1 //! Simple variable-time big integer implementation
4 use core::marker::PhantomData;
6 // **************************************
7 // * Implementations of math primitives *
8 // **************************************
10 macro_rules! debug_unwrap { ($v: expr) => { {
12 debug_assert!(v.is_ok());
15 Err(e) => return Err(e),
19 // Various const versions of existing slice utilities
20 /// Const version of `&a[start..end]`
21 const fn const_subslice<'a, T>(a: &'a [T], start: usize, end: usize) -> &'a [T] {
22 assert!(start <= a.len());
23 assert!(end <= a.len());
24 assert!(end >= start);
25 let mut startptr = a.as_ptr();
26 startptr = unsafe { startptr.add(start) };
27 let len = end - start;
28 // The docs for from_raw_parts do not mention any requirements that the pointer be valid if the
29 // length is zero, aside from requiring proper alignment (which is met here). Thus,
30 // one-past-the-end should be an acceptable pointer for a 0-length slice.
31 unsafe { alloc::slice::from_raw_parts(startptr, len) }
34 /// Const version of `dest[dest_start..dest_end].copy_from_slice(source)`
36 /// Once `const_mut_refs` is stable we can convert this to a function
37 macro_rules! copy_from_slice {
38 ($dest: ident, $dest_start: expr, $dest_end: expr, $source: ident) => { {
39 let dest_start = $dest_start;
40 let dest_end = $dest_end;
41 assert!(dest_start <= $dest.len());
42 assert!(dest_end <= $dest.len());
43 assert!(dest_end >= dest_start);
44 assert!(dest_end - dest_start == $source.len());
46 while i < $source.len() {
47 $dest[i + dest_start] = $source[i];
53 /// Const version of a > b
54 const fn slice_greater_than(a: &[u64], b: &[u64]) -> bool {
55 debug_assert!(a.len() == b.len());
56 let len = if a.len() <= b.len() { a.len() } else { b.len() };
59 if a[i] > b[i] { return true; }
60 else if a[i] < b[i] { return false; }
66 /// Const version of a == b
67 const fn slice_equal(a: &[u64], b: &[u64]) -> bool {
68 debug_assert!(a.len() == b.len());
69 let len = if a.len() <= b.len() { a.len() } else { b.len() };
72 if a[i] != b[i] { return false; }
78 /// Adds a single u64 valuein-place, returning an overflow flag, in which case one out-of-bounds
79 /// high bit is implicitly included in the result.
81 /// Once `const_mut_refs` is stable we can convert this to a function
82 macro_rules! add_u64 { ($a: ident, $b: expr) => { {
87 let (v, carry) = $a[i].overflowing_add(add);
90 if add == 0 { break; }
98 /// Negates the given u64 slice.
100 /// Once `const_mut_refs` is stable we can convert this to a function
101 macro_rules! negate { ($v: ident) => { {
104 $v[i] ^= 0xffff_ffff_ffff_ffff;
107 let _ = add_u64!($v, 1);
110 /// Doubles in-place, returning an overflow flag, in which case one out-of-bounds high bit is
111 /// implicitly included in the result.
113 /// Once `const_mut_refs` is stable we can convert this to a function
114 macro_rules! double { ($a: ident) => { {
115 { let _: &[u64] = &$a; } // Force type resolution
117 let mut carry = false;
120 let next_carry = ($a[i] & (1 << 63)) != 0;
121 let (v, _next_carry_2) = ($a[i] << 1).overflowing_add(carry as u64);
123 debug_assert!(!_next_carry_2, "Adding one to 0x7ffff..*2 is only 0xffff..");
125 // Note that we can ignore _next_carry_2 here as we never need it - it cannot be set if
126 // next_carry is not set and at max 0xffff..*2 + 1 is only 0x1ffff.. (i.e. we can not need
137 macro_rules! define_add { ($name: ident, $len: expr) => {
138 /// Adds two $len-64-bit integers together, returning a new $len-64-bit integer and an overflow
139 /// bit, with the same semantics as the std [`u64::overflowing_add`] method.
140 const fn $name(a: &[u64], b: &[u64]) -> ([u64; $len], bool) {
141 debug_assert!(a.len() == $len);
142 debug_assert!(b.len() == $len);
143 let mut r = [0; $len];
144 let mut carry = false;
145 let mut i = $len - 1;
147 let (v, mut new_carry) = a[i].overflowing_add(b[i]);
148 let (v2, new_new_carry) = v.overflowing_add(carry as u64);
149 new_carry |= new_new_carry;
160 define_add!(add_2, 2);
161 define_add!(add_3, 3);
162 define_add!(add_4, 4);
163 define_add!(add_6, 6);
164 define_add!(add_8, 8);
165 define_add!(add_12, 12);
166 define_add!(add_16, 16);
167 define_add!(add_32, 32);
168 define_add!(add_64, 64);
169 define_add!(add_128, 128);
171 macro_rules! define_sub { ($name: ident, $name_abs: ident, $len: expr) => {
172 /// Subtracts the `b` $len-64-bit integer from the `a` $len-64-bit integer, returning a new
173 /// $len-64-bit integer and an overflow bit, with the same semantics as the std
174 /// [`u64::overflowing_sub`] method.
175 const fn $name(a: &[u64], b: &[u64]) -> ([u64; $len], bool) {
176 debug_assert!(a.len() == $len);
177 debug_assert!(b.len() == $len);
178 let mut r = [0; $len];
179 let mut carry = false;
180 let mut i = $len - 1;
182 let (v, mut new_carry) = a[i].overflowing_sub(b[i]);
183 let (v2, new_new_carry) = v.overflowing_sub(carry as u64);
184 new_carry |= new_new_carry;
194 /// Subtracts the `b` $len-64-bit integer from the `a` $len-64-bit integer, returning a new
195 /// $len-64-bit integer representing the absolute value of the result, as well as a sign bit.
197 const fn $name_abs(a: &[u64], b: &[u64]) -> ([u64; $len], bool) {
198 let (mut res, neg) = $name(a, b);
206 define_sub!(sub_2, sub_abs_2, 2);
207 define_sub!(sub_3, sub_abs_3, 3);
208 define_sub!(sub_4, sub_abs_4, 4);
209 define_sub!(sub_6, sub_abs_6, 6);
210 define_sub!(sub_8, sub_abs_8, 8);
211 define_sub!(sub_12, sub_abs_12, 12);
212 define_sub!(sub_16, sub_abs_16, 16);
213 define_sub!(sub_32, sub_abs_32, 32);
214 define_sub!(sub_64, sub_abs_64, 64);
215 define_sub!(sub_128, sub_abs_128, 128);
217 /// Multiplies two 128-bit integers together, returning a new 256-bit integer.
219 /// This is the base case for our multiplication, taking advantage of Rust's native 128-bit int
220 /// types to do multiplication (potentially) natively.
221 const fn mul_2(a: &[u64], b: &[u64]) -> [u64; 4] {
222 debug_assert!(a.len() == 2);
223 debug_assert!(b.len() == 2);
225 // Gradeschool multiplication is way faster here.
226 let (a0, a1) = (a[0] as u128, a[1] as u128);
227 let (b0, b1) = (b[0] as u128, b[1] as u128);
231 let (z1, i_carry_a) = z1i.overflowing_add(z1j);
234 add_mul_2_parts(z2, z1, z0, i_carry_a)
237 /// Adds the gradeschool multiplication intermediate parts to a final 256-bit result
238 const fn add_mul_2_parts(z2: u128, z1: u128, z0: u128, i_carry_a: bool) -> [u64; 4] {
239 let z2a = ((z2 >> 64) & 0xffff_ffff_ffff_ffff) as u64;
240 let z1a = ((z1 >> 64) & 0xffff_ffff_ffff_ffff) as u64;
241 let z0a = ((z0 >> 64) & 0xffff_ffff_ffff_ffff) as u64;
242 let z2b = (z2 & 0xffff_ffff_ffff_ffff) as u64;
243 let z1b = (z1 & 0xffff_ffff_ffff_ffff) as u64;
244 let z0b = (z0 & 0xffff_ffff_ffff_ffff) as u64;
248 let (k, j_carry) = z0a.overflowing_add(z1b);
250 let (mut j, i_carry_b) = z1a.overflowing_add(z2b);
252 (j, i_carry_c) = j.overflowing_add(j_carry as u64);
254 let i_carry = i_carry_a as u64 + i_carry_b as u64 + i_carry_c as u64;
255 let (i, must_not_overflow) = z2a.overflowing_add(i_carry);
256 debug_assert!(!must_not_overflow, "Two 2*64 bit numbers, multiplied, will not use more than 4*64 bits");
261 const fn mul_3(a: &[u64], b: &[u64]) -> [u64; 6] {
262 debug_assert!(a.len() == 3);
263 debug_assert!(b.len() == 3);
265 let (a0, a1, a2) = (a[0] as u128, a[1] as u128, a[2] as u128);
266 let (b0, b1, b2) = (b[0] as u128, b[1] as u128, b[2] as u128);
278 let r5 = ((m4 >> 0) & 0xffff_ffff_ffff_ffff) as u64;
280 let r4a = ((m4 >> 64) & 0xffff_ffff_ffff_ffff) as u64;
281 let r4b = ((m3a >> 0) & 0xffff_ffff_ffff_ffff) as u64;
282 let r4c = ((m3b >> 0) & 0xffff_ffff_ffff_ffff) as u64;
284 let r3a = ((m3a >> 64) & 0xffff_ffff_ffff_ffff) as u64;
285 let r3b = ((m3b >> 64) & 0xffff_ffff_ffff_ffff) as u64;
286 let r3c = ((m2a >> 0 ) & 0xffff_ffff_ffff_ffff) as u64;
287 let r3d = ((m2b >> 0 ) & 0xffff_ffff_ffff_ffff) as u64;
288 let r3e = ((m2c >> 0 ) & 0xffff_ffff_ffff_ffff) as u64;
290 let r2a = ((m2a >> 64) & 0xffff_ffff_ffff_ffff) as u64;
291 let r2b = ((m2b >> 64) & 0xffff_ffff_ffff_ffff) as u64;
292 let r2c = ((m2c >> 64) & 0xffff_ffff_ffff_ffff) as u64;
293 let r2d = ((m1a >> 0 ) & 0xffff_ffff_ffff_ffff) as u64;
294 let r2e = ((m1b >> 0 ) & 0xffff_ffff_ffff_ffff) as u64;
296 let r1a = ((m1a >> 64) & 0xffff_ffff_ffff_ffff) as u64;
297 let r1b = ((m1b >> 64) & 0xffff_ffff_ffff_ffff) as u64;
298 let r1c = ((m0 >> 0 ) & 0xffff_ffff_ffff_ffff) as u64;
300 let r0a = ((m0 >> 64) & 0xffff_ffff_ffff_ffff) as u64;
302 let (r4, r3_ca) = r4a.overflowing_add(r4b);
303 let (r4, r3_cb) = r4.overflowing_add(r4c);
304 let r3_c = r3_ca as u64 + r3_cb as u64;
306 let (r3, r2_ca) = r3a.overflowing_add(r3b);
307 let (r3, r2_cb) = r3.overflowing_add(r3c);
308 let (r3, r2_cc) = r3.overflowing_add(r3d);
309 let (r3, r2_cd) = r3.overflowing_add(r3e);
310 let (r3, r2_ce) = r3.overflowing_add(r3_c);
311 let r2_c = r2_ca as u64 + r2_cb as u64 + r2_cc as u64 + r2_cd as u64 + r2_ce as u64;
313 let (r2, r1_ca) = r2a.overflowing_add(r2b);
314 let (r2, r1_cb) = r2.overflowing_add(r2c);
315 let (r2, r1_cc) = r2.overflowing_add(r2d);
316 let (r2, r1_cd) = r2.overflowing_add(r2e);
317 let (r2, r1_ce) = r2.overflowing_add(r2_c);
318 let r1_c = r1_ca as u64 + r1_cb as u64 + r1_cc as u64 + r1_cd as u64 + r1_ce as u64;
320 let (r1, r0_ca) = r1a.overflowing_add(r1b);
321 let (r1, r0_cb) = r1.overflowing_add(r1c);
322 let (r1, r0_cc) = r1.overflowing_add(r1_c);
323 let r0_c = r0_ca as u64 + r0_cb as u64 + r0_cc as u64;
325 let (r0, must_not_overflow) = r0a.overflowing_add(r0_c);
326 debug_assert!(!must_not_overflow, "Two 3*64 bit numbers, multiplied, will not use more than 6*64 bits");
328 [r0, r1, r2, r3, r4, r5]
331 macro_rules! define_mul { ($name: ident, $len: expr, $submul: ident, $add: ident, $subadd: ident, $sub: ident, $subsub: ident) => {
332 /// Multiplies two $len-64-bit integers together, returning a new $len*2-64-bit integer.
