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
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191

//! [`Uint`] subtraction modulus operations.

use crate::{Limb, SubMod, Uint};

impl<const LIMBS: usize> Uint<LIMBS> {
    /// Computes `self - rhs mod p`.
    ///
    /// Assumes `self - rhs` as unbounded signed integer is in `[-p, p)`.
    pub const fn sub_mod(&self, rhs: &Uint<LIMBS>, p: &Uint<LIMBS>) -> Uint<LIMBS> {
        let (out, borrow) = self.sbb(rhs, Limb::ZERO);

        // If underflow occurred on the final limb, borrow = 0xfff...fff, otherwise
        // borrow = 0x000...000. Thus, we use it as a mask to conditionally add the modulus.
        let mask = Uint::from_words([borrow.0; LIMBS]);

        out.wrapping_add(&p.bitand(&mask))
    }

    /// Returns `(self..., carry) - (rhs...) mod (p...)`, where `carry <= 1`.
    /// Assumes `-(p...) <= (self..., carry) - (rhs...) < (p...)`.
    #[inline(always)]
    pub(crate) const fn sub_mod_with_carry(&self, carry: Limb, rhs: &Self, p: &Self) -> Self {
        debug_assert!(carry.0 <= 1);

        let (out, borrow) = self.sbb(rhs, Limb::ZERO);

        // The new `borrow = Word::MAX` iff `carry == 0` and `borrow == Word::MAX`.
        let borrow = (!carry.0.wrapping_neg()) & borrow.0;

        // If underflow occurred on the final limb, borrow = 0xfff...fff, otherwise
        // borrow = 0x000...000. Thus, we use it as a mask to conditionally add the modulus.
        let mask = Uint::from_words([borrow; LIMBS]);

        out.wrapping_add(&p.bitand(&mask))
    }

    /// Computes `self - rhs mod p` for the special modulus
    /// `p = MAX+1-c` where `c` is small enough to fit in a single [`Limb`].
    ///
    /// Assumes `self - rhs` as unbounded signed integer is in `[-p, p)`.
    pub const fn sub_mod_special(&self, rhs: &Self, c: Limb) -> Self {
        let (out, borrow) = self.sbb(rhs, Limb::ZERO);

        // If underflow occurred, then we need to subtract `c` to account for
        // the underflow. This cannot underflow due to the assumption
        // `self - rhs >= -p`.
        let l = borrow.0 & c.0;
        out.wrapping_sub(&Uint::from_word(l))
    }
}

impl<const LIMBS: usize> SubMod for Uint<LIMBS> {
    type Output = Self;

    fn sub_mod(&self, rhs: &Self, p: &Self) -> Self {
        debug_assert!(self < p);
        debug_assert!(rhs < p);
        self.sub_mod(rhs, p)
    }
}

#[cfg(all(test, feature = "rand"))]
mod tests {
    use crate::{Limb, NonZero, Random, RandomMod, Uint};
    use rand_core::SeedableRng;

    macro_rules! test_sub_mod {
        ($size:expr, $test_name:ident) => {
            #[test]
            fn $test_name() {
                let mut rng = rand_chacha::ChaCha8Rng::seed_from_u64(1);
                let moduli = [
                    NonZero::<Uint<$size>>::random(&mut rng),
                    NonZero::<Uint<$size>>::random(&mut rng),
                ];

                for p in &moduli {
                    let base_cases = [
                        (1u64, 0u64, 1u64.into()),
                        (0, 1, p.wrapping_sub(&1u64.into())),
                        (0, 0, 0u64.into()),
                    ];
                    for (a, b, c) in &base_cases {
                        let a: Uint<$size> = (*a).into();
                        let b: Uint<$size> = (*b).into();

                        let x = a.sub_mod(&b, p);
                        assert_eq!(*c, x, "{} - {} mod {} = {} != {}", a, b, p, x, c);
                    }

                    if $size > 1 {
                        for _i in 0..100 {
                            let a: Uint<$size> = Limb::random(&mut rng).into();
                            let b: Uint<$size> = Limb::random(&mut rng).into();
                            let (a, b) = if a < b { (b, a) } else { (a, b) };

