", hex(&self.raw))
307 }
308}
309
310impl Display for Signature {
311 fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
312 write!(f, "{}", hex(&self.raw))
313 }
314}
315
316pub struct Batch {
318 verifier: ed25519_consensus::batch::Verifier,
319}
320
321impl BatchVerifier<PublicKey> for Batch {
322 fn new() -> Self {
323 Batch {
324 verifier: ed25519_consensus::batch::Verifier::new(),
325 }
326 }
327
328 fn add(
329 &mut self,
330 namespace: Option<&[u8]>,
331 message: &[u8],
332 public_key: &PublicKey,
333 signature: &Signature,
334 ) -> bool {
335 let payload = match namespace {
336 Some(namespace) => Cow::Owned(union_unique(namespace, message)),
337 None => Cow::Borrowed(message),
338 };
339 let item = ed25519_consensus::batch::Item::from((
340 public_key.key.into(),
341 signature.signature,
342 &payload,
343 ));
344 self.verifier.queue(item);
345 true
346 }
347
348 fn verify<R: RngCore + CryptoRng>(self, rng: &mut R) -> bool {
349 self.verifier.verify(rng).is_ok()
350 }
351}
352
353#[cfg(test)]
355mod tests {
356 use super::*;
357 use crate::{ed25519, Signer as _, Verifier as _};
358 use commonware_codec::{DecodeExt, Encode};
359 use rand::rngs::OsRng;
360
361 fn test_sign_and_verify(
362 private_key: PrivateKey,
363 public_key: PublicKey,
364 message: &[u8],
365 signature: Signature,
366 ) {
367 let computed_signature = private_key.sign(None, message);
368 assert_eq!(computed_signature, signature);
369 assert!(public_key.verify(None, message, &computed_signature));
370 }
371
372 fn parse_private_key(private_key: &str) -> PrivateKey {
373 PrivateKey::decode(
374 commonware_utils::from_hex_formatted(private_key)
375 .unwrap()
376 .as_ref(),
377 )
378 .unwrap()
379 }
380
381 fn parse_public_key(public_key: &str) -> PublicKey {
382 PublicKey::decode(
383 commonware_utils::from_hex_formatted(public_key)
384 .unwrap()
385 .as_ref(),
386 )
387 .unwrap()
388 }
389
390 fn parse_signature(signature: &str) -> Signature {
391 Signature::decode(
392 commonware_utils::from_hex_formatted(signature)
393 .unwrap()
394 .as_ref(),
395 )
396 .unwrap()
397 }
398
399 fn vector_1() -> (PrivateKey, PublicKey, Vec<u8>, Signature) {
400 (
401 parse_private_key(
403 "
404 9d61b19deffd5a60ba844af492ec2cc4
405 4449c5697b326919703bac031cae7f60
406 ",
407 ),
408 parse_public_key(
410 "
411 d75a980182b10ab7d54bfed3c964073a
412 0ee172f3daa62325af021a68f707511a
413 ",
414 ),
415 b"".to_vec(),
417 parse_signature(
419 "
420 e5564300c360ac729086e2cc806e828a
421 84877f1eb8e5d974d873e06522490155
422 5fb8821590a33bacc61e39701cf9b46b
423 d25bf5f0595bbe24655141438e7a100b
424 ",
425 ),
426 )
427 }
428
429 fn vector_2() -> (PrivateKey, PublicKey, Vec<u8>, Signature) {
430 (
431 parse_private_key(
433 "
434 4ccd089b28ff96da9db6c346ec114e0f
435 5b8a319f35aba624da8cf6ed4fb8a6fb
436 ",
437 ),
438 parse_public_key(
440 "
441 3d4017c3e843895a92b70aa74d1b7ebc
442 9c982ccf2ec4968cc0cd55f12af4660c
443 ",
444 ),
445 [0x72].to_vec(),
