babyjubjub-ark/src/lib.rs
src/lib.rsRaw
1#[macro_use]
2extern crate arrayref;
3extern crate generic_array;
4extern crate mimc_rs;
5extern crate num;
6extern crate num_bigint;
7extern crate num_traits;
8extern crate rand;
9
10use blake2::{Blake2b, Digest};
11use mimc_rs::Mimc7;
12use poseidon_rs::Poseidon;
13use std::cmp::min;
14
15use num_bigint::{BigInt, RandBigInt, Sign, ToBigInt};
16use num_traits::{One, Zero};
17
18use generic_array::GenericArray;
19
20mod utils;
21
22#[macro_use]
23extern crate lazy_static;
24
25lazy_static! {
26 static ref D: BigInt = BigInt::parse_bytes(b"168696", 10).unwrap();
27 static ref A: BigInt = BigInt::parse_bytes(b"168700", 10).unwrap();
28 static ref Q: BigInt = BigInt::parse_bytes(
29 b"21888242871839275222246405745257275088548364400416034343698204186575808495617",
30 10,
31 )
32 .unwrap();
33 static ref B8: Point = Point {
34 x: BigInt::parse_bytes(
35 b"5299619240641551281634865583518297030282874472190772894086521144482721001553",
36 10,
37 )
38 .unwrap(),
39 y: BigInt::parse_bytes(
40 b"16950150798460657717958625567821834550301663161624707787222815936182638968203",
41 10,
42 )
43 .unwrap(),
44 };
45 static ref ORDER: BigInt = BigInt::parse_bytes(
46 b"21888242871839275222246405745257275088614511777268538073601725287587578984328",
47 10,
48 )
49 .unwrap();
50 static ref SUBORDER: BigInt = &BigInt::parse_bytes(
51 b"21888242871839275222246405745257275088614511777268538073601725287587578984328",
52 10,
53 )
54 .unwrap()
55 >> 3;
56}
57
58#[derive(Clone, Debug)]
59pub struct Point {
60 pub x: BigInt,
61 pub y: BigInt,
62}
63
64impl Point {
65 pub fn add(&self, q: &Point) -> Result<Point, String> {
66 // x = (x1*y2+y1*x2)/(c*(1+d*x1*x2*y1*y2))
67 // y = (y1*y2-x1*x2)/(c*(1-d*x1*x2*y1*y2))
68
69 // x = (x1 * y2 + y1 * x2) / (1 + d * x1 * y1 * y2)
70 let one: BigInt = One::one();
71 let x_num: BigInt = &self.x * &q.y + &self.y * &q.x;
72 let x_den: BigInt = &one + &D.clone() * &self.x * &q.x * &self.y * &q.y;
73 let x_den_inv = utils::modinv(&x_den, &Q)?;
74 let x: BigInt = utils::modulus(&(&x_num * &x_den_inv), &Q);
75
76 // y = (y1 * y2 - a * x1 * x2) / (1 - d * x1 * x2 * y1 * y2)
77 let y_num = &self.y * &q.y - &A.clone() * &self.x * &q.x;
78 let y_den = utils::modulus(&(&one - &D.clone() * &self.x * &q.x * &self.y * &q.y), &Q);
79 let y_den_inv = utils::modinv(&y_den, &Q)?;
80 let y: BigInt = utils::modulus(&(&y_num * &y_den_inv), &Q);
81
82 Ok(Point { x: x, y: y })
83 }
84
85 pub fn mul_scalar(&self, n: &BigInt) -> Result<Point, String> {
86 let mut r: Point = Point {
87 x: Zero::zero(),
88 y: One::one(),
89 };
90 let mut rem: BigInt = n.clone();
91 let mut exp: Point = self.clone();
92
93 let zero: BigInt = Zero::zero();
94 let one: BigInt = One::one();
95 while rem != zero {
96 let is_odd = &rem & &one == one;
97 if is_odd == true {
98 r = r.add(&exp)?;
99 }
100 exp = exp.add(&exp)?;
101 rem = rem >> 1;
102 }
103 r.x = utils::modulus(&r.x, &Q);
104 r.y = utils::modulus(&r.y, &Q);
105 Ok(r)
106 }
107
108 pub fn compress(&self) -> [u8; 32] {
109 let mut r: [u8; 32] = [0; 32];
110 let (_, y_bytes) = self.y.to_bytes_le();
111 let len = min(y_bytes.len(), r.len());