333 const fn $name(a: &[u64], b: &[u64]) -> [u64; $len * 2] {
334 // We could probably get a bit faster doing gradeschool multiplication for some smaller
335 // sizes, but its easier to just have one variable-length multiplication, so we do
336 // Karatsuba always here.
337 debug_assert!(a.len() == $len);
338 debug_assert!(b.len() == $len);
340 let a0 = const_subslice(a, 0, $len / 2);
341 let a1 = const_subslice(a, $len / 2, $len);
342 let b0 = const_subslice(b, 0, $len / 2);
343 let b1 = const_subslice(b, $len / 2, $len);
345 let z2 = $submul(a0, b0);
346 let z0 = $submul(a1, b1);
348 let (z1a_max, z1a_min, z1a_sign) =
349 if slice_greater_than(&a1, &a0) { (a1, a0, true) } else { (a0, a1, false) };
350 let (z1b_max, z1b_min, z1b_sign) =
351 if slice_greater_than(&b1, &b0) { (b1, b0, true) } else { (b0, b1, false) };
353 let z1a = $subsub(z1a_max, z1a_min);
354 debug_assert!(!z1a.1, "z1a_max was selected to be greater than z1a_min");
355 let z1b = $subsub(z1b_max, z1b_min);
356 debug_assert!(!z1b.1, "z1b_max was selected to be greater than z1b_min");
357 let z1m_sign = z1a_sign == z1b_sign;
359 let z1m = $submul(&z1a.0, &z1b.0);
360 let z1n = $add(&z0, &z2);
361 let mut z1_carry = z1n.1;
362 let z1 = if z1m_sign {
363 let r = $sub(&z1n.0, &z1m);
364 if r.1 { z1_carry ^= true; }
367 let r = $add(&z1n.0, &z1m);
368 if r.1 { z1_carry = true; }
372 let l = const_subslice(&z0, $len / 2, $len);
373 let (k, j_carry) = $subadd(const_subslice(&z0, 0, $len / 2), const_subslice(&z1, $len / 2, $len));
374 let (mut j, i_carry_a) = $subadd(const_subslice(&z1, 0, $len / 2), const_subslice(&z2, $len / 2, $len));
375 let mut i_carry_b = false;
377 i_carry_b = add_u64!(j, 1);
379 let mut i = [0; $len / 2];
380 let i_source = const_subslice(&z2, 0, $len / 2);
381 copy_from_slice!(i, 0, $len / 2, i_source);
382 let i_carry = i_carry_a as u64 + i_carry_b as u64 + z1_carry as u64;
384 let must_not_overflow = add_u64!(i, i_carry);
385 debug_assert!(!must_not_overflow, "Two N*64 bit numbers, multiplied, will not use more than 2*N*64 bits");
388 let mut res = [0; $len * 2];
389 copy_from_slice!(res, $len * 2 * 0 / 4, $len * 2 * 1 / 4, i);
390 copy_from_slice!(res, $len * 2 * 1 / 4, $len * 2 * 2 / 4, j);
391 copy_from_slice!(res, $len * 2 * 2 / 4, $len * 2 * 3 / 4, k);
392 copy_from_slice!(res, $len * 2 * 3 / 4, $len * 2 * 4 / 4, l);
397 define_mul!(mul_4, 4, mul_2, add_4, add_2, sub_4, sub_2);
398 define_mul!(mul_6, 6, mul_3, add_6, add_3, sub_6, sub_3);
399 define_mul!(mul_8, 8, mul_4, add_8, add_4, sub_8, sub_4);
400 define_mul!(mul_16, 16, mul_8, add_16, add_8, sub_16, sub_8);
401 define_mul!(mul_32, 32, mul_16, add_32, add_16, sub_32, sub_16);
402 define_mul!(mul_64, 64, mul_32, add_64, add_32, sub_64, sub_32);
405 /// Squares a 128-bit integer, returning a new 256-bit integer.
407 /// This is the base case for our squaring, taking advantage of Rust's native 128-bit int
408 /// types to do multiplication (potentially) natively.
409 const fn sqr_2(a: &[u64]) -> [u64; 4] {
410 debug_assert!(a.len() == 2);
412 let (a0, a1) = (a[0] as u128, a[1] as u128);
414 let mut z1 = a0 * a1;
415 let i_carry_a = z1 & (1u128 << 127) != 0;
419 add_mul_2_parts(z2, z1, z0, i_carry_a)
422 macro_rules! define_sqr { ($name: ident, $len: expr, $submul: ident, $subsqr: ident, $subadd: ident) => {
423 /// Squares a $len-64-bit integers, returning a new $len*2-64-bit integer.
424 const fn $name(a: &[u64]) -> [u64; $len * 2] {
425 // Squaring is only 3 half-length multiplies/squares in gradeschool math, so use that.
426 debug_assert!(a.len() == $len);
428 let hi = const_subslice(a, 0, $len / 2);
429 let lo = const_subslice(a, $len / 2, $len);
431 let v0 = $subsqr(lo);
432 let mut v1 = $submul(hi, lo);
433 let i_carry_a = double!(v1);
434 let v2 = $subsqr(hi);
436 let l = const_subslice(&v0, $len / 2, $len);
437 let (k, j_carry) = $subadd(const_subslice(&v0, 0, $len / 2), const_subslice(&v1, $len / 2, $len));
438 let (mut j, i_carry_b) = $subadd(const_subslice(&v1, 0, $len / 2), const_subslice(&v2, $len / 2, $len));
440 let mut i = [0; $len / 2];
441 let i_source = const_subslice(&v2, 0, $len / 2);
442 copy_from_slice!(i, 0, $len / 2, i_source);
444 let mut i_carry_c = false;
446 i_carry_c = add_u64!(j, 1);
448 let i_carry = i_carry_a as u64 + i_carry_b as u64 + i_carry_c as u64;
450 let must_not_overflow = add_u64!(i, i_carry);
451 debug_assert!(!must_not_overflow, "Two N*64 bit numbers, multiplied, will not use more than 2*N*64 bits");
454 let mut res = [0; $len * 2];
455 copy_from_slice!(res, $len * 2 * 0 / 4, $len * 2 * 1 / 4, i);
456 copy_from_slice!(res, $len * 2 * 1 / 4, $len * 2 * 2 / 4, j);
457 copy_from_slice!(res, $len * 2 * 2 / 4, $len * 2 * 3 / 4, k);
458 copy_from_slice!(res, $len * 2 * 3 / 4, $len * 2 * 4 / 4, l);
463 // TODO: Write an optimized sqr_3 (though secp384r1 is barely used)
464 const fn sqr_3(a: &[u64]) -> [u64; 6] { mul_3(a, a) }
466 define_sqr!(sqr_4, 4, mul_2, sqr_2, add_2);
467 define_sqr!(sqr_6, 6, mul_3, sqr_3, add_3);
468 define_sqr!(sqr_8, 8, mul_4, sqr_4, add_4);
469 define_sqr!(sqr_16, 16, mul_8, sqr_8, add_8);
470 define_sqr!(sqr_32, 32, mul_16, sqr_16, add_16);
471 define_sqr!(sqr_64, 64, mul_32, sqr_32, add_32);
473 macro_rules! dummy_pre_push { ($name: ident, $len: expr) => {} }
474 macro_rules! vec_pre_push { ($name: ident, $len: expr) => { $name.push([0; $len]); } }
476 macro_rules! define_div_rem { ($name: ident, $len: expr, $sub: ident, $heap_init: expr, $pre_push: ident $(, $const_opt: tt)?) => {
477 /// Divides two $len-64-bit integers, `a` by `b`, returning the quotient and remainder
479 /// Fails iff `b` is zero.
480 $($const_opt)? fn $name(a: &[u64; $len], b: &[u64; $len]) -> Result<([u64; $len], [u64; $len]), ()> {
481 if slice_equal(b, &[0; $len]) { return Err(()); }
483 // Very naively divide `a` by `b` by calculating all the powers of two times `b` up to `a`,
484 // then subtracting the powers of two in decreasing order. What's left is the remainder.