447 parse_signature(
449 "
450 92a009a9f0d4cab8720e820b5f642540
451 a2b27b5416503f8fb3762223ebdb69da
452 085ac1e43e15996e458f3613d0f11d8c
453 387b2eaeb4302aeeb00d291612bb0c00
454 ",
455 ),
456 )
457 }
458
459 #[test]
460 fn test_codec_private_key() {
461 let private_key = parse_private_key(
462 "
463 9d61b19deffd5a60ba844af492ec2cc4
464 4449c5697b326919703bac031cae7f60
465 ",
466 );
467 let encoded = private_key.encode();
468 assert_eq!(encoded.len(), PRIVATE_KEY_LENGTH);
469 let decoded = PrivateKey::decode(encoded).unwrap();
470 assert_eq!(private_key, decoded);
471 }
472
473 #[test]
474 fn test_codec_public_key() {
475 let public_key = parse_public_key(
476 "
477 d75a980182b10ab7d54bfed3c964073a
478 0ee172f3daa62325af021a68f707511a
479 ",
480 );
481 let encoded = public_key.encode();
482 assert_eq!(encoded.len(), PUBLIC_KEY_LENGTH);
483 let decoded = PublicKey::decode(encoded).unwrap();
484 assert_eq!(public_key, decoded);
485 }
486
487 #[test]
488 fn test_codec_signature() {
489 let signature = parse_signature(
490 "
491 e5564300c360ac729086e2cc806e828a
492 84877f1eb8e5d974d873e06522490155
493 5fb8821590a33bacc61e39701cf9b46b
494 d25bf5f0595bbe24655141438e7a100b
495 ",
496 );
497 let encoded = signature.encode();
498 assert_eq!(encoded.len(), SIGNATURE_LENGTH);
499 let decoded = Signature::decode(encoded).unwrap();
500 assert_eq!(signature, decoded);
501 }
502
503 #[test]
504 fn rfc8032_test_vector_1() {
505 let (private_key, public_key, message, signature) = vector_1();
506 test_sign_and_verify(private_key, public_key, &message, signature)
507 }
508
509 #[test]
511 #[should_panic]
512 fn bad_signature() {
513 let (private_key, public_key, message, _) = vector_1();
514 let private_key_2 = PrivateKey::from_rng(&mut OsRng);
515 let bad_signature = private_key_2.sign(None, message.as_ref());
516 test_sign_and_verify(private_key, public_key, &message, bad_signature);
517 }
518
519 #[test]
521 #[should_panic]
522 fn different_message() {
523 let (private_key, public_key, _, signature) = vector_1();
524 let different_message = b"this is a different message".to_vec();
525 test_sign_and_verify(private_key, public_key, &different_message, signature);
526 }
527
528 #[test]
529 fn rfc8032_test_vector_2() {
530 let (private_key, public_key, message, signature) = vector_2();
531 test_sign_and_verify(private_key, public_key, &message, signature)
532 }
533
534 #[test]
535 fn rfc8032_test_vector_3() {
536 let private_key = parse_private_key(
537 "
538 c5aa8df43f9f837bedb7442f31dcb7b1
539 66d38535076f094b85ce3a2e0b4458f7
540 ",
541 );
542 let public_key = parse_public_key(
543 "
544 fc51cd8e6218a1a38da47ed00230f058
545 0816ed13ba3303ac5deb911548908025
546 ",
547 );
548 let message: [u8; 2] = [0xaf, 0x82];
549 let signature = parse_signature(
550 "
551 6291d657deec24024827e69c3abe01a3
552 0ce548a284743a445e3680d7db5ac3ac
553 18ff9b538d16f290ae67f760984dc659
554 4a7c15e9716ed28dc027beceea1ec40a
555 ",
556 );
