結果
問題 | No.8030 ミラー・ラビン素数判定法のテスト |
ユーザー | 👑 Mizar |
提出日時 | 2022-09-05 00:02:16 |
言語 | Rust (1.77.0 + proconio) |
結果 |
AC
|
実行時間 | 15 ms / 9,973 ms |
コード長 | 42,543 bytes |
コンパイル時間 | 13,530 ms |
コンパイル使用メモリ | 385,740 KB |
実行使用メモリ | 6,820 KB |
最終ジャッジ日時 | 2024-11-19 07:47:59 |
合計ジャッジ時間 | 14,868 ms |
ジャッジサーバーID (参考情報) |
judge3 / judge2 |
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テストケース
テストケース表示入力 | 結果 | 実行時間 実行使用メモリ |
---|---|---|
testcase_00 | AC | 1 ms
6,816 KB |
testcase_01 | AC | 1 ms
6,820 KB |
testcase_02 | AC | 1 ms
6,816 KB |
testcase_03 | AC | 1 ms
6,816 KB |
testcase_04 | AC | 11 ms
6,816 KB |
testcase_05 | AC | 12 ms
6,816 KB |
testcase_06 | AC | 10 ms
6,816 KB |
testcase_07 | AC | 10 ms
6,816 KB |
testcase_08 | AC | 10 ms
6,816 KB |
testcase_09 | AC | 15 ms
6,820 KB |
ソースコード
// -*- coding:utf-8-unix -*- use core::num::{NonZeroU128,NonZeroU64}; // 除数をNonZeroU型にした除算/剰余算の選択 macro_rules! cond_nzdiv { ($nzdiv:expr, $no_nzdiv:expr) => { $nzdiv // 1.51.0以降 feature = "nonzero_div" //$no_nzdiv // 1.50.0以前 } } // NonZeroU64/NonderoU128型を使用するか const USE_NZU64: bool = true; // モンゴメリ剰余乗算 (Montgomery modular multiplication) trait MontVal<T, BitCountType=u32> { fn n(&self) -> T; fn ni(&self) -> T; fn nh(&self) -> T; fn r(&self) -> T; fn rn(&self) -> T; fn r2(&self) -> T; fn d(&self) -> T; fn k(&self) -> BitCountType; } trait MontOps<T, BitCountType=u32> { fn add(&self, a: T, b: T) -> T; fn sub(&self, a: T, b: T) -> T; fn div2(&self, ar: T) -> T; fn mrmul(&self, ar: T, br: T) -> T; fn mr(&self, ar: T) -> T; fn ar(&self, a: T) -> T; fn pow(&self, ar: T, b: T) -> T; fn powodd(&self, ar: T, b: T) -> T; } trait MontModN<T, BitCountType=u32> { fn modn(&self, x: T) -> T; } trait MontNew<T, U=T, BitCountType=u32> { fn new(n: T) -> Self; } struct Mont<T, BitCountType=u32> { n: T, // n is odd, and n > 2 ni: T, // n * ni == 1 (mod 2**64) nh: T, // == (n + 1) / 2 r: T, // == 2**64 (mod n) rn: T, // == -(2**64) (mod n) r2: T, // == 2**128 (mod n) d: T, // == (n - 1) >> (n - 1).trailing_zeros() k: BitCountType, // == (n - 1).trailing_zeros() } impl MontVal<u64> for Mont<NonZeroU64> { #[inline] fn n(&self) -> u64 { self.n.get() } #[inline] fn ni(&self) -> u64 { self.ni.get() } #[inline] fn nh(&self) -> u64 { self.nh.get() } #[inline] fn r(&self) -> u64 { self.r.get() } #[inline] fn rn(&self) -> u64 { self.rn.get() } #[inline] fn r2(&self) -> u64 { self.r2.get() } #[inline] fn d(&self) -> u64 { self.d.get() } #[inline] fn k(&self) -> u32 { self.k } } impl MontVal<u64> for Mont<u64> { #[inline] fn n(&self) -> u64 { self.n } #[inline] fn ni(&self) -> u64 { self.ni } #[inline] fn nh(&self) -> u64 { self.nh } #[inline] fn r(&self) -> u64 { self.r } #[inline] fn rn(&self) -> u64 { self.rn } #[inline] fn r2(&self) -> u64 { self.r2 } #[inline] fn d(&self) -> u64 { self.d } #[inline] fn k(&self) -> u32 { self.k } } macro_rules! mont_ops_u64 { ($x:ty) => { impl MontOps<u64> for $x { #[inline] fn add(&self, a: u64, b: u64) -> u64 { // == a + b (mod n) debug_assert!(a < self.n()); debug_assert!(b < self.n()); let (t, fa) = a.overflowing_add(b); let (u, fs) = t.overflowing_sub(self.n()); if fa { u } else { if fs { t } else { u } } } #[inline] fn sub(&self, a: u64, b: u64) -> u64 { // == a - b (mod n) debug_assert!(a < self.n()); debug_assert!(b < self.n()); let (t, f) = a.overflowing_sub(b); if f { t.wrapping_add(self.n()) } else { t } } #[inline] fn div2(&self, ar: u64) -> u64 { // == ar / 2 (mod n) debug_assert!(ar < self.n()); let t = ar >> 1; if (ar & 1) == 0 { t } else { t + self.nh() } } #[inline] fn mrmul(&self, ar: u64, br: u64) -> u64 { // == (ar * br) / r (mod n) debug_assert!(ar < self.n()); debug_assert!(br < self.n()); let (n, ni) = (self.n(), self.ni()); let t: u128 = (ar as u128) * (br as u128); let (t, f) = ((t >> 64) as u64).overflowing_sub((((((t as u64).wrapping_mul(ni)) as u128) * (n as u128)) >> 64) as u64); if f { t.wrapping_add(n) } else { t } } #[inline] fn mr(&self, ar: u64) -> u64 { // == ar / r (mod n) debug_assert!(ar < self.n()); let (n, ni) = (self.n(), self.ni()); let (t, f) = (((((ar.wrapping_mul(ni)) as u128) * (n as u128)) >> 64) as u64).overflowing_neg(); if f { t.wrapping_add(n) } else { t } } #[inline] fn ar(&self, a: u64) -> u64 { // == a * r (mod n) debug_assert!