486 // This requires storing all the multiples of `b` in `pow2s`, which may be a vec or an
487 // array. `$pre_push!()` sets up the next element with zeros and then we can overwrite it.
489 let mut pow2s = $heap_init;
490 let mut pow2s_count = 0;
491 while slice_greater_than(a, &b_pow) {
492 $pre_push!(pow2s, $len);
493 pow2s[pow2s_count] = b_pow;
495 let double_overflow = double!(b_pow);
496 if double_overflow { break; }
498 let mut quot = [0; $len];
500 let mut pow2 = pow2s_count as isize - 1;
502 let b_pow = pow2s[pow2 as usize];
503 let overflow = double!(quot);
504 debug_assert!(!overflow, "quotient should be expressible in $len*64 bits");
505 if slice_greater_than(&rem, &b_pow) {
506 let (r, underflow) = $sub(&rem, &b_pow);
507 debug_assert!(!underflow, "rem was just checked to be > b_pow, so sub cannot underflow");
513 if slice_equal(&rem, b) {
514 let overflow = add_u64!(quot, 1);
515 debug_assert!(!overflow, "quotient should be expressible in $len*64 bits");
516 Ok((quot, [0; $len]))
524 define_div_rem!(div_rem_2, 2, sub_2, [[0; 2]; 2 * 64], dummy_pre_push, const);
525 define_div_rem!(div_rem_4, 4, sub_4, [[0; 4]; 4 * 64], dummy_pre_push, const); // Uses 8 KiB of stack
526 define_div_rem!(div_rem_6, 6, sub_6, [[0; 6]; 6 * 64], dummy_pre_push, const); // Uses 18 KiB of stack!
527 #[cfg(debug_assertions)]
528 define_div_rem!(div_rem_8, 8, sub_8, [[0; 8]; 8 * 64], dummy_pre_push, const); // Uses 32 KiB of stack!
529 #[cfg(debug_assertions)]
530 define_div_rem!(div_rem_12, 12, sub_12, [[0; 12]; 12 * 64], dummy_pre_push, const); // Uses 72 KiB of stack!
531 define_div_rem!(div_rem_64, 64, sub_64, Vec::new(), vec_pre_push); // Uses up to 2 MiB of heap
532 #[cfg(debug_assertions)]
533 define_div_rem!(div_rem_128, 128, sub_128, Vec::new(), vec_pre_push); // Uses up to 8 MiB of heap
535 macro_rules! define_mod_inv { ($name: ident, $len: expr, $div: ident, $add: ident, $sub_abs: ident, $mul: ident) => {
536 /// Calculates the modular inverse of a $len-64-bit number with respect to the given modulus,
538 const fn $name(a: &[u64; $len], m: &[u64; $len]) -> Result<[u64; $len], ()> {
539 if slice_equal(a, &[0; $len]) || slice_equal(m, &[0; $len]) { return Err(()); }
541 let (mut s, mut old_s) = ([0; $len], [0; $len]);
546 let (mut old_s_neg, mut s_neg) = (false, false);
548 while !slice_equal(&r, &[0; $len]) {
549 let (quot, new_r) = debug_unwrap!($div(&old_r, &r));
551 let new_sa = $mul(", &s);
552 debug_assert!(slice_equal(const_subslice(&new_sa, 0, $len), &[0; $len]), "S overflowed");
553 let (new_s, new_s_neg) = match (old_s_neg, s_neg) {
555 let (new_s, overflow) = $add(&old_s, const_subslice(&new_sa, $len, new_sa.len()));
556 debug_assert!(!overflow);
560 let (new_s, overflow) = $add(&old_s, const_subslice(&new_sa, $len, new_sa.len()));
561 debug_assert!(!overflow);
565 let (new_s, overflow) = $add(&old_s, const_subslice(&new_sa, $len, new_sa.len()));
566 debug_assert!(!overflow);
569 (false, false) => $sub_abs(&old_s, const_subslice(&new_sa, $len, new_sa.len())),
581 // At this point old_r contains our GCD and old_s our first Bézout's identity coefficient.
582 if !slice_equal(const_subslice(&old_r, 0, $len - 1), &[0; $len - 1]) || old_r[$len - 1] != 1 {
585 debug_assert!(slice_greater_than(m, &old_s));
587 let (modinv, underflow) = $sub_abs(m, &old_s);
588 debug_assert!(!underflow);
589 debug_assert!(slice_greater_than(m, &modinv));
598 define_mod_inv!(mod_inv_2, 2, div_rem_2, add_2, sub_abs_2, mul_2);
599 define_mod_inv!(mod_inv_4, 4, div_rem_4, add_4, sub_abs_4, mul_4);
600 define_mod_inv!(mod_inv_6, 6, div_rem_6, add_6, sub_abs_6, mul_6);
602 define_mod_inv!(mod_inv_8, 8, div_rem_8, add_8, sub_abs_8, mul_8);
604 // ******************
605 // * The public API *
606 // ******************
608 const WORD_COUNT_4096: usize = 4096 / 64;
609 const WORD_COUNT_256: usize = 256 / 64;
610 const WORD_COUNT_384: usize = 384 / 64;
612 // RFC 5702 indicates RSA keys can be up to 4096 bits, so we always use 4096-bit integers
613 #[derive(Clone, Debug, PartialEq, Eq, PartialOrd, Ord)]
614 pub(super) struct U4096([u64; WORD_COUNT_4096]);
616 #[derive(Clone, Debug, PartialEq, Eq, PartialOrd, Ord)]
617 pub(super) struct U256([u64; WORD_COUNT_256]);
619 #[derive(Clone, Debug, PartialEq, Eq, PartialOrd, Ord)]
620 pub(super) struct U384([u64; WORD_COUNT_384]);
622 pub(super) trait Int: Clone + Ord + Sized {
625 fn from_be_bytes(b: &[u8]) -> Result<Self, ()>;
626 fn limbs(&self) -> &[u64];
629 const ZERO: U256 = U256([0; 4]);
630 const BYTES: usize = 32;
631 fn from_be_bytes(b: &[u8]) -> Result<Self, ()> { Self::from_be_bytes(b) }
632 fn limbs(&self) -> &[u64] { &self.0 }
635 const ZERO: U384 = U384([0; 6]);
636 const BYTES: usize = 48;
637 fn from_be_bytes(b: &[u8]) -> Result<Self, ()> { Self::from_be_bytes(b) }
638 fn limbs(&self) -> &[u64] { &self.0 }
641 /// Defines a *PRIME* Modulus
642 pub(super) trait PrimeModulus<I: Int> {
644 const R_SQUARED_MOD_PRIME: I;
645 const NEGATIVE_PRIME_INV_MOD_R: I;
648 #[derive(Clone, Debug, PartialEq, Eq)] // Ord doesn't make sense cause we have an R factor
649 pub(super) struct U256Mod<M: PrimeModulus<U256>>(U256, PhantomData<M>);
651 #[derive(Clone, Debug, PartialEq, Eq)] // Ord doesn't make sense cause we have an R factor
652 pub(super) struct U384Mod<M: PrimeModulus<U384>>(U384, PhantomData<M>);
655 /// Constructs a new [`U4096`] from a variable number of big-endian bytes.
656 pub(super) fn from_be_bytes(bytes: &[u8]) -> Result<U4096, ()> {
657 if bytes.len() > 4096/8 { return Err(()); }
658 let u64s = (bytes.len() + 7) / 8;
659 let mut res = [0; WORD_COUNT_4096];
662 let pos = (u64s - i) * 8;
663 let start = bytes.len().saturating_sub(pos);
664 let end = bytes.len() + 8 - pos;
665 b[8 + start - end..].copy_from_slice(&bytes[start..end]);
666 res[i + WORD_COUNT_4096 - u64s] = u64::from_be_bytes(b);
671 /// Naively multiplies `self` * `b` mod `m`, returning a new [`U4096`].
673 /// Fails iff m is 0 or self or b are greater than m.
674 #[cfg(debug_assertions)]
675 fn mulmod_naive(&self, b: &U4096, m: &U4096) -> Result<U4096, ()> {
676 if m.0 == [0; WORD_COUNT_4096] { return Err(()); }
677 if self > m || b > m { return Err(()); }
679 let mul = mul_64(&self.0, &b.0);
681 let mut m_zeros = [0; 128];
682 m_zeros[WORD_COUNT_4096..].copy_from_slice(&m.0);
683 let (_, rem) = div_rem_128(&mul, &m_zeros)?;
684 let mut res = [0; WORD_COUNT_4096];
685 debug_assert_eq!(&rem[..WORD_COUNT_4096], &[0; WORD_COUNT_4096]);
686 res.copy_from_slice(&rem[WORD_COUNT_4096..]);
690 /// Calculates `self` ^ `exp` mod `m`, returning a new [`U4096`].
692 /// Fails iff m is 0, even, or self or b are greater than m.
693 pub(super) fn expmod_odd_mod(&self, mut exp: u32, m: &U4096) -> Result<U4096, ()> {
694 #![allow(non_camel_case_types)]
696 if m.0 == [0; WORD_COUNT_4096] { return Err(()); }
697 if m.0[WORD_COUNT_4096 - 1] & 1 == 0 { return Err(()); }
698 if self > m { return Err(()); }
700 let mut t = [0; WORD_COUNT_4096];
701 if &m.0[..WORD_COUNT_4096 - 1] == &[0; WORD_COUNT_4096 - 1] && m.0[WORD_COUNT_4096 - 1] == 1 {
704 t[WORD_COUNT_4096 - 1] = 1;
705 if exp == 0 { return Ok(U4096(t)); }
707 // Because m is not even, using 2^4096 as the Montgomery R value is always safe - it is
708 // guaranteed to be co-prime with any non-even integer.
710 // We use a single 4096-bit integer type for all our RSA operations, though in most cases
711 // we're actually dealing with 1024-bit or 2048-bit ints. Thus, we define sub-array math
712 // here which debug_assert's the required bits are 0s and then uses faster math primitives.