557 test_sign_and_verify(private_key, public_key, &message, signature)
558 }
559
560 #[test]
561 fn rfc8032_test_vector_1024() {
562 let private_key = parse_private_key(
563 "
564 f5e5767cf153319517630f226876b86c
565 8160cc583bc013744c6bf255f5cc0ee5
566 ",
567 );
568 let public_key = parse_public_key(
569 "
570 278117fc144c72340f67d0f2316e8386
571 ceffbf2b2428c9c51fef7c597f1d426e
572 ",
573 );
574 let message = commonware_utils::from_hex_formatted(
575 "
576 08b8b2b733424243760fe426a4b54908
577 632110a66c2f6591eabd3345e3e4eb98
578 fa6e264bf09efe12ee50f8f54e9f77b1
579 e355f6c50544e23fb1433ddf73be84d8
580 79de7c0046dc4996d9e773f4bc9efe57
581 38829adb26c81b37c93a1b270b20329d
582 658675fc6ea534e0810a4432826bf58c
583 941efb65d57a338bbd2e26640f89ffbc
584 1a858efcb8550ee3a5e1998bd177e93a
585 7363c344fe6b199ee5d02e82d522c4fe
586 ba15452f80288a821a579116ec6dad2b
587 3b310da903401aa62100ab5d1a36553e
588 06203b33890cc9b832f79ef80560ccb9
589 a39ce767967ed628c6ad573cb116dbef
590 efd75499da96bd68a8a97b928a8bbc10
591 3b6621fcde2beca1231d206be6cd9ec7
592 aff6f6c94fcd7204ed3455c68c83f4a4
593 1da4af2b74ef5c53f1d8ac70bdcb7ed1
594 85ce81bd84359d44254d95629e9855a9
595 4a7c1958d1f8ada5d0532ed8a5aa3fb2
596 d17ba70eb6248e594e1a2297acbbb39d
597 502f1a8c6eb6f1ce22b3de1a1f40cc24
598 554119a831a9aad6079cad88425de6bd
599 e1a9187ebb6092cf67bf2b13fd65f270
600 88d78b7e883c8759d2c4f5c65adb7553
601 878ad575f9fad878e80a0c9ba63bcbcc
602 2732e69485bbc9c90bfbd62481d9089b
603 eccf80cfe2df16a2cf65bd92dd597b07
604 07e0917af48bbb75fed413d238f5555a
605 7a569d80c3414a8d0859dc65a46128ba
606 b27af87a71314f318c782b23ebfe808b
607 82b0ce26401d2e22f04d83d1255dc51a
608 ddd3b75a2b1ae0784504df543af8969b
609 e3ea7082ff7fc9888c144da2af58429e
610 c96031dbcad3dad9af0dcbaaaf268cb8
611 fcffead94f3c7ca495e056a9b47acdb7
612 51fb73e666c6c655ade8297297d07ad1
613 ba5e43f1bca32301651339e22904cc8c
614 42f58c30c04aafdb038dda0847dd988d
615 cda6f3bfd15c4b4c4525004aa06eeff8
616 ca61783aacec57fb3d1f92b0fe2fd1a8
617 5f6724517b65e614ad6808d6f6ee34df
618 f7310fdc82aebfd904b01e1dc54b2927
619 094b2db68d6f903b68401adebf5a7e08
620 d78ff4ef5d63653a65040cf9bfd4aca7
621 984a74d37145986780fc0b16ac451649
622 de6188a7dbdf191f64b5fc5e2ab47b57
623 f7f7276cd419c17a3ca8e1b939ae49e4
624 88acba6b965610b5480109c8b17b80e1
625 b7b750dfc7598d5d5011fd2dcc5600a3
626 2ef5b52a1ecc820e308aa342721aac09
627 43bf6686b64b2579376504ccc493d97e
628 6aed3fb0f9cd71a43dd497f01f17c0e2
629 cb3797aa2a2f256656168e6c496afc5f
630 b93246f6b1116398a346f1a641f3b041
631 e989f7914f90cc2c7fff357876e506b5
632 0d334ba77c225bc307ba537152f3f161
633 0e4eafe595f6d9d90d11faa933a15ef1
634 369546868a7f3a45a96768d40fd9d034
635 12c091c6315cf4fde7cb68606937380d
636 b2eaaa707b4c4185c32eddcdd306705e
637 4dc1ffc872eeee475a64dfac86aba41c
638 0618983f8741c5ef68d3a101e8a3b8ca
639 c60c905c15fc910840b94c00a0b9d0
640 ",
641 )
642 .unwrap();
643 let signature = parse_signature(
644 "
645 0aab4c900501b3e24d7cdf4663326a3a
