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
|
extern crate num_bigint;
extern crate num_integer;
extern crate num_traits;
static BIG_B: &'static str = "\
efac3c0a_0de55551_fee0bfe4_67fa017a_1a898fa1_6ca57cb1\
ca9e3248_cacc09a9_b99d6abc_38418d0f_82ae4238_d9a68832\
aadec7c1_ac5fed48_7a56a71b_67ac59d5_afb28022_20d9592d\
247c4efc_abbd9b75_586088ee_1dc00dc4_232a8e15_6e8191dd\
675b6ae0_c80f5164_752940bc_284b7cee_885c1e10_e495345b\
8fbe9cfd_e5233fe1_19459d0b_d64be53c_27de5a02_a829976b\
33096862_82dad291_bd38b6a9_be396646_ddaf8039_a2573c39\
1b14e8bc_2cb53e48_298c047e_d9879e9c_5a521076_f0e27df3\
990e1659_d3d8205b_6443ebc0_9918ebee_6764f668_9f2b2be3\
b59cbc76_d76d0dfc_d737c3ec_0ccf9c00_ad0554bf_17e776ad\
b4edf9cc_6ce540be_76229093_5c53893b";
static BIG_E: &'static str = "\
be0e6ea6_08746133_e0fbc1bf_82dba91e_e2b56231_a81888d2\
a833a1fc_f7ff002a_3c486a13_4f420bf3_a5435be9_1a5c8391\
774d6e6c_085d8357_b0c97d4d_2bb33f7c_34c68059_f78d2541\
eacc8832_426f1816_d3be001e_b69f9242_51c7708e_e10efe98\
449c9a4a_b55a0f23_9d797410_515da00d_3ea07970_4478a2ca\
c3d5043c_bd9be1b4_6dce479d_4302d344_84a939e6_0ab5ada7\
12ae34b2_30cc473c_9f8ee69d_2cac5970_29f5bf18_bc8203e4\
f3e895a2_13c94f1e_24c73d77_e517e801_53661fdd_a2ce9e47\
a73dd7f8_2f2adb1e_3f136bf7_8ae5f3b8_08730de1_a4eff678\
e77a06d0_19a522eb_cbefba2a_9caf7736_b157c5c6_2d192591\
17946850_2ddb1822_117b68a0_32f7db88";
// This modulus is the prime from the 2048-bit MODP DH group:
// https://tools.ietf.org/html/rfc3526#section-3
static BIG_M: &'static str = "\
FFFFFFFF_FFFFFFFF_C90FDAA2_2168C234_C4C6628B_80DC1CD1\
29024E08_8A67CC74_020BBEA6_3B139B22_514A0879_8E3404DD\
EF9519B3_CD3A431B_302B0A6D_F25F1437_4FE1356D_6D51C245\
E485B576_625E7EC6_F44C42E9_A637ED6B_0BFF5CB6_F406B7ED\
EE386BFB_5A899FA5_AE9F2411_7C4B1FE6_49286651_ECE45B3D\
C2007CB8_A163BF05_98DA4836_1C55D39A_69163FA8_FD24CF5F\
83655D23_DCA3AD96_1C62F356_208552BB_9ED52907_7096966D\
670C354E_4ABC9804_F1746C08_CA18217C_32905E46_2E36CE3B\
E39E772C_180E8603_9B2783A2_EC07A28F_B5C55DF0_6F4C52C9\
DE2BCBF6_95581718_3995497C_EA956AE5_15D22618_98FA0510\
15728E5A_8AACAA68_FFFFFFFF_FFFFFFFF";
static BIG_R: &'static str = "\
a1468311_6e56edc9_7a98228b_5e924776_0dd7836e_caabac13\
eda5373b_4752aa65_a1454850_40dc770e_30aa8675_6be7d3a8\
9d3085e4_da5155cf_b451ef62_54d0da61_cf2b2c87_f495e096\
055309f7_77802bbb_37271ba8_1313f1b5_075c75d1_024b6c77\
fdb56f17_b05bce61_e527ebfd_2ee86860_e9907066_edd526e7\
93d289bf_6726b293_41b0de24_eff82424_8dfd374b_4ec59542\
35ced2b2_6b195c90_10042ffb_8f58ce21_bc10ec42_64fda779\
d352d234_3d4eaea6_a86111ad_a37e9555_43ca78ce_2885bed7\
5a30d182_f1cf6834_dc5b6e27_1a41ac34_a2e91e11_33363ff0\
f88a7b04_900227c9_f6e6d06b_7856b4bb_4e354d61_060db6c8\
109c4735_6e7db425_7b5d74c7_0b709508";
mod biguint {
use num_bigint::BigUint;