714 type mul_ty = fn(&[u64], &[u64]) -> [u64; WORD_COUNT_4096 * 2];
715 type sqr_ty = fn(&[u64]) -> [u64; WORD_COUNT_4096 * 2];
716 type add_double_ty = fn(&[u64], &[u64]) -> ([u64; WORD_COUNT_4096 * 2], bool);
717 type sub_ty = fn(&[u64], &[u64]) -> ([u64; WORD_COUNT_4096], bool);
718 let (word_count, log_bits, mul, sqr, add_double, sub) =
719 if m.0[..WORD_COUNT_4096 / 2] == [0; WORD_COUNT_4096 / 2] {
720 if m.0[..WORD_COUNT_4096 * 3 / 4] == [0; WORD_COUNT_4096 * 3 / 4] {
721 fn mul_16_subarr(a: &[u64], b: &[u64]) -> [u64; WORD_COUNT_4096 * 2] {
722 debug_assert_eq!(a.len(), WORD_COUNT_4096);
723 debug_assert_eq!(b.len(), WORD_COUNT_4096);
724 debug_assert_eq!(&a[..WORD_COUNT_4096 * 3 / 4], &[0; WORD_COUNT_4096 * 3 / 4]);
725 debug_assert_eq!(&b[..WORD_COUNT_4096 * 3 / 4], &[0; WORD_COUNT_4096 * 3 / 4]);
726 let mut res = [0; WORD_COUNT_4096 * 2];
727 res[WORD_COUNT_4096 + WORD_COUNT_4096 / 2..].copy_from_slice(
728 &mul_16(&a[WORD_COUNT_4096 * 3 / 4..], &b[WORD_COUNT_4096 * 3 / 4..]));
731 fn sqr_16_subarr(a: &[u64]) -> [u64; WORD_COUNT_4096 * 2] {
732 debug_assert_eq!(a.len(), WORD_COUNT_4096);
733 debug_assert_eq!(&a[..WORD_COUNT_4096 * 3 / 4], &[0; WORD_COUNT_4096 * 3 / 4]);
734 let mut res = [0; WORD_COUNT_4096 * 2];
735 res[WORD_COUNT_4096 + WORD_COUNT_4096 / 2..].copy_from_slice(
736 &sqr_16(&a[WORD_COUNT_4096 * 3 / 4..]));
739 fn add_32_subarr(a: &[u64], b: &[u64]) -> ([u64; WORD_COUNT_4096 * 2], bool) {
740 debug_assert_eq!(a.len(), WORD_COUNT_4096 * 2);
741 debug_assert_eq!(b.len(), WORD_COUNT_4096 * 2);
742 debug_assert_eq!(&a[..WORD_COUNT_4096 * 3 / 2], &[0; WORD_COUNT_4096 * 3 / 2]);
743 debug_assert_eq!(&b[..WORD_COUNT_4096 * 3 / 2], &[0; WORD_COUNT_4096 * 3 / 2]);
744 let (add, overflow) = add_32(&a[WORD_COUNT_4096 * 3 / 2..], &b[WORD_COUNT_4096 * 3 / 2..]);
745 let mut res = [0; WORD_COUNT_4096 * 2];
746 res[WORD_COUNT_4096 * 3 / 2..].copy_from_slice(&add);
749 fn sub_16_subarr(a: &[u64], b: &[u64]) -> ([u64; WORD_COUNT_4096], bool) {
750 debug_assert_eq!(a.len(), WORD_COUNT_4096);
751 debug_assert_eq!(b.len(), WORD_COUNT_4096);
752 debug_assert_eq!(&a[..WORD_COUNT_4096 * 3 / 4], &[0; WORD_COUNT_4096 * 3 / 4]);
753 debug_assert_eq!(&b[..WORD_COUNT_4096 * 3 / 4], &[0; WORD_COUNT_4096 * 3 / 4]);
754 let (sub, underflow) = sub_16(&a[WORD_COUNT_4096 * 3 / 4..], &b[WORD_COUNT_4096 * 3 / 4..]);
755 let mut res = [0; WORD_COUNT_4096];
756 res[WORD_COUNT_4096 * 3 / 4..].copy_from_slice(&sub);
759 (16, 10, mul_16_subarr as mul_ty, sqr_16_subarr as sqr_ty, add_32_subarr as add_double_ty, sub_16_subarr as sub_ty)
761 fn mul_32_subarr(a: &[u64], b: &[u64]) -> [u64; WORD_COUNT_4096 * 2] {
762 debug_assert_eq!(a.len(), WORD_COUNT_4096);
763 debug_assert_eq!(b.len(), WORD_COUNT_4096);
764 debug_assert_eq!(&a[..WORD_COUNT_4096 / 2], &[0; WORD_COUNT_4096 / 2]);
765 debug_assert_eq!(&b[..WORD_COUNT_4096 / 2], &[0; WORD_COUNT_4096 / 2]);
766 let mut res = [0; WORD_COUNT_4096 * 2];
767 res[WORD_COUNT_4096..].copy_from_slice(
768 &mul_32(&a[WORD_COUNT_4096 / 2..], &b[WORD_COUNT_4096 / 2..]));
771 fn sqr_32_subarr(a: &[u64]) -> [u64; WORD_COUNT_4096 * 2] {
772 debug_assert_eq!(a.len(), WORD_COUNT_4096);
773 debug_assert_eq!(&a[..WORD_COUNT_4096 / 2], &[0; WORD_COUNT_4096 / 2]);
774 let mut res = [0; WORD_COUNT_4096 * 2];
775 res[WORD_COUNT_4096..].copy_from_slice(
776 &sqr_32(&a[WORD_COUNT_4096 / 2..]));
779 fn add_64_subarr(a: &[u64], b: &[u64]) -> ([u64; WORD_COUNT_4096 * 2], bool) {
780 debug_assert_eq!(a.len(), WORD_COUNT_4096 * 2);
781 debug_assert_eq!(b.len(), WORD_COUNT_4096 * 2);
782 debug_assert_eq!(&a[..WORD_COUNT_4096], &[0; WORD_COUNT_4096]);
783 debug_assert_eq!(&b[..WORD_COUNT_4096], &[0; WORD_COUNT_4096]);
784 let (add, overflow) = add_64(&a[WORD_COUNT_4096..], &b[WORD_COUNT_4096..]);
785 let mut res = [0; WORD_COUNT_4096 * 2];
786 res[WORD_COUNT_4096..].copy_from_slice(&add);
789 fn sub_32_subarr(a: &[u64], b: &[u64]) -> ([u64; WORD_COUNT_4096], bool) {
790 debug_assert_eq!(a.len(), WORD_COUNT_4096);
791 debug_assert_eq!(b.len(), WORD_COUNT_4096);
792 debug_assert_eq!(&a[..WORD_COUNT_4096 / 2], &[0; WORD_COUNT_4096 / 2]);
793 debug_assert_eq!(&b[..WORD_COUNT_4096 / 2], &[0; WORD_COUNT_4096 / 2]);
794 let (sub, underflow) = sub_32(&a[WORD_COUNT_4096 / 2..], &b[WORD_COUNT_4096 / 2..]);
795 let mut res = [0; WORD_COUNT_4096];
796 res[WORD_COUNT_4096 / 2..].copy_from_slice(&sub);
799 (32, 11, mul_32_subarr as mul_ty, sqr_32_subarr as sqr_ty, add_64_subarr as add_double_ty, sub_32_subarr as sub_ty)
802 (64, 12, mul_64 as mul_ty, sqr_64 as sqr_ty, add_128 as add_double_ty, sub_64 as sub_ty)
805 // r is always the even value with one bit set above the word count we're using.
806 let mut r = [0; WORD_COUNT_4096 * 2];
807 r[WORD_COUNT_4096 * 2 - word_count - 1] = 1;
809 let mut m_inv_pos = [0; WORD_COUNT_4096];
810 m_inv_pos[WORD_COUNT_4096 - 1] = 1;
811 let mut two = [0; WORD_COUNT_4096];
812 two[WORD_COUNT_4096 - 1] = 2;
813 for _ in 0..log_bits {
814 let mut m_m_inv = mul(&m_inv_pos, &m.0);
815 m_m_inv[..WORD_COUNT_4096 * 2 - word_count].fill(0);
816 let m_inv = mul(&sub(&two, &m_m_inv[WORD_COUNT_4096..]).0, &m_inv_pos);
817 m_inv_pos[WORD_COUNT_4096 - word_count..].copy_from_slice(&m_inv[WORD_COUNT_4096 * 2 - word_count..]);
819 m_inv_pos[..WORD_COUNT_4096 - word_count].fill(0);
821 // `m_inv` is the negative modular inverse of m mod R, so subtract m_inv from R.
822 let mut m_inv = m_inv_pos;
824 m_inv[..WORD_COUNT_4096 - word_count].fill(0);
825 debug_assert_eq!(&mul(&m_inv, &m.0)[WORD_COUNT_4096 * 2 - word_count..],
827 &[0xffff_ffff_ffff_ffff; WORD_COUNT_4096][WORD_COUNT_4096 - word_count..]);
829 let mont_reduction = |mu: [u64; WORD_COUNT_4096 * 2]| -> [u64; WORD_COUNT_4096] {
830 debug_assert_eq!(&mu[..WORD_COUNT_4096 * 2 - word_count * 2],
831 &[0; WORD_COUNT_4096 * 2][..WORD_COUNT_4096 * 2 - word_count * 2]);
832 // Do a montgomery reduction of `mu`
834 // mu % R is just the bottom word_count bytes of mu
835 let mut mu_mod_r = [0; WORD_COUNT_4096];
836 mu_mod_r[WORD_COUNT_4096 - word_count..].copy_from_slice(&mu[WORD_COUNT_4096 * 2 - word_count..]);
838 // v = ((mu % R) * negative_modulus_inverse) % R
839 let mut v = mul(&mu_mod_r, &m_inv);
840 v[..WORD_COUNT_4096 * 2 - word_count].fill(0); // mod R
842 // t_on_r = (mu + v*modulus) / R
843 let t0 = mul(&v[WORD_COUNT_4096..], &m.0);
844 let (t1, t1_extra_bit) = add_double(&t0, &mu);
846 // Note that dividing t1 by R is simply a matter of shifting right by word_count bytes
847 // We only need to maintain word_count bytes (plus `t1_extra_bit` which is implicitly
848 // an extra bit) because t_on_r is guarantee to be, at max, 2*m - 1.
849 let mut t1_on_r = [0; WORD_COUNT_4096];
850 debug_assert_eq!(&t1[WORD_COUNT_4096 * 2 - word_count..], &[0; WORD_COUNT_4096][WORD_COUNT_4096 - word_count..],
851 "t1 should be divisible by r");
852 t1_on_r[WORD_COUNT_4096 - word_count..].copy_from_slice(&t1[WORD_COUNT_4096 * 2 - word_count * 2..WORD_COUNT_4096 * 2 - word_count]);
854 // The modulus has only word_count bytes, so if t1_extra_bit is set we are definitely
855 // larger than the modulus.