(a < self.n()); self.mrmul(a, self.r2()) } #[inline] fn pow(&self, mut ar: u64, mut b: u64) -> u64 { // == ((ar / r) ** b) * r (mod n) debug_assert!(ar < self.n()); let mut t = if (b & 1) == 0 { self.r() } else { ar }; loop { b >>= 1; if b == 0 { return t; } ar = self.mrmul(ar, ar); if (b & 1) != 0 { t = self.mrmul(t, ar); } } } #[inline] fn powodd(&self, mut ar: u64, mut b: u64) -> u64 { // == ((ar / r) ** b) * r (mod n) debug_assert!(ar < self.n()); debug_assert_eq!(b & 1, 1); // b is odd let mut t = ar; loop { b >>= 1; if b == 0 { return t; } ar = self.mrmul(ar, ar); if (b & 1) != 0 { t = self.mrmul(t, ar); } } } }}} mont_ops_u64!(Mont<NonZeroU64>); mont_ops_u64!(Mont<u64>); impl MontModN<u64> for Mont<NonZeroU64> { #[inline] fn modn(&self, x: u64) -> u64 { cond_nzdiv!(x % self.n, x % self.n()) } } impl MontModN<u64> for Mont<u64> { #[inline] fn modn(&self, x: u64) -> u64 { x % self.n() } } impl MontNew<NonZeroU64, u64> for Mont<NonZeroU64> { #[inline] fn new(n: NonZeroU64) -> Self { debug_assert_eq!(n.get() & 1, 1); // // n is odd number, n = 2*k+1, n >= 1, n < 2**64, k is non-negative integer, k >= 0, k < 2**63 // ni0 := n; // = 2*k+1 = (1+(2**3)*((k*(k+1)/2)**1))/(2*k+1) let mut ni = n.get(); // ni1 := ni0 * (2 - (n * ni0)); // = (1-(2**6)*((k*(k+1)/2)**2))/(2*k+1) // ni2 := ni1 * (2 - (n * ni1)); // = (1-(2**12)*((k*(k+1)/2)**4))/(2*k+1) // ni3 := ni2 * (2 - (n * ni2)); // = (1-(2**24)*((k*(k+1)/2)**8))/(2*k+1) // ni4 := ni3 * (2 - (n * ni3)); // = (1-(2**48)*((k*(k+1)/2)**16))/(2*k+1) // ni5 := ni4 * (2 - (n * ni4)); // = (1-(2**96)*((k*(k+1)/2)**32))/(2*k+1) // // (n * ni5) mod 2**64 = ((2*k+1) * ni5) mod 2**64 = 1 mod 2**64 for _ in 0..5 { ni = ni.wrapping_mul(2u64.wrapping_sub(n.get().wrapping_mul(ni))); } debug_assert_eq!(n.get().wrapping_mul(ni), 1); // n * ni == 1 (mod 2**64) let nh = (n.get() >> 1) + 1; let r = cond_nzdiv!( 1u64.wrapping_neg() % n, 1u64.wrapping_neg() % n.get() ) + 1; // == 2**64 (mod n) let rn = n.get() - r; let r2 = cond_nzdiv!( 1u128.wrapping_neg() % NonZeroU128::from(n), 1u128.wrapping_neg() % NonZeroU128::from(n).get() ) as u64 + 1; // == 2**128 (mod n) let mut d = n.get() - 1; let k = d.trailing_zeros(); d >>= k; let ni = NonZeroU64::new(ni).unwrap(); let nh = NonZeroU64::new(nh).unwrap(); let r = NonZeroU64::new(r).unwrap(); let rn = NonZeroU64::new(rn).unwrap(); let r2 = NonZeroU64::new(r2).unwrap(); let d = NonZeroU64::new(d).unwrap(); debug_assert_eq!(Self { n, ni, nh, r, rn, r2, d, k }.mr(r.get()), 1); // r / r == 1 (mod n) debug_assert_eq!(Self { n, ni, nh, r, rn, r2, d, k }.mrmul(1, r2.get()), r.get()); // r2 / r == r (mod n) Self { n, ni, nh, r, rn, r2, d, k } } } impl MontNew<u64> for Mont<u64> { #[inline] fn new(n: u64) -> Self { debug_assert_eq!(n & 1, 1); // // n is odd number, n = 2*k+1, n >= 1, n < 2**64, k is non-negative integer, k >= 0, k < 2**63 // ni0 := n; // = 2*k+1 = (1+(2**3)*((k*(k+1)/2)**1))/(2*k+1) let mut ni = n; // ni1 := ni0 * (2 - (n * ni0)); // = (1-(2**6)*((k*(k+1)/2)**2))/(2*k+1) // ni2 := ni1 * (2 - (n * ni1)); // = (1-(2**12)*((k*(k+1)/2)**4))/(2*k+1) // ni3 := ni2 * (2 - (n * ni2)); // = (1-(2**24)*((k*(k+1)/2)**8))/(2*k+1) // ni4 := ni3 * (2 - (n * ni3)); // = (1-(2**48)*((k*(k+1)/2)**16))/(2*k+1) // ni5 := ni4 * (2 - (n * ni4)); // = (1-(2**96)*((k*(k+1)/2)**32))/(2*k+1) // // (n * ni5) mod 2**64 = ((2*k+1) * ni5) mod 2**64 = 1 mod 2**64 for _ in 0..5 { ni = ni.wrapping_mul(2u64.wrapping_sub(n.wrapping_mul(ni))); } debug_assert_eq!(n.wrapping_mul(ni), 1); // n * ni == 1 (mod 2**64) let nh = (n >> 1) + 1; let r = (1u64.wrapping_neg() % n) + 1; // == 2**64 (mod n) let rn = n - r; let r2 = ((1u128.wrapping_neg() % (n as u128)) as u64) + 1; // == 2**128 (mod n) let mut d = n - 1; let k = d.trailing_zeros(); d >>= k; debug_assert_eq!(Self { n, ni, nh, r, rn, r2, d, k }.mr(r), 1); // r / r == 1 (mod n) debug_assert_eq!