646 87df5e4843b2cbdb67cbf6e460fec350
647 aa5371b1508f9f4528ecea23c436d94b
648 5e8fcd4f681e30a6ac00a9704a188a03
649 ",
650 );
651 test_sign_and_verify(private_key, public_key, &message, signature)
652 }
653
654 #[test]
655 fn rfc8032_test_vector_sha() {
656 let private_key = commonware_utils::from_hex_formatted(
657 "
658 833fe62409237b9d62ec77587520911e
659 9a759cec1d19755b7da901b96dca3d42
660 ",
661 )
662 .unwrap();
663 let public_key = commonware_utils::from_hex_formatted(
664 "
665 ec172b93ad5e563bf4932c70e1245034
666 c35467ef2efd4d64ebf819683467e2bf
667 ",
668 )
669 .unwrap();
670 let message = commonware_utils::from_hex_formatted(
671 "
672 ddaf35a193617abacc417349ae204131
673 12e6fa4e89a97ea20a9eeee64b55d39a
674 2192992a274fc1a836ba3c23a3feebbd
675 454d4423643ce80e2a9ac94fa54ca49f
676 ",
677 )
678 .unwrap();
679 let signature = commonware_utils::from_hex_formatted(
680 "
681 dc2a4459e7369633a52b1bf277839a00
682 201009a3efbf3ecb69bea2186c26b589
683 09351fc9ac90b3ecfdfbc7c66431e030
684 3dca179c138ac17ad9bef1177331a704
685 ",
686 )
687 .unwrap();
688 test_sign_and_verify(
689 PrivateKey::decode(private_key.as_ref()).unwrap(),
690 PublicKey::decode(public_key.as_ref()).unwrap(),
691 &message,
692 Signature::decode(signature.as_ref()).unwrap(),
693 )
694 }
695
696 #[test]
697 fn batch_verify_valid() {
698 let v1 = vector_1();
699 let v2 = vector_2();
700 let mut batch = ed25519::Batch::new();
701 assert!(batch.add(None, &v1.2, &v1.1, &v1.3));
702 assert!(batch.add(None, &v2.2, &v2.1, &v2.3));
703 assert!(batch.verify(&mut rand::thread_rng()));
704 }
705
706 #[test]
707 fn batch_verify_invalid() {
708 let v1 = vector_1();
709 let v2 = vector_2();
710 let mut bad_signature = v2.3.to_vec();
711 bad_signature[3] = 0xff;
712
713 let mut batch = Batch::new();
714 assert!(batch.add(None, &v1.2, &v1.1, &v1.3));
715 assert!(batch.add(
716 None,
717 &v2.2,
718 &v2.1,
719 &Signature::decode(bad_signature.as_ref()).unwrap()
720 ));
721 assert!(!batch.verify(&mut rand::thread_rng()));
722 }
723
724 #[test]
725 fn test_zero_signature_fails() {
726 let (_, public_key, message, _) = vector_1();
727 let zero_sig = Signature::decode(vec![0u8; Signature::SIZE].as_ref()).unwrap();
728 assert!(!public_key.verify(None, &message, &zero_sig));
729 }
730
731 #[test]
732 fn test_high_s_fails() {
733 let (_, public_key, message, signature) = vector_1();
734 let mut bad_signature = signature.to_vec();
735 bad_signature[63] |= 0x80; let bad_signature = Signature::decode(bad_signature.as_ref()).unwrap();
737 assert!(!public_key.verify(None, &message, &bad_signature));
738 }
739
740 #[test]
741 fn test_invalid_r_fails() {
742 let (_, public_key, message, signature) = vector_1();
743 let mut bad_signature = signature.to_vec();
744 for b in bad_signature.iter_mut().take(32) {
745 *b = 0xff; }
747 let bad_signature = Signature::decode(bad_signature.as_ref()).unwrap();
748 assert!(!public_key.verify(None, &message, &bad_signature));
749 }
750}