                            let c = a.sub_mod(&b, p);
                            assert!(c < **p, "not reduced");
                            assert_eq!(c, a.wrapping_sub(&b), "result incorrect");
                        }
                    }

                    for _i in 0..100 {
                        let a = Uint::<$size>::random_mod(&mut rng, p);
                        let b = Uint::<$size>::random_mod(&mut rng, p);

                        let c = a.sub_mod(&b, p);
                        assert!(c < **p, "not reduced: {} >= {} ", c, p);

                        let x = a.wrapping_sub(&b);
                        if a >= b && x < **p {
                            assert_eq!(c, x, "incorrect result");
                        }
                    }
                }
            }
        };
    }

    macro_rules! test_sub_mod_special {
        ($size:expr, $test_name:ident) => {
            #[test]
            fn $test_name() {
                let mut rng = rand_chacha::ChaCha8Rng::seed_from_u64(1);
                let moduli = [
                    NonZero::<Limb>::random(&mut rng),
                    NonZero::<Limb>::random(&mut rng),
                ];

                for special in &moduli {
                    let p = &NonZero::new(Uint::ZERO.wrapping_sub(&Uint::from_word(special.0)))
                        .unwrap();

                    let minus_one = p.wrapping_sub(&Uint::ONE);

                    let base_cases = [
                        (Uint::ZERO, Uint::ZERO, Uint::ZERO),
                        (Uint::ONE, Uint::ZERO, Uint::ONE),
                        (Uint::ZERO, Uint::ONE, minus_one),
                        (minus_one, minus_one, Uint::ZERO),
                        (Uint::ZERO, minus_one, Uint::ONE),
                    ];
                    for (a, b, c) in &base_cases {
                        let x = a.sub_mod_special(&b, *special.as_ref());
                        assert_eq!(*c, x, "{} - {} mod {} = {} != {}", a, b, p, x, c);
                    }

                    for _i in 0..100 {
                        let a = Uint::<$size>::random_mod(&mut rng, p);
                        let b = Uint::<$size>::random_mod(&mut rng, p);

                        let c = a.sub_mod_special(&b, *special.as_ref());
                        assert!(c < **p, "not reduced: {} >= {} ", c, p);

                        let expected = a.sub_mod(&b, p);
                        assert_eq!(c, expected, "incorrect result");
                    }
                }
            }
        };
    }

    // Test requires 1-limb is capable of representing a 64-bit integer
    #[cfg(target_pointer_width = "64")]
    test_sub_mod!(1, sub1);

    test_sub_mod!(2, sub2);
    test_sub_mod!(3, sub3);
    test_sub_mod!(4, sub4);
    test_sub_mod!(5, sub5);
    test_sub_mod!(6, sub6);
    test_sub_mod!(7, sub7);
    test_sub_mod!(8, sub8);
    test_sub_mod!(9, sub9);
    test_sub_mod!(10, sub10);
    test_sub_mod!(11, sub11);
    test_sub_mod!(12, sub12);

    test_sub_mod_special!(1, sub_mod_special_1);
    test_sub_mod_special!(2, sub_mod_special_2);
    test_sub_mod_special!(3, sub_mod_special_3);
    test_sub_mod_special!(4, sub_mod_special_4);
    test_sub_mod_special!(5, sub_mod_special_5);
    test_sub_mod_special!(6, sub_mod_special_6);
    test_sub_mod_special!(7, sub_mod_special_7);
    test_sub_mod_special!(8, sub_mod_special_8);
    test_sub_mod_special!(9, sub_mod_special_9);
    test_sub_mod_special!(10, sub_mod_special_10);
    test_sub_mod_special!(11, sub_mod_special_11);
    test_sub_mod_special!(12, sub_mod_special_12);
}