(Self { n, ni, nh, r, rn, r2, d, k }.mrmul(1, r2), r); // r2 / r == r (mod n) Self { n, ni, nh, r, rn, r2, d, k } } } #[inline] // ニュートン法による整数平方根 fn isqrt64n(x: u64) -> u64 { if x <= 1 { return x; } let k = 32 - ((x - 1).leading_zeros() >> 1); let mut s = 1u64 << k; // s = 2**k let mut t = (s + (x >> k)) >> 1; // t = (s + x/s)/2 // whileループ回数=除算回数は最大6回を想定 // s > floor(sqrt(x)) -> floor(sqrt(x)) <= t < s // s == floor(sqrt(x)) -> floor(sqrt(x)) <= t <= floor(sqrt(x)) + 1 while t < s { s = t; t = (s + (x / s)) >> 1; } s } #[inline] // 平方数判定 (true: 平方数である, false: 平方数ではない) fn issq_u64_n(x: u64) -> bool { let sqrt = isqrt64n(x); sqrt * sqrt == x } #[inline] // 擬平方数判定 mod64 12/64 0.1875 (true: 平方数かもしれない, false: 平方数ではない) fn issq_u64_mod64(x: u64) -> bool { (0x202021202030213u64 >> (x & 63)) & 1 == 1 } #[inline] // 擬平方数判定 mod4095 336/4095 0.0821 (true: 平方数かもしれない, false: 平方数ではない) fn issq_u64_mod4095(x: u64) -> bool { const SQTABLE_MOD4095: [u64; 64] = [0x2001002010213,0x4200001008028001,0x20000010004,0x80200082010,0x1800008200044029,0x120080000010,0x2200000080410400,0x8100041000200800,0x800004000020100,0x402000400082201,0x9004000040,0x800002000880,0x18002000012000,0x801208,0x26100000804010,0x80000080000002,0x108040040101045,0x20c00004000102,0x400000100c0010,0x1300000040208,0x804000020010000,0x1008402002400080,0x201001000200040,0x4402000000806000,0x10402000000,0x1040008001200801,0x4080000000020400,0x10083080000002,0x8220140000040000,0x800084020100000,0x80010400010000,0x1200020108008060,0x180000000,0x400002400000018,0x4241000200,0x100800000000,0x10201008400483,0xc008000208201000,0x800420000100,0x2010002000410,0x28041000000,0x4010080000024,0x400480010010080,0x200040028000008,0x100810084020,0x20c0401000080000,0x1000240000220000,0x4000020800,0x410000000480000,0x8004008000804201,0x806020000104000,0x2080002000211000,0x1001008001000,0x20000010024000,0x480200002040000,0x48200044008000,0x100000000010080,0x80090400042,0x41040200800200,0x4000020100110,0x2000400082200010,0x1008200000000040,0x2004800002,0x2002010000080]; let p = (x % 4095) as usize; (SQTABLE_MOD4095[p >> 6] >> (p & 63)) & 1 == 1 } // 平方数判定 (true: 平方数である, false: 平方数ではない) #[inline] fn issq_u64(x: u64) -> bool { issq_u64_mod64(x) && issq_u64_mod4095(x) && issq_u64_n(x) } // Jacobi symbol: ヤコビ記号 #[inline] fn jacobi(a: i64, mut n: u64) -> i32 { if n == 0 { return if a == 1 || a == -1 { 1 } else { 0 }; } let (mut a, mut j): (u64, i32) = if a >= 0 { (a as u64, 1) } else if (n & 3) == 3 { ((-a) as u64, -1) } else { ((-a) as u64, 1) }; while a != 0 { let ba = a.trailing_zeros(); a >>= ba; if ((n & 7) == 3 || (n & 7) == 5) && (ba & 1) != 0 { j = -j; } if (a & n & 3) == 3 { j = -j; } std::mem::swap(&mut n, &mut a); a %= n; if a > (n >> 1) { a = n - a; if (n & 3) == 3 { j = -j; } } } if n == 1 { j } else { 0 } } trait PrimeTestU64Trait<T> : MontOps<u64> + MontVal<u64> + MontModN<u64> { #[inline] fn primetest_base2(&self) -> bool { // Mirrer-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... let (r, rn, d, k) = (self.r(), self.rn(), self.d(), self.k()); let mut br = self.powodd(self.add(self.r(), self.r()), d); if br == r || br == rn { return true; } for _ in 1..k { br = self.mrmul(br, br); if br == rn { return true; } } false } #[inline] fn primetest_miller(&self, mut base: u64) -> bool { // Miller-Rabin primality test let (n, r, rn, d, k) = (self.n(), self.r(), self.rn(), self.d(), self.k()); if base > n { base = self.modn(base); if base == 0 { return true; } } let mut tr = self.pow(self.ar(base), d); if tr == r || tr == rn { return true; } for _ in 1..k { tr = self.mrmul(tr, tr); if tr == rn { return true; } } false } #[inline] fn primetest_lucas(&self) -> bool { // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... let n = self.n(); let mut d: i64 = 5; for i in 0u32.. { debug_assert!(i < 64); match jacobi(d, n) { -1 => break, 0 => if ((d.abs()) as u64) < n { return false; }, _ => {}, } if i == 8 && issq_u64(n) { return false; } if (i & 1) == 1 { d = 2 - d; } else { d = -(d + 2); } } let qm = self.ar(self.modn(if d < 0 {((1 - d) as u64) / 4} else {n - ((d - 1) as u64) / 4})); let mut k = (n + 1) << (n + 1).leading_zeros(); let mut um = self.r(); let mut vm = self.r(); let mut qn = qm; let dm: u64 = self.ar(if d < 0 { n - self.modn((-d) as u64) } else { self.modn(d as u64) }); k <<= 1; while k != 0 { um = self.mrmul(um, vm); vm = self.sub(self.mrmul(vm, vm), self.add(qn, qn)); qn = self.mrmul(qn, qn); if (k >> 63) != 0 { let uu = self.div2(self.add(um, vm)); vm = self.div2(self.add(self.mrmul(dm, um), vm)); um = uu; qn = self.mrmul(qn, qm); } k <<= 1; } if um == 0 || vm == 0 { return true; } let mut x = (n + 1) & (!n); x >>= 1; while x != 0 { um = self.mrmul(um, vm); vm = self.sub(self.mrmul(vm, vm), self.add(qn, qn)); if vm == 0 { return true; } qn = self.mrmul(qn, qn); x >>= 1; } false } #[inline] fn primetest_bpsw(&self) -> bool { // Mirrer-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... self.primetest_base2() && // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... self.primetest_lucas() } } impl PrimeTestU64Trait<NonZeroU64> for Mont<NonZeroU64> {} impl PrimeTestU64Trait<u64> for Mont<u64> {} // Deterministic variants of the Miller-Rabin primality test http://miller-rabin.appspot.com/ trait MillerU64<T> : PrimeTestU64Trait<T> { const MILLER_U64_1BASES: [u64; 1] = [ 9345883071009581737, ]; const MILLER_U64_2BASES: [u64; 2] = [ 336781006125, 9639812373923155, ]; const MILLER_U64_3BASES: [u64; 3] = [ 4230279247111683200, 14694767155120705706, 16641139526367750375, ]; const MILLER_U64_4BASES: [u64; 4] = [ 2, 141889084524735, 1199124725622454117, 11096072698276303650, ]; const MILLER_U64_5BASES: [u64; 5] = [ 2, 4130806001517, 149795463772692060, 186635894390467037, 3967304179347715805, ]; const MILLER_U64_6BASES: [u64; 6] = [ 2, 123635709730000, 9233062284813009, 43835965440333360, 761179012939631437, 1263739024124850375, ]; const MILLER_U64_7BASES: [u64; 7] = [ 2, 325, 9375, 28178, 450775, 9780504, 1795265022, ]; #[inline] fn primetest_miller_1bases(&self) -> bool { debug_assert!(self.n() < 341531); Self::MILLER_U64_1BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_2bases(&self) -> bool { debug_assert!(self.n() < 1050535501); Self::MILLER_U64_2BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_3bases(&self) -> bool { debug_assert!(self.n() < 350269456337); Self::MILLER_U64_3BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_4bases(&self) -> bool { debug_assert!(self.n() < 55245642489451); Self::MILLER_U64_4BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_5bases(&self) -> bool { debug_assert!(self.n() < 7999252175582851); Self::MILLER_U64_5BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_6bases(&self) -> bool { debug_assert!(self.n() < 585226005592931977); Self::MILLER_U64_6BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_7bases(&self) -> bool { Self::MILLER_U64_7BASES.iter().all(|&base| self.primetest_miller(base)) } #[inline] fn primetest_miller_mbases(&self) -> bool { // Deterministic variants of the Miller-Rabin primality test http://miller-rabin.appspot.com/ match self.n() { 0..=341530 => Self::MILLER_U64_1BASES.iter(), 0..=1050535500 => Self::MILLER_U64_2BASES.iter(), 0..=350269456336 => Self::MILLER_U64_3BASES.iter(), 0..=55245642489450 => Self::MILLER_U64_4BASES.iter(), 0..=7999252175582850 => Self::MILLER_U64_5BASES.iter(), 0..=585226005592931976 => Self::MILLER_U64_6BASES.iter(), _ => Self::MILLER_U64_7BASES.iter(), }.all(|&base| self.primetest_miller(base)) } } impl MillerU64<NonZeroU64> for Mont<NonZeroU64> {} impl MillerU64<u64> for Mont<u64> {} // Baillie–PSW primarity test (nonzerou64) pub fn primetest_nzu64_bpsw(n: NonZeroU64) -> bool { if n.get() == 2 { return true; } if n.get() == 1 || (n.get() & 1) == 0 { return false; } let mont = Mont::<NonZeroU64>::new(n); mont.primetest_bpsw() } // Baillie–PSW primarity test (u64) pub fn primetest_u64_bpsw(n: u64) -> bool { if n == 2 { return true; } if n == 1 || (n & 1) == 0 { return false; } let mont = Mont::<u64>::new(n); mont.primetest_bpsw() } // Miller-Rabin primarity test (nonzerou64) pub fn primetest_nzu64_miller_mbases(n: NonZeroU64) -> bool { if n.get() == 2 { return true; } if n.get() == 1 || (n.get() & 1) == 0 { return false; } let mont = Mont::<NonZeroU64>::new(n); mont.primetest_miller_mbases() } // Miller-Rabin primarity test (u64) pub fn primetest_u64_miller_mbases(n: u64) -> bool { if n == 2 { return true; } if n == 1 || (n & 1) == 0 { return false; } let mont = Mont::<u64>::new(n); mont.primetest_miller_mbases() } #[cfg(test)] mod tests { use crate::*; // ten least k's for which (2**n)-k is prime // https://primes.utm.edu/lists/2small/0bit.html const PRIMES_LAST10: [(u32,[u64;10]);57] = [ (8,[5,15,17,23,27,29,33,45,57,59]), (9,[3,9,13,21,25,33,45,49,51,55]), (10,[3,5,11,15,27,33,41,47,53,57]), (11,[9,19,21,31,37,45,49,51,55,61]), (12,[3,5,17,23,39,45,47,69,75,77]), (13,[1,13,21,25,31,45,69,75,81,91]), (14,[3,15,21,23,35,45,51,65,83,111]), (15,[19,49,51,55,61,75,81,115,121,135]), (16,[15,17,39,57,87,89,99,113,117,123]), (17,[1,9,13,31,49,61,63,85,91,99]), (18,[5,11,17,23,33,35,41,65,75,93]), (19,[1,19,27,31,45,57,67,69,85,87]), (20,[3,5,17,27,59,69,129,143,153,185]), (21,[9,19,21,55,61,69,105,111,121,129]), (22,[3,17,27,33,57,87,105,113,117,123]), (23,[15,21,27,37,61,69,135,147,157,159]), (24,[3,17,33,63,75,77,89,95,117,167]), (25,[39,49,61,85,91,115,141,159,165,183]), (26,[5,27,45,87,101,107,111,117,125,135]), (27,[39,79,111,115,135,187,199,219,231,235]), (28,[57,89,95,119,125,143,165,183,213,273]), (29,[3,33,43,63,73,75,93,99,121,133]), (30,[35,41,83,101,105,107,135,153,161,173]), (31,[1,19,61,69,85,99,105,151,159,171]), (32,[5,17,65,99,107,135,153,185,209,267]), (33,[9,25,49,79,105,285,301,303,321,355]), (34,[41,77,113,131,143,165,185,207,227,281]), (35,[31,49,61,69,79,121,141,247,309,325]), (36,[5,17,23,65,117,137,159,173,189,233]), (37,[25,31,45,69,123,141,199,201,351,375]), (38,[45,87,107,131,153,185,191,227,231,257]), (39,[7,19,67,91,135,165,219,231,241,301]), (40,[87,167,195,203,213,285,293,299,389,437]), (41,[21,31,55,63,73,75,91,111,133,139]), (42,[11,17,33,53,65,143,161,165,215,227]), (43,[57,67,117,175,255,267,291,309,319,369]), (44,[17,117,119,129,143,149,287,327,359,377]), (45,[55,69,81,93,121,133,139,159,193,229]), (46,[21,57,63,77,167,197,237,287,305,311]), (47,[115,127,147,279,297,339,435,541,619,649]), (48,[59,65,89,93,147,165,189,233,243,257]), (49,[81,111,123,139,181,201,213,265,283,339]), (50,[27,35,51,71,113,117,131,161,195,233]), (51,[129,139,165,231,237,247,355,391,397,439]), (52,[47,143,173,183,197,209,269,285,335,395]), (53,[111,145,231,265,315,339,343,369,379,421]), (54,[33,53,131,165,195,245,255,257,315,327]), (55,[55,67,99,127,147,169,171,199,207,267]), (56,[5,27,47,57,89,93,147,177,189,195]), (57,[13,25,49,61,69,111,195,273,363,423]), (58,[27,57,63,137,141,147,161,203,213,251]), (59,[55,99,225,427,517,607,649,687,861,871]), (60,[93,107,173,179,257,279,369,395,399,453]), (61,[1,31,45,229,259,283,339,391,403,465]), (62,[57,87,117,143,153,167,171,195,203,273]), (63,[25,165,259,301,375,387,391,409,457,471]), (64,[59,83,95,179,189,257,279,323,353,363]), ]; #[test] fn test_nzu64_nbits() { // ten least k's for which (2**n)-k is prime // https://primes.utm.edu/lists/2small/0bit.html let start_time = std::time::Instant::now(); for (bit_ref, kvec) in PRIMES_LAST10.iter() { let bit = *bit_ref; let &lastk = kvec.iter().last().unwrap(); for k in 1..=lastk { if (k & 1) == 0 { continue; } let n = (if bit < 64 { 1u64 << bit } else { 0u64 }).wrapping_sub(k); let n = NonZeroU64::new(n).unwrap(); let f = kvec.binary_search(&k).is_ok(); let mont = Mont::<NonZeroU64>::new(n); // Deterministic variants of the Miller-Rabin primality test // http://miller-rabin.appspot.com/ // n < 341531 : 1 base Miller-Rabin primality test if n.get() < 341531 { assert_eq!(mont.primetest_miller_1bases(), f); } // n < 1050535501 : 2 bases Miller-Rabin primality test if n.get() < 1050535501 { assert_eq!(mont.primetest_miller_2bases(), f); } // n < 350269456337 : 3 bases Miller-Rabin primality test if n.get() < 350269456337 { assert_eq!(mont.primetest_miller_3bases(), f); } // n < 55245642489451 : 4 bases Miller-Rabin primality test if n.get() < 55245642489451 { assert_eq!(mont.primetest_miller_4bases(), f); } // n < 7999252175582851 : 5 bases Miller-Rabin primality test if n.get() < 7999252175582851 { assert_eq!(mont.primetest_miller_5bases(), f); } // n < 585226005592931977 : 6 bases Miller-Rabin primality test if n.get() < 585226005592931977 { assert_eq!(mont.primetest_miller_6bases(), f); } // n for all 64bit integer : 7 bases Miller-Rabin primality test assert_eq!(mont.primetest_miller_7bases(), f); // n for all 64bit integer : multi bases Miller-Rabin primality test assert_eq!(mont.primetest_miller_mbases(), f); // Baillie–PSW primality test assert_eq!(mont.primetest_bpsw(), f); // Lucas test assert_eq!(mont.primetest_lucas(), mont.primetest_lucas()); } } eprint!("test_nzu64_nbits {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_u64_nbits() { // ten least k's for which (2**n)-k is prime // https://primes.utm.edu/lists/2small/0bit.html let start_time = std::time::Instant::now(); for (bit_ref, kvec) in PRIMES_LAST10.iter() { let bit = *bit_ref; let &lastk = kvec.iter().last().unwrap(); for k in 1..