1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145

//! [`UInt`] square root operations.

use super::UInt;
use crate::{Limb, LimbUInt};
use subtle::{ConstantTimeEq, CtOption};

impl<const LIMBS: usize> UInt<LIMBS> {
    /// Computes √(`self`)
    /// Uses Brent & Zimmermann, Modern Computer Arithmetic, v0.5.9, Algorithm 1.13
    ///
    /// Callers can check if `self` is a square by squaring the result
    pub const fn sqrt(&self) -> Self {
        let max_bits = (self.bits() + 1) >> 1;
        let cap = Self::ONE.shl_vartime(max_bits);
        let mut guess = cap; // ≥ √(`self`)
        let mut xn = {
            let q = self.wrapping_div(&guess);
            let t = guess.wrapping_add(&q);
            t.shr_vartime(1)
        };

        // If guess increased, the initial guess was low.
        // Repeat until reverse course.
        while guess.ct_cmp(&xn) == -1 {
            // Sometimes an increase is too far, especially with large
            // powers, and then takes a long time to walk back.  The upper
            // bound is based on bit size, so saturate on that.
            let res = Limb::ct_cmp(Limb(xn.bits() as LimbUInt), Limb(max_bits as LimbUInt)) - 1;
            let le = Limb::is_nonzero(Limb(res as LimbUInt));
            guess = Self::ct_select(cap, xn, le);
            xn = {
                let q = self.wrapping_div(&guess);
                let t = guess.wrapping_add(&q);
                t.shr_vartime(1)
            };
        }

        // Repeat while guess decreases.
        while guess.ct_cmp(&xn) == 1 && xn.ct_is_nonzero() == LimbUInt::MAX {
            guess = xn;
            xn = {
                let q = self.wrapping_div(&guess);
                let t = guess.wrapping_add(&q);
                t.shr_vartime(1)
            };
        }

        Self::ct_select(Self::ZERO, guess, self.ct_is_nonzero())
    }

    /// Wrapped sqrt is just normal √(`self`)
    /// There’s no way wrapping could ever happen.
    /// This function exists, so that all operations are accounted for in the wrapping operations.
    pub const fn wrapping_sqrt(&self) -> Self {
        self.sqrt()
    }

    /// Perform checked sqrt, returning a [`CtOption`] which `is_some`
    /// only if the √(`self`)² == self
    pub fn checked_sqrt(&self) -> CtOption<Self> {
        let r = self.sqrt();
        let s = r.wrapping_mul(&r);
        CtOption::new(r, self.ct_eq(&s))
    }
}

#[cfg(test)]
mod tests {
    use crate::{Limb, U256};

    #[cfg(feature = "rand")]
    use {
        crate::{CheckedMul, Random, U512},
        rand_chacha::ChaChaRng,
        rand_core::{RngCore, SeedableRng},
    };

    #[test]
    fn edge() {
        assert_eq!(U256::ZERO.sqrt(), U256::ZERO);
        assert_eq!(U256::ONE.sqrt(), U256::ONE);
        let mut half = U256::ZERO;
        for i in 0..half.limbs.len() / 2 {
            half.limbs[i] = Limb::MAX;
        }
        assert_eq!(U256::MAX.sqrt(), half,);
    }

    #[test]
    fn simple() {
        let tests = [
            (4u8, 2u8),
            (9, 3),
            (16, 4),
            (25, 5),
            (36, 6),
            (49, 7),
            (64, 8),
            (81, 9),
            (100, 10),
            (121, 11),
            (144, 12),
            (169, 13),
        ];
        for (a, e) in &tests {
            let l = U256::from(*a);
            let r = U256::from(*e);
            assert_eq!(l.sqrt(), r);
            assert_eq!(l.checked_sqrt().is_some().unwrap_u8(), 1u8);
        }
    }

    #[test]
    fn nonsquares() {
        assert_eq!(U256::from(2u8).sqrt(), U256::from(1u8));
        assert_eq!(U256::from(2u8).checked_sqrt().is_some().unwrap_u8(), 0);
        assert_eq!(U256::from(3u8).sqrt(), U256::from(1u8));
        assert_eq!(U256::from(3u8).checked_sqrt().is_some().unwrap_u8(), 0);
        assert_eq!(U256::from(5u8).sqrt(), U256::from(2u8));
        assert_eq!(U256::from(6u8).sqrt(), U256::from(2u8));
        assert_eq!(U256::from(7u8).sqrt(), U256::from(2u8));
        assert_eq!(U256::from(8u8).sqrt(), U256::from(2u8));
        assert_eq!(U256::from(10u8).sqrt(), U256::from(3u8));
    }

    #[cfg(feature = "rand")]
    #[test]
    fn fuzz() {
        let mut rng = ChaChaRng::from_seed([7u8; 32]);
        for _ in 0..50 {
            let t = rng.next_u32() as u64;
            let s = U256::from(t);
            let s2 = s.checked_mul(&s).unwrap();
            assert_eq!(s2.sqrt(), s);
            assert_eq!(s2.checked_sqrt().is_some().unwrap_u8(), 1);
        }

        for _ in 0..50 {
            let s = U256::random(&mut rng);
            let mut s2 = U512::ZERO;
            s2.limbs[..s.limbs.len()].copy_from_slice(&s.limbs);
            assert_eq!(s.square().sqrt(), s2);
        }
    }
}