112 r[..len].copy_from_slice(&y_bytes[..len]);
113 if &self.x > &(&Q.clone() >> 1) {
114 r[31] = r[31] | 0x80;
115 }
116 r
117 }
118
119 pub fn equals(&self, p: Point) -> bool {
120 if self.x == p.x && self.y == p.y {
121 return true;
122 }
123 false
124 }
125}
126
127pub fn decompress_point(bb: [u8; 32]) -> Result<Point, String> {
128 // https://tools.ietf.org/html/rfc8032#section-5.2.3
129 let mut sign: bool = false;
130 let mut b = bb.clone();
131 if b[31] & 0x80 != 0x00 {
132 sign = true;
133 b[31] = b[31] & 0x7F;
134 }
135 let y: BigInt = BigInt::from_bytes_le(Sign::Plus, &b[..]);
136 if y >= Q.clone() {
137 return Err("y outside the Finite Field over R".to_string());
138 }
139 let one: BigInt = One::one();
140
141 // x^2 = (1 - y^2) / (a - d * y^2) (mod p)
142 let den = utils::modinv(
143 &utils::modulus(
144 &(&A.clone() - utils::modulus(&(&D.clone() * (&y * &y)), &Q)),
145 &Q,
146 ),
147 &Q,
148 )?;
149 let mut x: BigInt = utils::modulus(&((one - utils::modulus(&(&y * &y), &Q)) * den), &Q);
150 x = utils::modsqrt(&x, &Q)?;
151
152 if sign && !(&x > &(&Q.clone() >> 1)) || (!sign && (&x > &(&Q.clone() >> 1))) {
153 x = x * -1.to_bigint().unwrap();
154 }
155 x = utils::modulus(&x, &Q);
156 Ok(Point { x: x, y: y })
157}
158
159pub struct Signature {
160 r_b8: Point,
161 s: BigInt,
162}
163
164impl Signature {
165 pub fn compress(&self) -> [u8; 64] {
166 let mut b: Vec<u8> = Vec::new();
167 b.append(&mut self.r_b8.compress().to_vec());
168 let (_, s_bytes) = self.s.to_bytes_le();
169 let mut s_32bytes: [u8; 32] = [0; 32];
170 let len = min(s_bytes.len(), s_32bytes.len());
171 s_32bytes[..len].copy_from_slice(&s_bytes[..len]);
172 b.append(&mut s_32bytes.to_vec());
173 let mut r: [u8; 64] = [0; 64];
174 r[..].copy_from_slice(&b[..]);
175 r
176 }
177}
178
179pub fn decompress_signature(b: &[u8; 64]) -> Result<Signature, String> {
180 let r_b8_bytes: [u8; 32] = *array_ref!(b[..32], 0, 32);
181 let s: BigInt = BigInt::from_bytes_le(Sign::Plus, &b[32..]);
182 let r_b8 = decompress_point(r_b8_bytes);
183 match r_b8 {
184 Result::Err(err) => return Err(err.to_string()),
185 Result::Ok(res) => Ok(Signature {
186 r_b8: res.clone(),
187 s: s,
188 }),
189 }
190}
191
192pub struct PrivateKey {
193 key: BigInt,
194}
195
196impl PrivateKey {
197 pub fn public(&self) -> Result<Point, String> {
198 // https://tools.ietf.org/html/rfc8032#section-5.1.5
199 let pk = B8.mul_scalar(&self.key)?;
200 Ok(pk.clone())
201 }
202
203 pub fn sign_mimc(&self, msg: BigInt) -> Result<Signature, String> {
204 // https://tools.ietf.org/html/rfc8032#section-5.1.6
205 let mut hasher = Blake2b::new();
206 let (_, sk_bytes) = self.key.to_bytes_be();
207 hasher.input(sk_bytes);
208 let mut h = hasher.result(); // h: hash(sk)
209 // s: h[32:64]
210 let s = GenericArray::<u8, generic_array::typenum::U32>::from_mut_slice(&mut h[32..64]);
211 let (_, msg_bytes) = msg.to_bytes_be();
212 let r_bytes = utils::concatenate_arrays(s, &msg_bytes);
213 let mut r = BigInt::from_bytes_be(Sign::Plus, &r_bytes[..]);
214 r = utils::modulus(&r, &SUBORDER);
215 let r8: Point = B8.mul_scalar(&r)?;
216 let a = &self.public()?;
217
218 let hm_input = vec![r8.x.clone(), r8.y.clone(), a.x.clone(), a.y.clone(), msg];