=lastk { if (k & 1) == 0 { continue; } let n = (if bit < 64 { 1u64 << bit } else { 0u64 }).wrapping_sub(k); let f = kvec.binary_search(&k).is_ok(); let mont = Mont::<u64>::new(n); // Deterministic variants of the Miller-Rabin primality test // http://miller-rabin.appspot.com/ // n < 341531 : 1 base Miller-Rabin primality test if n < 341531 { assert_eq!(mont.primetest_miller_1bases(), f); } // n < 1050535501 : 2 bases Miller-Rabin primality test if n < 1050535501 { assert_eq!(mont.primetest_miller_2bases(), f); } // n < 350269456337 : 3 bases Miller-Rabin primality test if n < 350269456337 { assert_eq!(mont.primetest_miller_3bases(), f); } // n < 55245642489451 : 4 bases Miller-Rabin primality test if n < 55245642489451 { assert_eq!(mont.primetest_miller_4bases(), f); } // n < 7999252175582851 : 5 bases Miller-Rabin primality test if n < 7999252175582851 { assert_eq!(mont.primetest_miller_5bases(), f); } // n < 585226005592931977 : 6 bases Miller-Rabin primality test if n < 585226005592931977 { assert_eq!(mont.primetest_miller_6bases(), f); } // n for all 64bit integer : 7 bases Miller-Rabin primality test assert_eq!(mont.primetest_miller_7bases(), f); // n for all 64bit integer : multi bases Miller-Rabin primality test assert_eq!(mont.primetest_miller_mbases(), f); // Baillie–PSW primality test assert_eq!(mont.primetest_bpsw(), f); // Lucas test assert_eq!(mont.primetest_lucas(), mont.primetest_lucas()); } } eprint!("test_u64_nbits {}us\n", start_time.elapsed().as_micros()); } // #SPSP-2 Miller-Rabin base 2 (up to 1e7) const SPRP_2: [u64; 162] = [ 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,65281,74665,80581,85489,88357,90751,104653, 130561,196093,220729,233017,252601,253241,256999,271951,280601,314821,357761,390937,458989,476971, 486737,489997,514447,580337,635401,647089,741751,800605,818201,838861,873181,877099,916327,976873, 983401,1004653,1016801,1023121,1082401,1145257,1194649,1207361,1251949,1252697,1302451,1325843,1357441, 1373653,1397419,1441091,1493857,1507963,1509709,1530787,1678541,1730977,1811573,1876393,1907851,1909001, 1969417,1987021,2004403,2081713,2181961,2205967,2264369,2269093,2284453,2304167,2387797,2419385,2510569, 2746477,2748023,2757241,2811271,2909197,2953711,2976487,3090091,3116107,3125281,3375041,3400013,3429037, 3539101,3567481,3581761,3605429,3898129,4181921,4188889,4335241,4360621,4469471,4502485,4513841,4682833, 4835209,4863127,5016191,5044033,5049001,5173169,5173601,5256091,5310721,5444489,5489641,5590621,5599765, 5672041,5681809,5919187,6140161,6226193,6233977,6334351,6368689,6386993,6787327,6836233,6952037,7177105, 7306261,7306561,7462001,7674967,7759937,7820201,7883731,8036033,8095447,8384513,8388607,8534233,8725753, 8727391,9006401,9056501,9069229,9073513,9371251,9564169,9567673,9588151,9729301,9774181,9863461,9995671 ]; #[test] fn test_base2_nzu64_1e7() { // Miller-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... // composite 2-SPRP list up to 2**64 ( http://miller-rabin.appspot.com/#links ) // Pseudoprime Statistics, Tables, and Data ( http://ntheory.org/pseudoprimes.html ) // Miller-Rabin base 2 data (up to 1e15) ( http://ntheory.org/data/spsps.txt ) let assumed: Vec<u64> = Vec::from(SPRP_2); let start_time = std::time::Instant::now(); let result: Vec<u64> = (3..10_000_000).filter(|&n| { if (n & 1) == 0 { return false; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_3bases = mont.primetest_miller_3bases(); let res_base2 = mont.primetest_base2(); assert!(!res_3bases || res_base2); res_3bases != res_base2 }).collect(); assert_eq!(assumed, result); eprint!("test_base2_nzu64_1e7 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_base2_u64_1e7() { // Miller-Rabin primality test (base 2) // strong pseudoprimes to base 2 ( https://oeis.org/A001262 ): 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,... // composite 2-SPRP list up to 2**64 ( http://miller-rabin.appspot.com/#links ) // Pseudoprime Statistics, Tables, and Data ( http://ntheory.org/pseudoprimes.html ) // Miller-Rabin base 2 data (up to 1e15) ( http://ntheory.org/data/spsps.txt ) let assumed: Vec<u64> = Vec::from(SPRP_2); let start_time = std::time::Instant::now(); let result: Vec<u64> = (3..10_000_000).filter(|&n| { if (n & 1) == 0 { return false; } let mont = Mont::<u64>::new(n); let res_3bases = mont.primetest_miller_3bases(); let res_base2 = mont.primetest_base2(); res_3bases != res_base2 }).collect(); assert_eq!