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Файл:Введение в криптологию. В 4 частях. Ч.3. Учебное пособие.pdf
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2.1.2. ПРОЕКТИВНАЯ ТОЧКА ПРОЕКТИВНОЙ ПЛОСКОСТИ
Проективной точкой проективной плоскости
)(2KP
над полем K называется
класс эквивалентности какой-либо последовательности вида (X, Y, Z), состоящей
из не равных одновременно нулю элементов
KZYX ,,
.
При этом, если в качестве поля K выступает
p
F
(поле вычетов по модулю p),
проективная точка будет иметь мощность, равную p – 1.
Рассмотрим, например, классы эквивалентности проективной плоскости
)(2KP
=
)(
2
p
FP
=
)(
3
2
FP
над полем K=
p
F
=
3
F
= {0, 1, 2}. Заметим, что при этом
}0{\}0{\}0{\
3
**
FFF
pp
KK
= {0, 1, 2}\{0} = {1, 2}.
Так, элемент (0, 0, 1) проективной плоскости
)(
3
2
FP
при λ = 1 в соответствии
с (2.1) эквивалентен сам себе:
(0, 0, 1) (1 ‧ 0 (mod 3), 1 ‧ 0 (mod 3), 1 ‧ 1 (mod 3)) (0, 0, 1),
а при λ = 2 эквивалентен элементу (0, 0, 2):
(0, 0, 1) (2 ‧ 0 (mod 3), 2 ‧ 0 (mod 3), 2 ‧ 1 (mod 3)) (0, 0, 2).
Элемент (0, 0, 2) проективной плоскости
)(
3
2
FP
при λ = 1 эквивалентен сам себе:
(0, 0, 2) (1 ‧ 0 (mod 3), 1 ‧ 0 (mod 3), 1 ‧ 2 (mod 3)) (0, 0, 2),
а при λ = 2 эквивалентен элементу (0, 0, 1):
(0, 0, 2) (2 ‧ 0 (mod 3), 2 ‧ 0 (mod 3), 2 ‧ 2 (mod 3)) (0, 0, 1).
Следовательно, классом эквивалентности элемента (0, 0, 1) является множество
эквивалентных элементов C1 = {(0, 0, 1), (0, 0, 2)}.
Аналогично, классом эквивалентности элемента (0, 1, 0) является множество
эквивалентных элементов C2 = {(0, 1, 0), (0, 2, 0)}.
Далее, классом эквивалентности элемента (0, 1, 1) является множество эквивалентных элементов C3 = {(0, 1, 1), (0, 2, 2)}.
Остальными классами эквивалентности (проективными точками) проективной
плоскости
)(
3
2
FP
являются множества:
C4 = {(0, 1, 2), (0, 2, 1)},
C5 = {(1, 0, 0), (2, 0, 0)},
C6 = {(1, 0, 1), (2, 0, 2)},
C7 = {(1, 0, 2), (2, 0, 1)},
C8 = {(1, 1, 0), (2, 2, 0)},
C9 = {(1, 1, 1), (2, 2, 2)},
C10 = {(1, 1, 2), (2, 2, 1)},
C11 = {(1, 2, 0), (2, 1, 0)},
C12 = {(1, 2, 1), (2, 1, 2)},
C13 = {(1, 2, 2), (2, 1, 1)}.
12
Заметим, что каждая из этих тринадцати проективных точек проективной плоско-
сти
)(
2
p
FP
=
)(
3
2
FP
, имеет мощность, равную p – 1 = 3 – 1 = 2.