856 if t1_extra_bit || t1_on_r >= m.0 {
858 (t1_on_r, underflow) = sub(&t1_on_r, &m.0);
859 debug_assert_eq!(t1_extra_bit, underflow,
860 "The number (t1_extra_bit, t1_on_r) is at most 2m-1, so underflowing t1_on_r - m should happen iff t1_extra_bit is set.");
865 // Calculate R^2 mod m as ((2^DOUBLES * R) mod m)^(log_bits - LOG2_DOUBLES) mod R
866 let mut r_minus_one = [0xffff_ffff_ffff_ffffu64; WORD_COUNT_4096];
867 r_minus_one[..WORD_COUNT_4096 - word_count].fill(0);
868 // While we do a full div here, in general R should be less than 2x m (assuming the RSA
869 // modulus used its full bit range and is 1024, 2048, or 4096 bits), so it should be cheap.
870 // In cases with a nonstandard RSA modulus we may end up being pretty slow here, but we'll
872 // If we ever find a problem with this we should reduce R to be tigher on m, as we're
873 // wasting extra bits of calculation if R is too far from m.
874 let (_, mut r_mod_m) = debug_unwrap!(div_rem_64(&r_minus_one, &m.0));
875 let r_mod_m_overflow = add_u64!(r_mod_m, 1);
876 if r_mod_m_overflow || r_mod_m >= m.0 {
877 (r_mod_m, _) = sub_64(&r_mod_m, &m.0);
880 let mut r2_mod_m: [u64; 64] = r_mod_m;
881 const DOUBLES: usize = 32;
882 const LOG2_DOUBLES: usize = 5;
884 for _ in 0..DOUBLES {
885 let overflow = double!(r2_mod_m);
886 if overflow || r2_mod_m > m.0 {
887 (r2_mod_m, _) = sub_64(&r2_mod_m, &m.0);
890 for _ in 0..log_bits - LOG2_DOUBLES {
891 r2_mod_m = mont_reduction(sqr(&r2_mod_m));
893 // Clear excess high bits
894 for (m_limb, r2_limb) in m.0.iter().zip(r2_mod_m.iter_mut()) {
895 let clear_bits = m_limb.leading_zeros();
896 if clear_bits == 0 { break; }
897 *r2_limb &= !(0xffff_ffff_ffff_ffffu64 << (64 - clear_bits));
898 if *m_limb != 0 { break; }
900 debug_assert!(r2_mod_m < m.0);
902 // Finally, actually do the exponentiation...
904 // Calculate t * R and a * R as mont multiplications by R^2 mod m
905 let mut tr = mont_reduction(mul(&r2_mod_m, &t));
906 let mut ar = mont_reduction(mul(&r2_mod_m, &self.0));
908 #[cfg(debug_assertions)] {
909 debug_assert_eq!(r2_mod_m, U4096(r_mod_m).mulmod_naive(&U4096(r_mod_m), &m).unwrap().0);
910 debug_assert_eq!(&tr, &U4096(t).mulmod_naive(&U4096(r_mod_m), &m).unwrap().0);
911 debug_assert_eq!(&ar, &self.mulmod_naive(&U4096(r_mod_m), &m).unwrap().0);
916 tr = mont_reduction(mul(&tr, &ar));
919 ar = mont_reduction(sqr(&ar));
922 ar = mont_reduction(mul(&ar, &tr));
923 let mut resr = [0; WORD_COUNT_4096 * 2];
924 resr[WORD_COUNT_4096..].copy_from_slice(&ar);
925 Ok(U4096(mont_reduction(resr)))
929 // In a const context we can't subslice a slice, so instead we pick the eight bytes we want and
930 // pass them here to build u64s from arrays.
931 const fn eight_bytes_to_u64_be(a: u8, b: u8, c: u8, d: u8, e: u8, f: u8, g: u8, h: u8) -> u64 {
932 let b = [a, b, c, d, e, f, g, h];
933 u64::from_be_bytes(b)
937 /// Constructs a new [`U256`] from a variable number of big-endian bytes.
938 pub(super) fn from_be_bytes(bytes: &[u8]) -> Result<U256, ()> {
939 if bytes.len() > 256/8 { return Err(()); }
940 let u64s = (bytes.len() + 7) / 8;
941 let mut res = [0; WORD_COUNT_256];
944 let pos = (u64s - i) * 8;
945 let start = bytes.len().saturating_sub(pos);
946 let end = bytes.len() + 8 - pos;
947 b[8 + start - end..].copy_from_slice(&bytes[start..end]);
948 res[i + WORD_COUNT_256 - u64s] = u64::from_be_bytes(b);
953 /// Constructs a new [`U256`] from a fixed number of big-endian bytes.
954 pub(super) const fn from_32_be_bytes_panicking(bytes: &[u8; 32]) -> U256 {
956 eight_bytes_to_u64_be(bytes[0*8 + 0], bytes[0*8 + 1], bytes[0*8 + 2], bytes[0*8 + 3],
957 bytes[0*8 + 4], bytes[0*8 + 5], bytes[0*8 + 6], bytes[0*8 + 7]),
958 eight_bytes_to_u64_be(bytes[1*8 + 0], bytes[1*8 + 1], bytes[1*8 + 2], bytes[1*8 + 3],
959 bytes[1*8 + 4], bytes[1*8 + 5], bytes[1*8 + 6], bytes[1*8 + 7]),
960 eight_bytes_to_u64_be(bytes[2*8 + 0], bytes[2*8 + 1], bytes[2*8 + 2], bytes[2*8 + 3],
961 bytes[2*8 + 4], bytes[2*8 + 5], bytes[2*8 + 6], bytes[2*8 + 7]),
962 eight_bytes_to_u64_be(bytes[3*8 + 0], bytes[3*8 + 1], bytes[3*8 + 2], bytes[3*8 + 3],
963 bytes[3*8 + 4], bytes[3*8 + 5], bytes[3*8 + 6], bytes[3*8 + 7]),
968 pub(super) const fn zero() -> U256 { U256([0, 0, 0, 0]) }
969 pub(super) const fn one() -> U256 { U256([0, 0, 0, 1]) }
970 pub(super) const fn three() -> U256 { U256([0, 0, 0, 3]) }
973 // Values modulus M::PRIME.0, stored in montgomery form.
974 impl<M: PrimeModulus<U256>> U256Mod<M> {
975 const fn mont_reduction(mu: [u64; 8]) -> Self {
976 #[cfg(debug_assertions)] {
977 // Check NEGATIVE_PRIME_INV_MOD_R is correct. Since this is all const, the compiler
978 // should be able to do it at compile time alone.
979 let minus_one_mod_r = mul_4(&M::PRIME.0, &M::NEGATIVE_PRIME_INV_MOD_R.0);
980 assert!(slice_equal(const_subslice(&minus_one_mod_r, 4, 8), &[0xffff_ffff_ffff_ffff; 4]));
983 #[cfg(debug_assertions)] {
984 // Check R_SQUARED_MOD_PRIME is correct. Since this is all const, the compiler
985 // should be able to do it at compile time alone.
986 let r_minus_one = [0xffff_ffff_ffff_ffff; 4];
987 let (mut r_mod_prime, _) = sub_4(&r_minus_one, &M::PRIME.0);
988 let r_mod_prime_overflow = add_u64!(r_mod_prime, 1);
989 assert!(!r_mod_prime_overflow);
990 let r_squared = sqr_4(&r_mod_prime);
991 let mut prime_extended = [0; 8];
992 let prime = M::PRIME.0;
993 copy_from_slice!(prime_extended, 4, 8, prime);
994 let (_, r_squared_mod_prime) = if let Ok(v) = div_rem_8(&r_squared, &prime_extended) { v } else { panic!() };
995 assert!(slice_greater_than(&prime_extended, &r_squared_mod_prime));
996 assert!(slice_equal(const_subslice(&r_squared_mod_prime, 4, 8), &M::R_SQUARED_MOD_PRIME.0));
999 // mu % R is just the bottom 4 bytes of mu
1000 let mu_mod_r = const_subslice(&mu, 4, 8);
1001 // v = ((mu % R) * negative_modulus_inverse) % R
1002 let mut v = mul_4(&mu_mod_r, &M::NEGATIVE_PRIME_INV_MOD_R.0);
1003 const ZEROS: &[u64; 4] = &[0; 4];
1004 copy_from_slice!(v, 0, 4, ZEROS); // mod R
1006 // t_on_r = (mu + v*modulus) / R
1007 let t0 = mul_4(const_subslice(&v, 4, 8), &M::PRIME.0);
1008 let (t1, t1_extra_bit) = add_8(&t0, &mu);
1010 // Note that dividing t1 by R is simply a matter of shifting right by 4 bytes.
1011 // We only need to maintain 4 bytes (plus `t1_extra_bit` which is implicitly an extra bit)
1012 // because t_on_r is guarantee to be, at max, 2*m - 1.
1013 let t1_on_r = const_subslice(&t1, 0, 4);
1015 let mut res = [0; 4];
1016 // The modulus is only 4 bytes, so t1_extra_bit implies we're definitely larger than the
1018 if t1_extra_bit || slice_greater_than(&t1_on_r, &M::PRIME.0) {
1020 (res, underflow) = sub_4(&t1_on_r, &M::PRIME.0);
1021 debug_assert!(t1_extra_bit == underflow,
1022 "The number (t1_extra_bit, t1_on_r) is at most 2m-1, so underflowing t1_on_r - m should happen iff t1_extra_bit is set.");
1024 copy_from_slice!(res, 0, 4, t1_on_r);
1026 Self(U256(res), PhantomData)
1029 pub(super) const fn from_u256_panicking(v: U256) -> Self {
1030 assert!(v.0[0] <= M::PRIME.0[0]);
1031 if v.0[0] == M::PRIME.0[0] {
1032 assert!(v.0[1] <= M::PRIME.0[1]);
1033 if v.0[1] == M::PRIME.0[1] {
1034 assert!(v.0[2] <= M::PRIME.0[2]);
1035 if v.0[2] == M::PRIME.0[2] {
1036 assert!(v.0[3] < M::PRIME.0[3]);
1040 assert!(M::PRIME.0[0] != 0 || M::PRIME.0[1] != 0 || M::PRIME.0[2] != 0 || M::PRIME.0[3] != 0);
1041 Self::mont_reduction(mul_4(&M::R_SQUARED_MOD_PRIME.0, &v.0))
1044 pub(super) fn from_u256(mut v: U256) -> Self {
1045 debug_assert!(M::PRIME.0 != [0; 4]);
1046 debug_assert!(M::PRIME.0[0] > (1 << 63), "PRIME should have the top bit set");
1047 while v >= M::PRIME {
1048 let (new_v, spurious_underflow) = sub_4(&v.0, &M::PRIME.0);
1049 debug_assert!(!spurious_underflow, "v was > M::PRIME.0");
1052 Self::mont_reduction(mul_4(&M::R_SQUARED_MOD_PRIME.0, &v.0))
1055 pub(super) fn from_modinv_of(v: U256) -> Result<Self, ()> {
1056 Ok(Self::from_u256(U256(mod_inv_4(&v.0, &M::PRIME.0)?)))
1059 /// Multiplies `self` * `b` mod `m`.