(assumed, result); eprint!("test_base2_u64_1e7 {}us\n", start_time.elapsed().as_micros()); } // #SLPSP Strong Lucas-Selfridge (up to 1e7) const SLPRP: [u64; 178] = [ 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,75077,97439,100127,113573,115639,130139,155819, 158399,161027,162133,176399,176471,189419,192509,197801,224369,230691,231703,243629,253259,268349,288919, 313499,324899,353219,366799,391169,430127,436409,455519,487199,510479,572669,611399,622169,635627,636199, 701999,794611,835999,839159,851927,871859,875879,887879,895439,950821,960859,1033997,1106327,1241099, 1256293,1308119,1311389,1388903,1422319,1501439,1697183,1711469,1777159,1981559,2003579,2263127,2435423, 2461211,2518889,2566409,2624399,2662277,2666711,2690759,2738969,2782079,2828699,2942081,2952071,3109049, 3165119,3175883,3179609,3204599,3373649,3399527,3410531,3441239,3452147,3479111,3498879,3579599,3684251, 3694079,3700559,3706169,3735521,3774377,3776219,3785699,3802499,3813011,3865319,3892529,3900797,3903791, 4067279,4109363,4226777,4309631,4322399,4368869,4403027,4563719,4828277,4870847,5133281,5208377,5299139, 5396999,5450201,5479109,5514479,5720219,5762629,5807759,5879411,5942627,6001379,6003923,6296291,6562891, 6641189,6668099,6784721,6784861,6863291,6893531,6965639,7017949,7163441,7199399,7241639,7353917,7453619, 7621499,8112899,8159759,8221121,8234159,8361989,8372849,8518127,8530559,8555009,8574551,8581219,8711699, 8817899,8990279,9049319,9335969,9401893,9485951,9587411,9713027,9793313,9800981,9827711,9922337,9965069 ]; #[test] fn test_lucas_nzu64_1e7() { // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... // Pseudoprime Statistics, Tables, and Data ( http://ntheory.org/pseudoprimes.html ) // Strong Lucas-Selfridge data (up to 1e15) ( http://ntheory.org/data/slpsps-baillie.txt ) let assumed: Vec<u64> = Vec::from(SLPRP); let start_time = std::time::Instant::now(); let result: Vec<u64> = (3..10_000_000).filter(|&n| { if (n & 1) == 0 { return false; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_3bases = mont.primetest_miller_3bases(); let res_lucas = mont.primetest_lucas(); assert!(!res_3bases || res_lucas); res_3bases != res_lucas }).collect(); assert_eq!(assumed, result); eprint!("test_lucas_nzu64_1e7 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_lucas_u64_1e7() { // Lucas primality test // strong Lucas pseudoprimes ( https://oeis.org/A217255 ): 5459,5777,10877,16109,18971,22499,24569,25199,40309,58519,... // Pseudoprime Statistics, Tables, and Data ( http://ntheory.org/pseudoprimes.html ) // Strong Lucas-Selfridge data (up to 1e15) ( http://ntheory.org/data/slpsps-baillie.txt ) let assumed: Vec<u64> = Vec::from(SLPRP); let start_time = std::time::Instant::now(); let result: Vec<u64> = (3..10_000_000).filter(|&n| { if (n & 1) == 0 { return false; } let mont = Mont::<u64>::new(n); let res_3bases = mont.primetest_miller_3bases(); let res_lucas = mont.primetest_lucas(); assert!(!res_3bases || res_lucas); res_3bases != res_lucas }).collect(); assert_eq!(assumed, result); eprint!("test_lucas_u64_1e7 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_nzu64_1e7() { // 24bit let start_time = std::time::Instant::now(); for n in 3..10_000_000 { if (n & 1) == 0 { continue; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_3bases = mont.primetest_miller_3bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_3bases, res_bpsw); } eprint!("test_bpsw_nzu64_1e7 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_u64_1e7() { // 24bit let start_time = std::time::Instant::now(); for n in 3..10_000_000 { if (n & 1) == 0 { continue; } let mont = Mont::<u64>::new(n); let res_3bases = mont.primetest_miller_3bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_3bases, res_bpsw); } eprint!("test_bpsw_u64_1e7 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_nzu64_4e9() { // 32bit let start_time = std::time::Instant::now(); for n in 4_000_000_000..4_010_000_000 { if (n & 1) == 0 { continue; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_3bases = mont.primetest_miller_3bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_3bases, res_bpsw); } eprint!