219 let mimc7 = Mimc7::new();
220 let hm = mimc7.hash(hm_input)?;
221
222 let mut s = &self.key << 3;
223 s = hm * s;
224 s = r + s;
225 s = s % &SUBORDER.clone();
226
227 Ok(Signature {
228 r_b8: r8.clone(),
229 s: s,
230 })
231 }
232 pub fn sign_poseidon(&self, msg: BigInt) -> Result<Signature, String> {
233 // https://tools.ietf.org/html/rfc8032#section-5.1.6
234 let mut hasher = Blake2b::new();
235 let (_, sk_bytes) = self.key.to_bytes_be();
236 hasher.input(sk_bytes);
237 let mut h = hasher.result(); // h: hash(sk)
238 // s: h[32:64]
239 let s = GenericArray::<u8, generic_array::typenum::U32>::from_mut_slice(&mut h[32..64]);
240 let (_, msg_bytes) = msg.to_bytes_be();
241 let r_bytes = utils::concatenate_arrays(s, &msg_bytes);
242 let mut r = BigInt::from_bytes_be(Sign::Plus, &r_bytes[..]);
243 r = utils::modulus(&r, &SUBORDER);
244 let r8: Point = B8.mul_scalar(&r)?;
245 let a = &self.public()?;
246
247 let hm_input = vec![r8.x.clone(), r8.y.clone(), a.x.clone(), a.y.clone(), msg];
248 let poseidon = Poseidon::new();
249 let hm = poseidon.hash(hm_input)?;
250
251 let mut s = &self.key << 3;
252 s = hm * s;
253 s = r + s;
254 s = s % &SUBORDER.clone();
255
256 Ok(Signature {
257 r_b8: r8.clone(),
258 s: s,
259 })
260 }
261
262 pub fn sign_schnorr(&self, m: Vec<u8>) -> Result<(Point, BigInt), String> {
263 // random r
264 let mut rng = rand::thread_rng();
265 let k = rng.gen_biguint(1024).to_bigint().unwrap();
266
267 // r = k·G
268 let r = B8.mul_scalar(&k)?;
269
270 // h = H(x, r, m)
271 let pk = &self.public()?;
272 let h = schnorr_hash(&pk, m, &r)?;
273
274 // s= k+x·h
275 let s = k + &self.key * &h;
276 Ok((r, s))
277 }
278}
279
280pub fn schnorr_hash(pk: &Point, m: Vec<u8>, c: &Point) -> Result<BigInt, String> {
281 let b: &mut Vec<u8> = &mut Vec::new();
282
283 // other option could be to do it without compressing the points, and concatenating x|y
284 b.append(&mut pk.compress().to_vec());
285 b.append(&mut c.compress().to_vec());
286 b.append(&mut m.clone());
287
288 let poseidon = Poseidon::new();
289 let h = poseidon.hash_bytes(b.to_vec())?;
290 println!("h {:?}", h.to_string());
291 Ok(h)
292}
293
294pub fn verify_schnorr(pk: Point, m: Vec<u8>, r: Point, s: BigInt) -> Result<bool, String> {
295 // sG = s·G
296 let sg = B8.mul_scalar(&s)?;
297
298 // r + h · x
299 let h = schnorr_hash(&pk, m, &r)?;
300 let pk_h = pk.mul_scalar(&h)?;
301 let right = r.add(&pk_h)?;
302
303 Ok(sg.equals(right))
304}
305
306pub fn new_key() -> PrivateKey {
307 // https://tools.ietf.org/html/rfc8032#section-5.1.5
308 let mut rng = rand::thread_rng();
309 let sk_raw = rng.gen_biguint(1024).to_bigint().unwrap();
310
311 let mut hasher = Blake2b::new();
312 let (_, sk_raw_bytes) = sk_raw.to_bytes_be();
313 hasher.input(sk_raw_bytes);
314 let mut h = hasher.result();
315
316 h[0] = h[0] & 0xF8;
317 h[31] = h[31] & 0x7F;
318 h[31] = h[31] | 0x40;
319
320 let sk = BigInt::from_bytes_le(Sign::Plus, &h[..]);
321
322 PrivateKey { key: sk }
323}
324
325pub fn verify_mimc(pk: Point, sig: Signature, msg: BigInt) -> bool {
326 let hm_input = vec![
327 sig.r_b8.x.clone(),
328 sig.r_b8.y.clone(),