1061 /// Panics if `self`'s modulus is not equal to `b`'s
1062 pub(super) fn mul(&self, b: &Self) -> Self {
1063 Self::mont_reduction(mul_4(&self.0.0, &b.0.0))
1066 /// Doubles `self` mod `m`.
1067 pub(super) fn double(&self) -> Self {
1068 let mut res = self.0.0;
1069 let overflow = double!(res);
1070 if overflow || !slice_greater_than(&M::PRIME.0, &res) {
1072 (res, underflow) = sub_4(&res, &M::PRIME.0);
1073 debug_assert_eq!(overflow, underflow);
1075 Self(U256(res), PhantomData)
1078 /// Multiplies `self` by 3 mod `m`.
1079 pub(super) fn times_three(&self) -> Self {
1080 // TODO: Optimize this a lot
1081 self.mul(&U256Mod::from_u256(U256::three()))
1084 /// Multiplies `self` by 4 mod `m`.
1085 pub(super) fn times_four(&self) -> Self {
1086 // TODO: Optimize this somewhat?
1087 self.double().double()
1090 /// Multiplies `self` by 8 mod `m`.
1091 pub(super) fn times_eight(&self) -> Self {
1092 // TODO: Optimize this somewhat?
1093 self.double().double().double()
1096 /// Multiplies `self` by 8 mod `m`.
1097 pub(super) fn square(&self) -> Self {
1098 Self::mont_reduction(sqr_4(&self.0.0))
1101 /// Subtracts `b` from `self` % `m`.
1102 pub(super) fn sub(&self, b: &Self) -> Self {
1103 let (mut val, underflow) = sub_4(&self.0.0, &b.0.0);
1106 (val, overflow) = add_4(&val, &M::PRIME.0);
1107 debug_assert_eq!(overflow, underflow);
1109 Self(U256(val), PhantomData)
1112 /// Adds `b` to `self` % `m`.
1113 pub(super) fn add(&self, b: &Self) -> Self {
1114 let (mut val, overflow) = add_4(&self.0.0, &b.0.0);
1115 if overflow || !slice_greater_than(&M::PRIME.0, &val) {
1117 (val, underflow) = sub_4(&val, &M::PRIME.0);
1118 debug_assert_eq!(overflow, underflow);
1120 Self(U256(val), PhantomData)
1123 /// Returns the underlying [`U256`].
1124 pub(super) fn into_u256(self) -> U256 {
1125 let mut expanded_self = [0; 8];
1126 expanded_self[4..].copy_from_slice(&self.0.0);
1127 Self::mont_reduction(expanded_self).0
1131 // Values modulus M::PRIME.0, stored in montgomery form.
1133 /// Constructs a new [`U384`] from a variable number of big-endian bytes.
1134 pub(super) fn from_be_bytes(bytes: &[u8]) -> Result<U384, ()> {
1135 if bytes.len() > 384/8 { return Err(()); }
1136 let u64s = (bytes.len() + 7) / 8;
1137 let mut res = [0; WORD_COUNT_384];
1140 let pos = (u64s - i) * 8;
1141 let start = bytes.len().saturating_sub(pos);
1142 let end = bytes.len() + 8 - pos;
1143 b[8 + start - end..].copy_from_slice(&bytes[start..end]);
1144 res[i + WORD_COUNT_384 - u64s] = u64::from_be_bytes(b);
1149 /// Constructs a new [`U384`] from a fixed number of big-endian bytes.
1150 pub(super) const fn from_48_be_bytes_panicking(bytes: &[u8; 48]) -> U384 {
1152 eight_bytes_to_u64_be(bytes[0*8 + 0], bytes[0*8 + 1], bytes[0*8 + 2], bytes[0*8 + 3],
1153 bytes[0*8 + 4], bytes[0*8 + 5], bytes[0*8 + 6], bytes[0*8 + 7]),
1154 eight_bytes_to_u64_be(bytes[1*8 + 0], bytes[1*8 + 1], bytes[1*8 + 2], bytes[1*8 + 3],
1155 bytes[1*8 + 4], bytes[1*8 + 5], bytes[1*8 + 6], bytes[1*8 + 7]),
1156 eight_bytes_to_u64_be(bytes[2*8 + 0], bytes[2*8 + 1], bytes[2*8 + 2], bytes[2*8 + 3],
1157 bytes[2*8 + 4], bytes[2*8 + 5], bytes[2*8 + 6], bytes[2*8 + 7]),
1158 eight_bytes_to_u64_be(bytes[3*8 + 0], bytes[3*8 + 1], bytes[3*8 + 2], bytes[3*8 + 3],
1159 bytes[3*8 + 4], bytes[3*8 + 5], bytes[3*8 + 6], bytes[3*8 + 7]),
1160 eight_bytes_to_u64_be(bytes[4*8 + 0], bytes[4*8 + 1], bytes[4*8 + 2], bytes[4*8 + 3],
1161 bytes[4*8 + 4], bytes[4*8 + 5], bytes[4*8 + 6], bytes[4*8 + 7]),
1162 eight_bytes_to_u64_be(bytes[5*8 + 0], bytes[5*8 + 1], bytes[5*8 + 2], bytes[5*8 + 3],
1163 bytes[5*8 + 4], bytes[5*8 + 5], bytes[5*8 + 6], bytes[5*8 + 7]),
1168 pub(super) const fn zero() -> U384 { U384([0, 0, 0, 0, 0, 0]) }
1169 pub(super) const fn one() -> U384 { U384([0, 0, 0, 0, 0, 1]) }
1170 pub(super) const fn three() -> U384 { U384([0, 0, 0, 0, 0, 3]) }
1173 impl<M: PrimeModulus<U384>> U384Mod<M> {
1174 const fn mont_reduction(mu: [u64; 12]) -> Self {
1175 #[cfg(debug_assertions)] {
1176 // Check NEGATIVE_PRIME_INV_MOD_R is correct. Since this is all const, the compiler
1177 // should be able to do it at compile time alone.
1178 let minus_one_mod_r = mul_6(&M::PRIME.0, &M::NEGATIVE_PRIME_INV_MOD_R.0);
1179 assert!(slice_equal(const_subslice(&minus_one_mod_r, 6, 12), &[0xffff_ffff_ffff_ffff; 6]));
1182 #[cfg(debug_assertions)] {
1183 // Check R_SQUARED_MOD_PRIME is correct. Since this is all const, the compiler
1184 // should be able to do it at compile time alone.
1185 let r_minus_one = [0xffff_ffff_ffff_ffff; 6];
1186 let (mut r_mod_prime, _) = sub_6(&r_minus_one, &M::PRIME.0);
1187 let r_mod_prime_overflow = add_u64!(r_mod_prime, 1);
1188 assert!(!r_mod_prime_overflow);
1189 let r_squared = sqr_6(&r_mod_prime);
1190 let mut prime_extended = [0; 12];
1191 let prime = M::PRIME.0;
1192 copy_from_slice!(prime_extended, 6, 12, prime);
1193 let (_, r_squared_mod_prime) = if let Ok(v) = div_rem_12(&r_squared, &prime_extended) { v } else { panic!() };
1194 assert!(slice_greater_than(&prime_extended, &r_squared_mod_prime));
1195 assert!(slice_equal(const_subslice(&r_squared_mod_prime, 6, 12), &M::R_SQUARED_MOD_PRIME.0));
1198 // mu % R is just the bottom 4 bytes of mu
1199 let mu_mod_r = const_subslice(&mu, 6, 12);
1200 // v = ((mu % R) * negative_modulus_inverse) % R
1201 let mut v = mul_6(&mu_mod_r, &M::NEGATIVE_PRIME_INV_MOD_R.0);
1202 const ZEROS: &[u64; 6] = &[0; 6];
1203 copy_from_slice!(v, 0, 6, ZEROS); // mod R
1205 // t_on_r = (mu + v*modulus) / R
1206 let t0 = mul_6(const_subslice(&v, 6, 12), &M::PRIME.0);
1207 let (t1, t1_extra_bit) = add_12(&t0, &mu);
1209 // Note that dividing t1 by R is simply a matter of shifting right by 4 bytes.
1210 // We only need to maintain 4 bytes (plus `t1_extra_bit` which is implicitly an extra bit)
1211 // because t_on_r is guarantee to be, at max, 2*m - 1.