use num_integer::Integer;
use num_traits::Num;
fn check_modpow<T: Into<BigUint>>(b: T, e: T, m: T, r: T) {
let b: BigUint = b.into();
let e: BigUint = e.into();
let m: BigUint = m.into();
let r: BigUint = r.into();
assert_eq!(b.modpow(&e, &m), r);
let even_m = &m << 1;
let even_modpow = b.modpow(&e, &even_m);
assert!(even_modpow < even_m);
assert_eq!(even_modpow.mod_floor(&m), r);
}
#[test]
fn test_modpow() {
check_modpow::<u32>(1, 0, 11, 1);
check_modpow::<u32>(0, 15, 11, 0);
check_modpow::<u32>(3, 7, 11, 9);
check_modpow::<u32>(5, 117, 19, 1);
check_modpow::<u32>(20, 1, 2, 0);
check_modpow::<u32>(20, 1, 3, 2);
}
#[test]
fn test_modpow_small() {
for b in 0u64..11 {
for e in 0u64..11 {
for m in 1..11 {
check_modpow::<u64>(b, e, m, b.pow(e as u32) % m);
}
}
}
}
#[test]
fn test_modpow_big() {
let b = BigUint::from_str_radix(super::BIG_B, 16).unwrap();
let e = BigUint::from_str_radix(super::BIG_E, 16).unwrap();
let m = BigUint::from_str_radix(super::BIG_M, 16).unwrap();
let r = BigUint::from_str_radix(super::BIG_R, 16).unwrap();
assert_eq!(b.modpow(&e, &m), r);
let even_m = &m << 1;
let even_modpow = b.modpow(&e, &even_m);
assert!(even_modpow < even_m);
assert_eq!(even_modpow % m, r);
}
}
mod bigint {
use num_bigint::BigInt;
use num_integer::Integer;
use num_traits::{Num, One, Signed};
fn check_modpow<T: Into<BigInt>>(b: T, e: T, m: T, r: T) {
fn check(b: &BigInt, e: &BigInt, m: &BigInt, r: &BigInt) {
assert_eq!(&b.modpow(e, m), r);
let even_m = m << 1;
let even_modpow = b.modpow(e, m);
assert!(even_modpow.abs() < even_m.abs());
assert_eq!(&even_modpow.mod_floor(&m), r);
// the sign of the result follows the modulus like `mod_floor`, not `rem`
assert_eq!(b.modpow(&BigInt::one(), m), b.mod_floor(m));
}
let b: BigInt = b.into();
let e: BigInt = e.into();
let m: BigInt = m.into();
let r: BigInt = r.into();
let neg_b_r = if e.is_odd() {
(-&r).mod_floor(&m)
} else {
r.clone()
};
let neg_m_r = r.mod_floor(&-&m);
let neg_bm_r = neg_b_r.mod_floor(&-&m);
check(&b, &e, &m, &r);
check(&-&b, &e, &m, &neg_b_r);
check(&b, &e, &-&m, &neg_m_r);
check(&-b, &e, &-&m, &neg_bm_r);
}
#[test]
fn test_modpow() {
check_modpow(1, 0, 11, 1);
check_modpow(0, 15, 11, 0);
check_modpow(3, 7, 11, 9);
check_modpow(5, 117, 19, 1);
check_modpow(-20, 1, 2, 0);
check_modpow(-20, 1, 3, 1);
}
#[test]
fn test_modpow_small() {
for b in -10i64..11 {
for e in 0i64..11 {
for m in -10..11 {
if m == 0 {
continue;
}
check_modpow(b, e, m, b.pow(e as u32).mod_floor(&m));
}
}
}
}
#[test]
fn test_modpow_big() {
let b = BigInt::from_str_radix(super::BIG_B, 16).unwrap();
let e = BigInt::from_str_radix(super::BIG_E, 16).unwrap();
let m = BigInt::from_str_radix(super::BIG_M, 16).unwrap();
let r = BigInt::from_str_radix(super::BIG_R, 16).unwrap();
check_modpow(b, e, m, r);
}
}
|
