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
|
module Number.F2m (tests) where
import Imports hiding ((.&.))
import Data.Bits
import Data.Maybe
import Crypto.Number.Basic (log2)
import Crypto.Number.F2m
addTests = testGroup "addF2m"
[ testProperty "commutative"
$ \a b -> a `addF2m` b == b `addF2m` a
, testProperty "associative"
$ \a b c -> (a `addF2m` b) `addF2m` c == a `addF2m` (b `addF2m` c)
, testProperty "0 is neutral"
$ \a -> a `addF2m` 0 == a
, testProperty "nullable"
$ \a -> a `addF2m` a == 0
, testProperty "works per bit"
$ \a b -> (a `addF2m` b) .&. b == (a .&. b) `addF2m` b
]
modTests = testGroup "modF2m"
[ testProperty "idempotent"
$ \(Positive m) (NonNegative a) -> modF2m m a == modF2m m (modF2m m a)
, testProperty "upper bound"
$ \(Positive m) (NonNegative a) -> modF2m m a < 2 ^ log2 m
, testProperty "reach upper"
$ \(Positive m) -> let a = 2 ^ log2 m - 1 in modF2m m (m `addF2m` a) == a
, testProperty "lower bound"
$ \(Positive m) (NonNegative a) -> modF2m m a >= 0
, testProperty "reach lower"
$ \(Positive m) -> modF2m m m == 0
, testProperty "additive"
$ \(Positive m) (NonNegative a) (NonNegative b)
-> modF2m m a `addF2m` modF2m m b == modF2m m (a `addF2m` b)
]
mulTests = testGroup "mulF2m"
[ testProperty "commutative"
$ \(Positive m) (NonNegative a) (NonNegative b) -> mulF2m m a b == mulF2m m b a
, testProperty "associative"
$ \(Positive m) (NonNegative a) (NonNegative b) (NonNegative c)
-> mulF2m m (mulF2m m a b) c == mulF2m m a (mulF2m m b c)
, testProperty "1 is neutral"
$ \(Positive m) (NonNegative a) -> mulF2m m a 1 == modF2m m a
, testProperty "0 is annihilator"
$ \(Positive m) (NonNegative a) -> mulF2m m a 0 == 0
, testProperty "distributive"
$ \(Positive m) (NonNegative a) (NonNegative b) (NonNegative c)
-> mulF2m m a (b `addF2m` c) == mulF2m m a b `addF2m` mulF2m m a c
]
squareTests = testGroup "squareF2m"
[ testProperty "sqr(a) == a * a"
$ \(Positive m) (NonNegative a) -> mulF2m m a a == squareF2m m a
-- disabled because we require @m@ to be a suitable modulus and there is no
-- way to guarantee this
-- , testProperty "sqrt(a) * sqrt(a) = a"
-- $ \(Positive m) (NonNegative aa) -> let a = sqrtF2m m aa in mulF2m m a a == modF2m m aa
, testProperty "sqrt(a) * sqrt(a) = a in GF(2^16)"
$ let m = 65581 :: Integer -- x^16 + x^5 + x^3 + x^2 + 1
nums = [0 .. 65535 :: Integer]
in nums == [let y = sqrtF2m m x in squareF2m m y | x <- nums]
]
powTests = testGroup "powF2m"
[ testProperty "2 is square"
$ \(Positive m) (NonNegative a) -> powF2m m a 2 == squareF2m m a
, testProperty "1 is identity"
$ \(Positive m) (NonNegative a) -> powF2m m a 1 == modF2m m a
, testProperty "0 is annihilator"
$ \(Positive m) (NonNegative a) -> powF2m m a 0 == modF2m m 1
, testProperty "(a * b) ^ c == (a ^ c) * (b ^ c)"
$ \(Positive m) (NonNegative a) (NonNegative b) (NonNegative c)
-> powF2m m (mulF2m m a b) c == mulF2m m (powF2m m a c) (powF2m m b c)
, testProperty "a ^ (b + c) == (a ^ b) * (a ^ c)"
$ \(Positive m) (NonNegative a) (NonNegative b) (NonNegative c)
-> powF2m m a (b + c) == mulF2m m (powF2m m a b) (powF2m m a c)
, testProperty "a ^ (b * c) == (a ^ b) ^ c"
$ \(Positive m) (NonNegative a) (NonNegative b) (NonNegative c)
-> powF2m m a (b * c) == powF2m m (powF2m m a b) c
]
invTests = testGroup "invF2m"
[ testProperty "1 / a * a == 1"
$ \(Positive m) (NonNegative a)
-> maybe True (\c -> mulF2m m c a == modF2m m 1) (invF2m m a)
, testProperty "1 / a == a (mod a^2-1)"
$ \(NonNegative a) -> a < 2 || invF2m (squareF2m' a `addF2m` 1) a == Just a
]
divTests = testGroup "divF2m"
[ testProperty "1 / a == inv a"
$ \(Positive m) (NonNegative a) -> divF2m m 1 a == invF2m m a
, testProperty "a / b == a * inv b"
$ \(Positive m) (NonNegative a) (NonNegative b)
-> divF2m m a b == (mulF2m m a <$> invF2m m b)
, testProperty "a * b / b == a"
$ \(Positive m) (NonNegative a) (NonNegative b)
-> isNothing (invF2m m b) || divF2m m (mulF2m m a b) b == Just (modF2m m a)
]
tests = testGroup "number.F2m"
[ addTests
, modTests
, mulTests
, squareTests
, powTests
, invTests
, divTests
]
|