Во втором примере классами эквивалентности (проективными точками) проектив-
ной плоскости
)(2KP
=
)(
2
p
FP
=
)(
5
2
FP
над полем K=
p
F
=
5
F
= {0, 1, 2, 3, 4} при
*
5
**
FF pK
= {1, 2, 3, 4} являются множества:
C1 = {(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4)},
C2 = {(0, 1, 0), (0, 2, 0), (0, 3, 0), (0, 4, 0)},
C3 = {(0, 1, 1), (0, 2, 2), (0, 3, 3), (0, 4, 4)},
C4 = {(0, 1, 2), (0, 2, 4), (0, 3, 1), (0, 4, 3)},
C5 = {(0, 1, 3), (0, 2, 1), (0, 3, 4), (0, 4, 2)},
C6 = {(0, 1, 4), (0, 2, 3), (0, 3, 2), (0, 4, 1)},
C7 = {(1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0)},
C8 = {(1, 0, 1), (2, 0, 2), (3, 0, 3), (4, 0, 4)},
C9 = {(1, 0, 2), (2, 0, 4), (3, 0, 1), (4, 0, 3)},
C10 = {(1, 0, 3), (2, 0, 1), (3, 0, 4), (4, 0, 2)},
C11 = {(1, 0, 4), (2, 0, 3), (3, 0, 2), (4, 0, 1)},
C12 = {(1, 1, 0), (2, 2, 0), (3, 3, 0), (4, 4, 0)},
C13 = {(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4)},
C14 = {(1, 1, 2), (2, 2, 4), (3, 3, 1), (4, 4, 3)},
C15 = {(1, 1, 3), (2, 2, 1), (3, 3, 4), (4, 4, 2)},
C16 = {(1, 1, 4), (2, 2, 3), (3, 3, 2), (4, 4, 1)},
C17 = {(1, 2, 0), (2, 4, 0), (3, 1, 0), (4, 3, 0)},
C18 = {(1, 2, 1), (2, 4, 2), (3, 1, 3), (4, 3, 4)},
C19 = {(1, 2, 2), (2, 4, 4), (3, 1, 1), (4, 3, 3)},
C20 = {(1, 2, 3), (2, 4, 1), (3, 1, 4), (4, 3, 2)},
C21 = {(1, 2, 4), (2, 4, 3), (3, 1, 2), (4, 3, 1)},
C22 = {(1, 3, 0), (2, 1, 0), (3, 4, 0), (4, 2, 0)},
C23 = {(1, 3, 1), (2, 1, 2), (3, 4, 3), (4, 2, 4)},
C24 = {(1, 3, 2), (2, 1, 4), (3, 4, 1), (4, 2, 3)},
C25 = {(1, 3, 3), (2, 1, 1), (3, 4, 4), (4, 2, 2)},
C26 = {(1, 3, 4), (2, 1, 3), (3, 4, 2), (4, 2, 1)},
C27 = {(1, 4, 0), (2, 3, 0), (3, 2, 0), (4, 1, 0)},
C28 = {(1, 4, 1), (2, 3, 2), (3, 2, 3), (4, 1, 4)},
C29 = {(1, 4, 2), (2, 3, 4), (3, 2, 1), (4, 1, 3)},
C30 = {(1, 4, 3), (2, 3, 1), (3, 2, 4), (4, 1, 2)},
C31 = {(1, 4, 4), (2, 3, 3), (3, 2, 2), (4, 1, 1)}.
13
Заметим, что каждая из этих тридцати одной проективной точки проективной
плоскости
)(
2
p
FP
=
)(
5
2
FP
, имеет мощность, равную p – 1 = 5 – 1 = 4.
В этой проективной плоскости, например, точки (2, 1, 3) и (3, 4, 2) эквивалентны,
поскольку существует значение (λ = 4)
*
5
**
FF
p
K
= {1, 2, 3, 4}, при котором:
(2, 1, 3) (4 ‧ 2 (mod 5), 4 ‧ 1 (mod 5), 4 ‧ 3 (mod 5)) (3, 4, 2),
а точки, например, (2, 1, 3) и (4, 2, 3) эквивалентными не являются, поскольку не существует значения λ
*
5
**
FF
p
K
= {1, 2, 3, 4}, обеспечивающего эквивалентность дан-
ных элементов проективной плоскости:
(2, 1, 3) (1 ‧ 2 (mod 5), 1 ‧ 1 (mod 5), 1 ‧ 3 (mod 5)) (2, 1, 3) (4, 2, 3),
(2, 1, 3) (2 ‧ 2 (mod 5), 2 ‧ 1 (mod 5), 2 ‧ 3 (mod 5)) (4, 2, 1) (4, 2, 3),
(2, 1, 3) (3 ‧ 2 (mod 5), 3 ‧ 1 (mod 5), 3 ‧ 3 (mod 5)) (1, 3, 4) (4, 2, 3),
(2, 1, 3) (4 ‧ 2 (mod 5), 4 ‧ 1 (mod 5), 4 ‧ 3 (mod 5)) (3, 4, 2) (4, 2, 3).