1212 let t1_on_r = const_subslice(&t1, 0, 6);
1214 let mut res = [0; 6];
1215 // The modulus is only 4 bytes, so t1_extra_bit implies we're definitely larger than the
1217 if t1_extra_bit || slice_greater_than(&t1_on_r, &M::PRIME.0) {
1219 (res, underflow) = sub_6(&t1_on_r, &M::PRIME.0);
1220 debug_assert!(t1_extra_bit == underflow);
1222 copy_from_slice!(res, 0, 6, t1_on_r);
1224 Self(U384(res), PhantomData)
1227 pub(super) const fn from_u384_panicking(v: U384) -> Self {
1228 assert!(v.0[0] <= M::PRIME.0[0]);
1229 if v.0[0] == M::PRIME.0[0] {
1230 assert!(v.0[1] <= M::PRIME.0[1]);
1231 if v.0[1] == M::PRIME.0[1] {
1232 assert!(v.0[2] <= M::PRIME.0[2]);
1233 if v.0[2] == M::PRIME.0[2] {
1234 assert!(v.0[3] <= M::PRIME.0[3]);
1235 if v.0[3] == M::PRIME.0[3] {
1236 assert!(v.0[4] <= M::PRIME.0[4]);
1237 if v.0[4] == M::PRIME.0[4] {
1238 assert!(v.0[5] < M::PRIME.0[5]);
1244 assert!(M::PRIME.0[0] != 0 || M::PRIME.0[1] != 0 || M::PRIME.0[2] != 0
1245 || M::PRIME.0[3] != 0|| M::PRIME.0[4] != 0|| M::PRIME.0[5] != 0);
1246 Self::mont_reduction(mul_6(&M::R_SQUARED_MOD_PRIME.0, &v.0))
1249 pub(super) fn from_u384(mut v: U384) -> Self {
1250 debug_assert!(M::PRIME.0 != [0; 6]);
1251 debug_assert!(M::PRIME.0[0] > (1 << 63), "PRIME should have the top bit set");
1252 while v >= M::PRIME {
1253 let (new_v, spurious_underflow) = sub_6(&v.0, &M::PRIME.0);
1254 debug_assert!(!spurious_underflow);
1257 Self::mont_reduction(mul_6(&M::R_SQUARED_MOD_PRIME.0, &v.0))
1260 pub(super) fn from_modinv_of(v: U384) -> Result<Self, ()> {
1261 Ok(Self::from_u384(U384(mod_inv_6(&v.0, &M::PRIME.0)?)))
1264 /// Multiplies `self` * `b` mod `m`.
1266 /// Panics if `self`'s modulus is not equal to `b`'s
1267 pub(super) fn mul(&self, b: &Self) -> Self {
1268 Self::mont_reduction(mul_6(&self.0.0, &b.0.0))
1271 /// Doubles `self` mod `m`.
1272 pub(super) fn double(&self) -> Self {
1273 let mut res = self.0.0;
1274 let overflow = double!(res);
1275 if overflow || !slice_greater_than(&M::PRIME.0, &res) {
1277 (res, underflow) = sub_6(&res, &M::PRIME.0);
1278 debug_assert_eq!(overflow, underflow);
1280 Self(U384(res), PhantomData)
1283 /// Multiplies `self` by 3 mod `m`.
1284 pub(super) fn times_three(&self) -> Self {
1285 // TODO: Optimize this a lot
1286 self.mul(&U384Mod::from_u384(U384::three()))
1289 /// Multiplies `self` by 4 mod `m`.
1290 pub(super) fn times_four(&self) -> Self {
1291 // TODO: Optimize this somewhat?
1292 self.double().double()
1295 /// Multiplies `self` by 8 mod `m`.
1296 pub(super) fn times_eight(&self) -> Self {
1297 // TODO: Optimize this somewhat?
1298 self.double().double().double()
1301 /// Multiplies `self` by 8 mod `m`.
1302 pub(super) fn square(&self) -> Self {
1303 Self::mont_reduction(sqr_6(&self.0.0))
1306 /// Subtracts `b` from `self` % `m`.
1307 pub(super) fn sub(&self, b: &Self) -> Self {
1308 let (mut val, underflow) = sub_6(&self.0.0, &b.0.0);
1311 (val, overflow) = add_6(&val, &M::PRIME.0);
1312 debug_assert_eq!(overflow, underflow);
1314 Self(U384(val), PhantomData)
1317 /// Adds `b` to `self` % `m`.
1318 pub(super) fn add(&self, b: &Self) -> Self {
1319 let (mut val, overflow) = add_6(&self.0.0, &b.0.0);
1320 if overflow || !slice_greater_than(&M::PRIME.0, &val) {
1322 (val, underflow) = sub_6(&val, &M::PRIME.0);
1323 debug_assert_eq!(overflow, underflow);
1325 Self(U384(val), PhantomData)
1328 /// Returns the underlying [`U384`].
1329 pub(super) fn into_u384(self) -> U384 {
1330 let mut expanded_self = [0; 12];
1331 expanded_self[6..].copy_from_slice(&self.0.0);
1332 Self::mont_reduction(expanded_self).0
1341 impl PrimeModulus<U256> for P256 {
1342 const PRIME: U256 = U256::from_32_be_bytes_panicking(&hex_lit::hex!(
1343 "ffffffff00000001000000000000000000000000ffffffffffffffffffffffff"));
1344 const R_SQUARED_MOD_PRIME: U256 = U256::from_32_be_bytes_panicking(&hex_lit::hex!(
1345 "00000004fffffffdfffffffffffffffefffffffbffffffff0000000000000003"));
1346 const NEGATIVE_PRIME_INV_MOD_R: U256 = U256::from_32_be_bytes_panicking(&hex_lit::hex!(
1347 "ffffffff00000002000000000000000000000001000000000000000000000001"));
1351 impl PrimeModulus<U384> for P384 {
1352 const PRIME: U384 = U384::from_48_be_bytes_panicking(&hex_lit::hex!(
1353 "fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000ffffffff"));
1354 const R_SQUARED_MOD_PRIME: U384 = U384::from_48_be_bytes_panicking(&hex_lit::hex!(
1355 "000000000000000000000000000000010000000200000000fffffffe000000000000000200000000fffffffe00000001"));
1356 const NEGATIVE_PRIME_INV_MOD_R: U384 = U384::from_48_be_bytes_panicking(&hex_lit::hex!(
1357 "00000014000000140000000c00000002fffffffcfffffffafffffffbfffffffe00000000000000010000000100000001"));
1364 /// Read some bytes and use them to test bigint math by comparing results against the `ibig` crate.
1365 pub fn fuzz_math(input: &[u8]) {
1366 if input.len() < 32 || input.len() % 16 != 0 { return; }
1367 let split = core::cmp::min(input.len() / 2, 512);
1368 let (a, b) = input.split_at(core::cmp::min(input.len() / 2, 512));
1369 let b = &b[..split];
1371 let ai = ibig::UBig::from_be_bytes(&a);
1372 let bi = ibig::UBig::from_be_bytes(&b);
1374 let mut a_u64s = Vec::with_capacity(split / 8);
1375 for chunk in a.chunks(8) {
1376 a_u64s.push(u64::from_be_bytes(chunk.try_into().unwrap()));
1378 let mut b_u64s = Vec::with_capacity(split / 8);
1379 for chunk in b.chunks(8) {
1380 b_u64s.push(u64::from_be_bytes(chunk.try_into().unwrap()));
1383 macro_rules! test { ($mul: ident, $sqr: ident, $add: ident, $sub: ident, $div_rem: ident, $mod_inv: ident) => {
1384 let res = $mul(&a_u64s, &b_u64s);
1385 let mut res_bytes = Vec::with_capacity(input.len() / 2);
1387 res_bytes.extend_from_slice(&i.to_be_bytes());
1389 assert_eq!(ibig::UBig::from_be_bytes(&res_bytes), ai.clone() * bi.clone());
1391 debug_assert_eq!($mul(&a_u64s, &a_u64s), $sqr(&a_u64s));
1392 debug_assert_eq!($mul(&b_u64s, &b_u64s), $sqr(&b_u64s));
1394 let (res, carry) = $add(&a_u64s, &b_u64s);
1395 let mut res_bytes = Vec::with_capacity(input.len() / 2 + 1);
1396 if carry { res_bytes.push(1); } else { res_bytes.push(0); }
1398 res_bytes.extend_from_slice(&i.to_be_bytes());
1400 assert_eq!(ibig::UBig::from_be_bytes(&res_bytes), ai.clone() + bi.clone());
1402 let mut add_u64s = a_u64s.clone();
1403 let carry = add_u64!(add_u64s, 1);
1404 let mut res_bytes = Vec::with_capacity(input.len() / 2 + 1);
1405 if carry { res_bytes.push(1); } else { res_bytes.push(0); }
1406 for i in &add_u64s {
1407 res_bytes.extend_from_slice(&i.to_be_bytes());
1409 assert_eq!(ibig::UBig::from_be_bytes(&res_bytes), ai.clone() + 1);
1411 let mut double_u64s = b_u64s.clone();
1412 let carry = double!(double_u64s);
1413 let mut res_bytes = Vec::with_capacity(input.len() / 2 + 1);
1414 if carry { res_bytes.push(1); } else { res_bytes.push(0); }
1415 for i in &double_u64s {
1416 res_bytes.extend_from_slice(&i.to_be_bytes());
1418 assert_eq!(ibig::UBig::from_be_bytes(&res_bytes), bi.clone() * 2);
1420 let (quot, rem) = if let Ok(res) =
1421 $div_rem(&a_u64s[..].try_into().unwrap(), &b_u64s[..].try_into().unwrap()) {
1424 let mut quot_bytes = Vec::with_capacity(input.len() / 2);
1426 quot_bytes.extend_from_slice(&i.to_be_bytes());
1428 let mut rem_bytes = Vec::with_capacity(input.len() / 2);
1430 rem_bytes.extend_from_slice(&i.to_be_bytes());
1432 let (quoti, remi) = ibig::ops::DivRem::div_rem(ai.clone(), &bi);
1433 assert_eq!(ibig::UBig::from_be_bytes("_bytes), quoti);
1434 assert_eq!(ibig::UBig::from_be_bytes(&rem_bytes), remi);
1436 if ai != ibig::UBig::from(0u32) { // ibig provides a spurious modular inverse for 0
1437 let ring = ibig::modular::ModuloRing::new(&bi);
1438 let ar = ring.from(ai.clone());
1439 let invi = ar.inverse().map(|i| i.residue());
1441 if let Ok(modinv) = $mod_inv(&a_u64s[..].try_into().unwrap(), &b_u64s[..].try_into().unwrap()) {