("test_bpsw_nzu64_4e9 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_u64_4e9() { // 32bit let start_time = std::time::Instant::now(); for n in 4_000_000_000..4_010_000_000 { if (n & 1) == 0 { continue; } let mont = Mont::<u64>::new(n); let res_3bases = mont.primetest_miller_3bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_3bases, res_bpsw); } eprint!("test_bpsw_u64_4e9 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_nzu64_1e16() { // 54bit let start_time = std::time::Instant::now(); for n in 10_000_000_000_000_000..10_000_000_010_000_000 { if (n & 1) == 0 { continue; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_nzu64_1e16 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_u64_1e16() { // 54bit let start_time = std::time::Instant::now(); for n in 10_000_000_000_000_000..10_000_000_010_000_000 { if (n & 1) == 0 { continue; } let mont = Mont::<u64>::new(n); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_u64_1e16 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_nzu64_9e18() { // 63bit let start_time = std::time::Instant::now(); for n in 9_000_000_000_000_000_000..9_000_000_000_010_000_000 { if (n & 1) == 0 { continue; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_nzu64_9e18 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_u64_9e18() { // 63bit let start_time = std::time::Instant::now(); for n in 9_000_000_000_000_000_000..9_000_000_000_010_000_000 { if (n & 1) == 0 { continue; } let mont = Mont::<u64>::new(n); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_u64_9e18 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_nzu64_10e18() { // 64bit let start_time = std::time::Instant::now(); for n in 10_000_000_000_000_000_000..10_000_000_000_010_000_000 { if (n & 1) == 0 { continue; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_nzu64_10e18 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_u64_10e18() { // 64bit let start_time = std::time::Instant::now(); for n in 10_000_000_000_000_000_000..10_000_000_000_010_000_000 { if (n & 1) == 0 { continue; } let mont = Mont::<u64>::new(n); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_u64_10e18 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_nzu64_18e18() { // 64bit let start_time = std::time::Instant::now(); for n in 18_000_000_000_000_000_000..18_000_000_000_010_000_000 { if (n & 1) == 0 { continue; } let nn = NonZeroU64::new(n).unwrap(); let mont = Mont::<NonZeroU64>::new(nn); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_nzu64_18e18 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_bpsw_u64_18e18() { // 64bit let start_time = std::time::Instant::now(); for n in 18_000_000_000_000_000_000..18_000_000_000_010_000_000 { if (n & 1) == 0 { continue; } let mont = Mont::<u64>::new(n); let res_7bases = mont.primetest_miller_7bases(); let res_bpsw = mont.primetest_bpsw(); assert_eq!(res_7bases, res_bpsw); } eprint!("test_bpsw_u64_18e18 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_issq_u64() { let start_time = std::time::Instant::now(); for i in 0xffff_0000u64..=0xffff_ffff { let n = i * i; assert!(issq_u64(n)); } eprint!("test_issq_u64 {}us\n", start_time.elapsed().as_micros()); } #[test] fn test_issq_u64_p1() { let start_time = std::time::Instant::now(); for i in 0xffff_0000u64..=0xffff_ffff { let n = i * i + 1; assert!(!issq_u64(n)); } eprint!("test_issq_u64_p1 {}us\n", start_time.elapsed().as_micros()); } } fn main() { use std::io::{BufRead, Write}; let start_time = std::time::Instant::now(); let out = std::io::stdout(); let mut out = std::io::BufWriter::new(out.lock()); let input = std::io::stdin(); let mut buf = String::new(); let mut reader = std::io::BufReader::new(input.lock()); reader.read_line(&mut buf).unwrap(); let n = buf.trim().parse::<usize>().unwrap(); for _ in 0..n { buf.clear(); reader.read_line(&mut buf).unwrap(); let xs = buf.trim(); let res = if USE_NZU64 { primetest_nzu64_bpsw(xs.parse::<NonZeroU64>().unwrap()) } else { primetest_u64_bpsw(xs.parse::<u64>().unwrap()) }; out.write(xs.as_bytes()).unwrap(); out.write(if res { b" 1\n" } else { b" 0\n" }).unwrap(); } eprintln!("{}us", start_time.elapsed().as_micros()); }