В третьем примере классами эквивалентности (проективными точками) проектив-
ной плоскости
)(2KP
=
)(
2
p
FP
=
)(
7
2
FP
над полем K=
p
F
=
7
F
= {0, 1, 2, 3, 4, 5, 6} при
*
7
**
FF pK
= {1, 2, 3, 4, 5, 6} являются множества:
C1 = {(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5), (0, 0, 6)},
C2 = {(0, 1, 0), (0, 2, 0), (0, 3, 0), (0, 4, 0), (0, 5, 0), (0, 6, 0)},
C3 = {(0, 1, 1), (0, 2, 2), (0, 3, 3), (0, 4, 4), (0, 5, 5), (0, 6, 6)},
C4 = {(0, 1, 2), (0, 2, 4), (0, 3, 6), (0, 4, 1), (0, 5, 3), (0, 6, 5)},
C5 = {(0, 1, 3), (0, 2, 6), (0, 3, 2), (0, 4, 5), (0, 5, 1), (0, 6, 4)},
C6 = {(0, 1, 4), (0, 2, 1), (0, 3, 5), (0, 4, 2), (0, 5, 6), (0, 6, 3)},
C7 = {(0, 1, 5), (0, 2, 3), (0, 3, 1), (0, 4, 6), (0, 5, 4), (0, 6, 2)},
C8 = {(0, 1, 6), (0, 2, 5), (0, 3, 4), (0, 4, 3), (0, 5, 2), (0, 6, 1)},
C9 = {(1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0), (5, 0, 0), (6, 0, 0)},
C10 = {(1, 0, 1), (2, 0, 2), (3, 0, 3), (4, 0, 4), (5, 0, 5), (6, 0, 6)},
C11 = {(1, 0, 2), (2, 0, 4), (3, 0, 6), (4, 0, 1), (5, 0, 3), (6, 0, 5)},
C12 = {(1, 0, 3), (2, 0, 6), (3, 0, 2), (4, 0, 5), (5, 0, 1), (6, 0, 4)},
C13 = {(1, 0, 4), (2, 0, 1), (3, 0, 5), (4, 0, 2), (5, 0, 6), (6, 0, 3)},
C14 = {(1, 0, 5), (2, 0, 3), (3, 0, 1), (4, 0, 6), (5, 0, 4), (6, 0, 2)},
C15 = {(1, 0, 6), (2, 0, 5), (3, 0, 4), (4, 0, 3), (5, 0, 2), (6, 0, 1)},
C16 = {(1, 1, 0), (2, 2, 0), (3, 3, 0), (4, 4, 0), (5, 5, 0), (6, 6, 0)},
C17 = {(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4), (5, 5, 5), (6, 6, 6)},
C18 = {(1, 1, 2), (2, 2, 4), (3, 3, 6), (4, 4, 1), (5, 5, 3), (6, 6, 5)},
C19 = {(1, 1, 3), (2, 2, 6), (3, 3, 2), (4, 4, 5), (5, 5, 1), (6, 6, 4)},
C20 = {(1, 1, 4), (2, 2, 1), (3, 3, 5), (4, 4, 2), (5, 5, 6), (6, 6, 3)},
14
C21 = {(1, 1, 5), (2, 2, 3), (3, 3, 1), (4, 4, 6), (5, 5, 4), (6, 6, 2)},
C22 = {(1, 1, 6), (2, 2, 5), (3, 3, 4), (4, 4, 3), (5, 5, 2), (6, 6, 1)},
C23 = {(1, 2, 0), (2, 4, 0), (3, 6, 0), (4, 1, 0), (5, 3, 0), (6, 5, 0)},
C24 = {(1, 2, 1), (2, 4, 2), (3, 6, 3), (4, 1, 4), (5, 3, 5), (6, 5, 6)},
C25 = {(1, 2, 2), (2, 4, 4), (3, 6, 6), (4, 1, 1), (5, 3, 3), (6, 5, 5)},
C26 = {(1, 2, 3), (2, 4, 6), (3, 6, 2), (4, 1, 5), (5, 3, 1), (6, 5, 4)},
C27 = {(1, 2, 4), (2, 4, 1), (3, 6, 5), (4, 1, 2), (5, 3, 6), (6, 5, 3)},
C28 = {(1, 2, 5), (2, 4, 3), (3, 6, 1), (4, 1, 6), (5, 3, 4), (6, 5, 2)},
C29 = {(1, 2, 6), (2, 4, 5), (3, 6, 4), (4, 1, 3), (5, 3, 2), (6, 5, 1)},
C30 = {(1, 3, 0), (2, 6, 0), (3, 2, 0), (4, 5, 0), (5, 1, 0), (6, 4, 0)},
C31 = {(1, 3, 1), (2, 6, 2), (3, 2, 3), (4, 5, 4), (5, 1, 5), (6, 4, 6)},
C32 = {(1, 3, 2), (2, 6, 4), (3, 2, 6), (4, 5, 1), (5, 1, 3), (6, 4, 5)},
C33 = {(1, 3, 3), (2, 6, 6), (3, 2, 2), (4, 5, 5), (5, 1, 1), (6, 4, 4)},
C34 = {(1, 3, 4), (2, 6, 1), (3, 2, 5), (4, 5, 2), (5, 1, 6), (6, 4, 3)},
C35 = {(1, 3, 5), (2, 6, 3), (3, 2, 1), (4, 5, 6), (5, 1, 4), (6, 4, 2)},