329 pk.x.clone(),
330 pk.y.clone(),
331 msg,
332 ];
333 let mimc7 = Mimc7::new();
334 let hm = match mimc7.hash(hm_input) {
335 Result::Err(_) => return false,
336 Result::Ok(hm) => hm,
337 };
338 let l = match B8.mul_scalar(&sig.s) {
339 Result::Err(_) => return false,
340 Result::Ok(l) => l,
341 };
342 let r = match sig
343 .r_b8
344 .add(&pk.mul_scalar(&(8.to_bigint().unwrap() * hm)).unwrap())
345 {
346 Result::Err(_) => return false,
347 Result::Ok(r) => r,
348 };
349 l.equals(r)
350}
351pub fn verify_poseidon(pk: Point, sig: Signature, msg: BigInt) -> bool {
352 let hm_input = vec![
353 sig.r_b8.x.clone(),
354 sig.r_b8.y.clone(),
355 pk.x.clone(),
356 pk.y.clone(),
357 msg,
358 ];
359 let poseidon = Poseidon::new();
360 let hm = match poseidon.hash(hm_input) {
361 Result::Err(_) => return false,
362 Result::Ok(hm) => hm,
363 };
364 let l = match B8.mul_scalar(&sig.s) {
365 Result::Err(_) => return false,
366 Result::Ok(l) => l,
367 };
368 let r = match sig
369 .r_b8
370 .add(&pk.mul_scalar(&(8.to_bigint().unwrap() * hm)).unwrap())
371 {
372 Result::Err(_) => return false,
373 Result::Ok(r) => r,
374 };
375 l.equals(r)
376}
377
378#[cfg(test)]
379mod tests {
380 use super::*;
381 extern crate rustc_hex;
382 use rustc_hex::{FromHex, ToHex};
383
384 #[test]
385 fn test_add_same_point() {
386 let p: Point = Point {
387 x: BigInt::parse_bytes(
388 b"17777552123799933955779906779655732241715742912184938656739573121738514868268",
389 10,
390 )
391 .unwrap(),
392 y: BigInt::parse_bytes(
393 b"2626589144620713026669568689430873010625803728049924121243784502389097019475",
394 10,
395 )
396 .unwrap(),
397 };
398 let q: Point = Point {
399 x: BigInt::parse_bytes(
400 b"17777552123799933955779906779655732241715742912184938656739573121738514868268",
401 10,
402 )
403 .unwrap(),
404 y: BigInt::parse_bytes(
405 b"2626589144620713026669568689430873010625803728049924121243784502389097019475",
406 10,
407 )
408 .unwrap(),
409 };
410 let res = p.add(&q).unwrap();
411 assert_eq!(
412 res.x.to_string(),
413 "6890855772600357754907169075114257697580319025794532037257385534741338397365"
414 );
415 assert_eq!(
416 res.y.to_string(),
417 "4338620300185947561074059802482547481416142213883829469920100239455078257889"
418 );
419 }
420 #[test]
421 fn test_add_different_points() {
422 let p: Point = Point {
423 x: BigInt::parse_bytes(
424 b"17777552123799933955779906779655732241715742912184938656739573121738514868268",
425 10,
426 )
427 .unwrap(),
428 y: BigInt::parse_bytes(
429 b"2626589144620713026669568689430873010625803728049924121243784502389097019475",
430 10,
431 )
432 .unwrap(),
433 };
434 let q: Point = Point {
435 x: BigInt::parse_bytes(
436 b"16540640123574156134436876038791482806971768689494387082833631921987005038935",
437 10,
438 )
439 .unwrap(),
440 y: BigInt::parse_bytes(
441 b"20819045374670962167435360035096875258406992893633759881276124905556507972311",
442 10,
443 )
444 .unwrap(),
445 };
446 let res = p.add(&q).unwrap();
447 assert_eq!(
448 res.x.to_string(),
449 "7916061937171219682591368294088513039687205273691143098332585753343424131937"
450 );
451 assert_eq!(
452 res.y.to_string(),