1442 let mut modinv_bytes = Vec::with_capacity(input.len() / 2);
1444 modinv_bytes.extend_from_slice(&i.to_be_bytes());
1446 assert_eq!(invi.unwrap(), ibig::UBig::from_be_bytes(&modinv_bytes));
1448 assert!(invi.is_none());
1453 macro_rules! test_mod { ($amodp: expr, $bmodp: expr, $PRIME: expr, $len: expr, $into: ident, $div_rem_double: ident, $div_rem: ident, $mul: ident, $add: ident, $sub: ident) => {
1454 // Test the U256/U384Mod wrapper, which operates in Montgomery representation
1455 let mut p_extended = [0; $len * 2];
1456 p_extended[$len..].copy_from_slice(&$PRIME);
1458 let amodp_squared = $div_rem_double(&$mul(&a_u64s, &a_u64s), &p_extended).unwrap().1;
1459 assert_eq!(&amodp_squared[..$len], &[0; $len]);
1460 assert_eq!(&$amodp.square().$into().0, &amodp_squared[$len..]);
1462 let abmodp = $div_rem_double(&$mul(&a_u64s, &b_u64s), &p_extended).unwrap().1;
1463 assert_eq!(&abmodp[..$len], &[0; $len]);
1464 assert_eq!(&$amodp.mul(&$bmodp).$into().0, &abmodp[$len..]);
1466 let (aplusb, aplusb_overflow) = $add(&a_u64s, &b_u64s);
1467 let mut aplusb_extended = [0; $len * 2];
1468 aplusb_extended[$len..].copy_from_slice(&aplusb);
1469 if aplusb_overflow { aplusb_extended[$len - 1] = 1; }
1470 let aplusbmodp = $div_rem_double(&aplusb_extended, &p_extended).unwrap().1;
1471 assert_eq!(&aplusbmodp[..$len], &[0; $len]);
1472 assert_eq!(&$amodp.add(&$bmodp).$into().0, &aplusbmodp[$len..]);
1474 let (mut aminusb, aminusb_underflow) = $sub(&a_u64s, &b_u64s);
1475 if aminusb_underflow {
1477 (aminusb, overflow) = $add(&aminusb, &$PRIME);
1479 (aminusb, overflow) = $add(&aminusb, &$PRIME);
1483 let aminusbmodp = $div_rem(&aminusb, &$PRIME).unwrap().1;
1484 assert_eq!(&$amodp.sub(&$bmodp).$into().0, &aminusbmodp);
1487 if a_u64s.len() == 2 {
1488 test!(mul_2, sqr_2, add_2, sub_2, div_rem_2, mod_inv_2);
1489 } else if a_u64s.len() == 4 {
1490 test!(mul_4, sqr_4, add_4, sub_4, div_rem_4, mod_inv_4);
1491 let amodp = U256Mod::<fuzz_moduli::P256>::from_u256(U256(a_u64s[..].try_into().unwrap()));
1492 let bmodp = U256Mod::<fuzz_moduli::P256>::from_u256(U256(b_u64s[..].try_into().unwrap()));
1493 test_mod!(amodp, bmodp, fuzz_moduli::P256::PRIME.0, 4, into_u256, div_rem_8, div_rem_4, mul_4, add_4, sub_4);
1494 } else if a_u64s.len() == 6 {
1495 test!(mul_6, sqr_6, add_6, sub_6, div_rem_6, mod_inv_6);
1496 let amodp = U384Mod::<fuzz_moduli::P384>::from_u384(U384(a_u64s[..].try_into().unwrap()));
1497 let bmodp = U384Mod::<fuzz_moduli::P384>::from_u384(U384(b_u64s[..].try_into().unwrap()));
1498 test_mod!(amodp, bmodp, fuzz_moduli::P384::PRIME.0, 6, into_u384, div_rem_12, div_rem_6, mul_6, add_6, sub_6);
1499 } else if a_u64s.len() == 8 {
1500 test!(mul_8, sqr_8, add_8, sub_8, div_rem_8, mod_inv_8);
1501 } else if input.len() == 512*2 + 4 {
1502 let mut e_bytes = [0; 4];
1503 e_bytes.copy_from_slice(&input[512 * 2..512 * 2 + 4]);
1504 let e = u32::from_le_bytes(e_bytes);
1505 let a = U4096::from_be_bytes(&a).unwrap();
1506 let b = U4096::from_be_bytes(&b).unwrap();
1508 let res = if let Ok(r) = a.expmod_odd_mod(e, &b) { r } else { return };
1509 let mut res_bytes = Vec::with_capacity(512);
1511 res_bytes.extend_from_slice(&i.to_be_bytes());
1514 let ring = ibig::modular::ModuloRing::new(&bi);
1515 let ar = ring.from(ai.clone());
1516 assert_eq!(ar.pow(&e.into()).residue(), ibig::UBig::from_be_bytes(&res_bytes));
1524 fn u64s_to_u128(v: [u64; 2]) -> u128 {
1527 r |= (v[0] as u128) << 64;
1531 fn u64s_to_i128(v: [u64; 2]) -> i128 {
1534 r |= (v[0] as i128) << 64;
1540 let mut zero = [0u64; 2];
1542 assert_eq!(zero, [0; 2]);
1544 let mut one = [0u64, 1u64];
1546 assert_eq!(u64s_to_i128(one), -1);
1548 let mut minus_one: [u64; 2] = [u64::MAX, u64::MAX];
1550 assert_eq!(minus_one, [0, 1]);
1555 let mut zero = [0u64; 2];
1556 assert!(!double!(zero));
1557 assert_eq!(zero, [0; 2]);
1559 let mut one = [0u64, 1u64];
1560 assert!(!double!(one));
1561 assert_eq!(one, [0, 2]);
1563 let mut u64_max = [0, u64::MAX];
1564 assert!(!double!(u64_max));
1565 assert_eq!(u64_max, [1, u64::MAX - 1]);
1567 let mut u64_carry_overflow = [0x7fff_ffff_ffff_ffffu64, 0x8000_0000_0000_0000];
1568 assert!(!double!(u64_carry_overflow));
1569 assert_eq!(u64_carry_overflow, [u64::MAX, 0]);
1571 let mut max = [u64::MAX; 4];
1572 assert!(double!(max));
1573 assert_eq!(max, [u64::MAX, u64::MAX, u64::MAX, u64::MAX - 1]);
1577 fn mul_min_simple_tests() {
1580 let res = mul_2(&a, &b);
1581 assert_eq!(res, [0, 3, 10, 8]);
1583 let a = [0x1bad_cafe_dead_beef, 2424];
1584 let b = [0x2bad_beef_dead_cafe, 4242];
1585 let res = mul_2(&a, &b);
1586 assert_eq!(res, [340296855556511776, 15015369169016130186, 4248480538569992542, 10282608]);
1588 let a = [0xf6d9_f8eb_8b60_7a6d, 0x4b93_833e_2194_fc2e];
1589 let b = [0xfdab_0000_6952_8ab4, 0xd302_0000_8282_0000];
1590 let res = mul_2(&a, &b);
1591 assert_eq!(res, [17625486516939878681, 18390748118453258282, 2695286104209847530, 1510594524414214144]);
1593 let a = [0x8b8b_8b8b_8b8b_8b8b, 0x8b8b_8b8b_8b8b_8b8b];
1594 let b = [0x8b8b_8b8b_8b8b_8b8b, 0x8b8b_8b8b_8b8b_8b8b];
1595 let res = mul_2(&a, &b);
1596 assert_eq!(res, [5481115605507762349, 8230042173354675923, 16737530186064798, 15714555036048702841]);
1598 let a = [0x0000_0000_0000_0020, 0x002d_362c_005b_7753];
1599 let b = [0x0900_0000_0030_0003, 0xb708_00fe_0000_00cd];
1600 let res = mul_2(&a, &b);
1601 assert_eq!(res, [1, 2306290405521702946, 17647397529888728169, 10271802099389861239]);
1603 let a = [0x0000_0000_7fff_ffff, 0xffff_ffff_0000_0000];
1604 let b = [0x0000_0800_0000_0000, 0x0000_1000_0000_00e1];
1605 let res = mul_2(&a, &b);
1606 assert_eq!(res, [1024, 0, 483183816703, 18446743107341910016]);
1608 let a = [0xf6d9_f8eb_ebeb_eb6d, 0x4b93_83a0_bb35_0680];
1609 let b = [0xfd02_b9b9_b9b9_b9b9, 0xb9b9_b9b9_b9b9_b9b9];
1610 let res = mul_2(&a, &b);
1611 assert_eq!(res, [17579814114991930107, 15033987447865175985, 488855932380801351, 5453318140933190272]);
1613 let a = [u64::MAX; 2];
1614 let b = [u64::MAX; 2];
1615 let res = mul_2(&a, &b);
1616 assert_eq!(res, [18446744073709551615, 18446744073709551614, 0, 1]);
1621 fn test(a: [u64; 2], b: [u64; 2]) {
1622 let a_int = u64s_to_u128(a);
1623 let b_int = u64s_to_u128(b);
1625 let res = add_2(&a, &b);
1626 assert_eq!((u64s_to_u128(res.0), res.1), a_int.overflowing_add(b_int));
1628 let res = sub_2(&a, &b);
1629 assert_eq!((u64s_to_u128(res.0), res.1), a_int.overflowing_sub(b_int));
1632 test([0; 2], [0; 2]);
1633 test([0x1bad_cafe_dead_beef, 2424], [0x2bad_cafe_dead_cafe, 4242]);
1634 test([u64::MAX; 2], [u64::MAX; 2]);
1635 test([u64::MAX, 0x8000_0000_0000_0000], [0, 0x7fff_ffff_ffff_ffff]);
1636 test([0, 0x7fff_ffff_ffff_ffff], [u64::MAX, 0x8000_0000_0000_0000]);
1637 test([u64::MAX, 0], [0, u64::MAX]);
1638 test([0, u64::MAX], [u64::MAX, 0]);
1639 test([u64::MAX; 2], [0; 2]);
1640 test([0; 2], [u64::MAX; 2]);
1644 fn mul_4_simple_tests() {
1647 assert_eq!(mul_4(&a, &b),
1648 [0, 2, 4, 6, 8, 6, 4, 2]);
1650 let a = [0x1bad_cafe_dead_beef, 2424, 0x1bad_cafe_dead_beef, 2424];
1651 let b = [0x2bad_beef_dead_cafe, 4242, 0x2bad_beef_dead_cafe, 4242];
1652 assert_eq!(mul_4(&a, &b),
1653 [340296855556511776, 15015369169016130186, 4929074249683016095, 11583994264332991364,
1654 8837257932696496860, 15015369169036695402, 4248480538569992542, 10282608]);
1656 let a = [u64::MAX; 4];
1657 let b = [u64::MAX; 4];
1658 assert_eq!(mul_4(&a, &b),
1659 [18446744073709551615, 18446744073709551615, 18446744073709551615,
1660 18446744073709551614, 0, 0, 0, 1]);
1664 fn double_simple_tests() {
1665 let mut a = [0xfff5_b32d_01ff_0000, 0x00e7_e7e7_e7e7_e7e7];
1666 assert!(double!(a));
1667 assert_eq!(a, [18440945635998695424, 130551405668716494]);
1669 let mut a = [u64::MAX, u64::MAX];
1670 assert!(double!(a));
1671 assert_eq!(a, [18446744073709551615, 18446744073709551614]);