453 "14035240266687799601661095864649209771790948434046947201833777492504781204499"
454 );
455 }
456
457 #[test]
458 fn test_mul_scalar() {
459 let p: Point = Point {
460 x: BigInt::parse_bytes(
461 b"17777552123799933955779906779655732241715742912184938656739573121738514868268",
462 10,
463 )
464 .unwrap(),
465 y: BigInt::parse_bytes(
466 b"2626589144620713026669568689430873010625803728049924121243784502389097019475",
467 10,
468 )
469 .unwrap(),
470 };
471 let res_m = p.mul_scalar(&3.to_bigint().unwrap()).unwrap();
472 let res_a = p.add(&p).unwrap();
473 let res_a = res_a.add(&p).unwrap();
474 assert_eq!(res_m.x, res_a.x);
475 assert_eq!(
476 res_m.x.to_string(),
477 "19372461775513343691590086534037741906533799473648040012278229434133483800898"
478 );
479 assert_eq!(
480 res_m.y.to_string(),
481 "9458658722007214007257525444427903161243386465067105737478306991484593958249"
482 );
483
484 let n = BigInt::parse_bytes(
485 b"14035240266687799601661095864649209771790948434046947201833777492504781204499",
486 10,
487 )
488 .unwrap();
489 let res2 = p.mul_scalar(&n).unwrap();
490 assert_eq!(
491 res2.x.to_string(),
492 "17070357974431721403481313912716834497662307308519659060910483826664480189605"
493 );
494 assert_eq!(
495 res2.y.to_string(),
496 "4014745322800118607127020275658861516666525056516280575712425373174125159339"
497 );
498 }
499
500 #[test]
501 fn test_new_key_sign_verify_mimc_0() {
502 let sk = new_key();
503 let pk = sk.public().unwrap();
504 let msg = 5.to_bigint().unwrap();
505 let sig = sk.sign_mimc(msg.clone()).unwrap();
506 let v = verify_mimc(pk, sig, msg);
507 assert_eq!(v, true);
508 }
509
510 #[test]
511 fn test_new_key_sign_verify_mimc_1() {
512 let sk = new_key();
513 let pk = sk.public().unwrap();
514 let msg = BigInt::parse_bytes(b"123456789012345678901234567890", 10).unwrap();
515 let sig = sk.sign_mimc(msg.clone()).unwrap();
516 let v = verify_mimc(pk, sig, msg);
517 assert_eq!(v, true);
518 }
519 #[test]
520 fn test_new_key_sign_verify_poseidon_0() {
521 let sk = new_key();
522 let pk = sk.public().unwrap();
523 let msg = 5.to_bigint().unwrap();
524 let sig = sk.sign_poseidon(msg.clone()).unwrap();
525 let v = verify_poseidon(pk, sig, msg);
526 assert_eq!(v, true);
527 }
528
529 #[test]
530 fn test_new_key_sign_verify_poseidon_1() {
531 let sk = new_key();
532 let pk = sk.public().unwrap();
533 let msg = BigInt::parse_bytes(b"123456789012345678901234567890", 10).unwrap();
534 let sig = sk.sign_poseidon(msg.clone()).unwrap();
535 let v = verify_poseidon(pk, sig, msg);
536 assert_eq!(v, true);
537 }
538
539 #[test]
540 fn test_point_compress_decompress() {
541 let p: Point = Point {
542 x: BigInt::parse_bytes(
543 b"17777552123799933955779906779655732241715742912184938656739573121738514868268",
544 10,
545 )
546 .unwrap(),
547 y: BigInt::parse_bytes(
548 b"2626589144620713026669568689430873010625803728049924121243784502389097019475",
549 10,
550 )
551 .unwrap(),
552 };
553 let p_comp = p.compress();
554 assert_eq!(
555 p_comp[..].to_hex(),
556 "53b81ed5bffe9545b54016234682e7b2f699bd42a5e9eae27ff4051bc698ce85"
557 );
558 let p2 = decompress_point(p_comp).unwrap();
559 assert_eq!(p.x, p2.x);
560 assert_eq!(p.y, p2.y);
561 }
562
563 #[test]
564 fn test_point_decompress0() {
565 let y_bytes_raw = "b5328f8791d48f20bec6e481d91c7ada235f1facf22547901c18656b6c3e042f"
566 .from_hex()
567 .unwrap();
568 let mut y_bytes: [u8; 32] = [0; 32];
569 y_bytes.copy_from_slice(&y_bytes_raw);
570 let p = decompress_point(y_bytes).unwrap();
571
572 let expected_px_raw = "b86cc8d9c97daef0afe1a4753c54fb2d8a530dc74c7eee4e72b3fdf2496d2113"
573 .from_hex()
574 .unwrap();
575 let mut e_px_bytes: [u8; 32] = [0; 32];
576 e_px_bytes.copy_from_slice(&expected_px_raw);
577 let expected_px: BigInt = BigInt::from_bytes_le(Sign::Plus, &e_px_bytes);
578 assert_eq!(&p.x, &expected_px);
579 }
580
581 #[test]
582 fn test_point_decompress1() {
583 let y_bytes_raw = "70552d3ff548e09266ded29b33ce75139672b062b02aa66bb0d9247ffecf1d0b"
584 .from_hex()
585 .unwrap();
586 let mut y_bytes: [u8; 32] = [0; 32];
587 y_bytes.copy_from_slice(&y_bytes_raw);
588 let p = decompress_point(y_bytes).unwrap();
589
590 let expected_px_raw = "30f1635ba7d56f9cb32c3ffbe6dca508a68c7f43936af11a23c785ce98cb3404"
591 .from_hex()
592 .unwrap();
593 let mut e_px_bytes: [u8; 32] = [0; 32];
594 e_px_bytes.copy_from_slice(&expected_px_raw);
595 let expected_px: BigInt = BigInt::from_bytes_le(Sign::Plus, &e_px_bytes);
596 assert_eq!(&p.x, &expected_px);
597 }
598
599 #[test]
600 fn test_point_decompress_loop() {
601 for _ in 0..5 {
602 let mut rng = rand::thread_rng();
603 let sk_raw = rng.gen_biguint(1024).to_bigint().unwrap();
604 let mut hasher = Blake2b::new();
605 let (_, sk_raw_bytes) = sk_raw.to_bytes_be();
606 hasher.input(sk_raw_bytes);
607 let mut h = hasher.result();
608
609 h[0] = h[0] & 0xF8;
610 h[31] = h[31] & 0x7F;
611 h[31] = h[31] | 0x40;
612
613 let sk = BigInt::from_bytes_le(Sign::Plus, &h[..]);
614 let point = B8.mul_scalar(&sk).unwrap();
615 let cmp_point = point.compress();
616 let dcmp_point = decompress_point(cmp_point).unwrap();
617
618 assert_eq!(&point.x, &dcmp_point.x);
619 assert_eq!(&point.y, &dcmp_point.y);
620 }
621 }
622
623 #[test]
624 fn test_signature_compress_decompress() {
625 let sk = new_key();
626 let pk = sk.public().unwrap();
627
628 for i in 0..5 {
629 let msg_raw = "123456".to_owned() + &i.to_string();
630 let msg = BigInt::parse_bytes(msg_raw.as_bytes(), 10).unwrap();
631 let sig = sk.sign_mimc(msg.clone()).unwrap();
632
633 let compressed_sig = sig.compress();
634 let decompressed_sig = decompress_signature(&compressed_sig).unwrap();
635 assert_eq!(&sig.r_b8.x, &decompressed_sig.r_b8.x);
636 assert_eq!(&sig.r_b8.y, &decompressed_sig.r_b8.y);
637 assert_eq!(&sig.s, &decompressed_sig.s);
638
639 let v = verify_mimc(pk.clone(), decompressed_sig, msg);
640 assert_eq!(v, true);
641 }
642 }
643
644 #[test]
645 fn test_schnorr_signature() {
646 let sk = new_key();
647 let pk = sk.public().unwrap();
648
649 let msg: Vec<u8> = ("123456".to_owned() + &1.to_string()).as_bytes().to_vec();
650 let (s, e) = sk.sign_schnorr(msg.clone()).unwrap();
651 println!("s {:?}", s.x.to_string());
652 println!("s {:?}", s.y.to_string());
653 println!("e {:?}", e.to_string());
654 let verification = verify_schnorr(pk, msg, s, e).unwrap();
655 assert_eq!(true, verification);
656 }
657}