C36 = {(1, 3, 6), (2, 6, 5), (3, 2, 4), (4, 5, 3), (5, 1, 2), (6, 4, 1)},
C37 = {(1, 4, 0), (2, 1, 0), (3, 5, 0), (4, 2, 0), (5, 6, 0), (6, 3, 0)},
C38 = {(1, 4, 1), (2, 1, 2), (3, 5, 3), (4, 2, 4), (5, 6, 5), (6, 3, 6)},
C39 = {(1, 4, 2), (2, 1, 4), (3, 5, 6), (4, 2, 1), (5, 6, 3), (6, 3, 5)},
C40 = {(1, 4, 3), (2, 1, 6), (3, 5, 2), (4, 2, 5), (5, 6, 1), (6, 3, 4)},
C41 = {(1, 4, 4), (2, 1, 1), (3, 5, 5), (4, 2, 2), (5, 6, 6), (6, 3, 3)},
C42 = {(1, 4, 5), (2, 1, 3), (3, 5, 1), (4, 2, 6), (5, 6, 4), (6, 3, 2)},
C43 = {(1, 4, 6), (2, 1, 5), (3, 5, 4), (4, 2, 3), (5, 6, 2), (6, 3, 1)},
C44 = {(1, 5, 0), (2, 3, 0), (3, 1, 0), (4, 6, 0), (5, 4, 0), (6, 2, 0)},
C45 = {(1, 5, 1), (2, 3, 2), (3, 1, 3), (4, 6, 4), (5, 4, 5), (6, 2, 6)},
C46 = {(1, 5, 2), (2, 3, 4), (3, 1, 6), (4, 6, 1), (5, 4, 3), (6, 2, 5)},
C47 = {(1, 5, 3), (2, 3, 6), (3, 1, 2), (4, 6, 5), (5, 4, 1), (6, 2, 4)},
C48 = {(1, 5, 4), (2, 3, 1), (3, 1, 5), (4, 6, 2), (5, 4, 6), (6, 2, 3)},
C49 = {(1, 5, 5), (2, 3, 3), (3, 1, 1), (4, 6, 6), (5, 4, 4), (6, 2, 2)},
C50 = {(1, 5, 6), (2, 3, 5), (3, 1, 4), (4, 6, 3), (5, 4, 2), (6, 2, 1)},
C51 = {(1, 6, 0), (2, 5, 0), (3, 4, 0), (4, 3, 0), (5, 2, 0), (6, 1, 0)},
C52 = {(1, 6, 1), (2, 5, 2), (3, 4, 3), (4, 3, 4), (5, 2, 5), (6, 1, 6)},
C53 = {(1, 6, 2), (2, 5, 4), (3, 4, 6), (4, 3, 1), (5, 2, 3), (6, 1, 5)},
C54 = {(1, 6, 3), (2, 5, 6), (3, 4, 2), (4, 3, 5), (5, 2, 1), (6, 1, 4)},
C55 = {(1, 6, 4), (2, 5, 1), (3, 4, 5), (4, 3, 2), (5, 2, 6), (6, 1, 3)},
C56 = {(1, 6, 5), (2, 5, 3), (3, 4, 1), (4, 3, 6), (5, 2, 4), (6, 1, 2)},
C57 = {(1, 6, 6), (2, 5, 5), (3, 4, 4), (4, 3, 3), (5, 2, 2), (6, 1, 1)}.
15
Заметим, что каждая из этих пятидесяти семи проективных точек проективной
плоскости
)(
2
p
FP
=
)(
7
2
FP
имеет мощность, равную p – 1 = 7 – 1 = 6.
В четвертом примере классами эквивалентности (проективными точками) проек-
тивной плоскости
)(2KP
=
)(
2
p
FP
=
)(
11
2
FP
над полем K=
p
F
=
11
F
= {0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10} при
*
11
**
FF pK
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} являются множества:
C1 = {(0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5), (0, 0, 6), (0, 0, 7),
(0, 0, 8), (0, 0, 9), (0, 0, 10)},
C2 = {(0, 1, 0), (0, 2, 0), (0, 3, 0), (0, 4, 0), (0, 5, 0), (0, 6, 0), (0, 7, 0),
(0, 8, 0), (0, 9, 0), (0, 10, 0)},
C3 = {(0, 1, 1), (0, 2, 2), (0, 3, 3), (0, 4, 4), (0, 5, 5), (0, 6, 6), (0, 7, 7),
(0, 8, 8), (0, 9, 9), (0, 10, 10)},
C4 = {(0, 1, 2), (0, 2, 4), (0, 3, 6), (0, 4, 8), (0, 5, 10), (0, 6, 1), (0, 7, 3),
(0, 8, 5), (0, 9, 7), (0, 10, 9)},
C5 = {(0, 1, 3), (0, 2, 6), (0, 3, 9), (0, 4, 1), (0, 5, 4), (0, 6, 7), (0, 7, 10),
(0, 8, 2), (0, 9, 5), (0, 10, 8)},
C6 = {(0, 1, 4), (0, 2, 8), (0, 3, 1), (0, 4, 5), (0, 5, 9), (0, 6, 2), (0, 7, 6),
(0, 8, 10), (0, 9, 3), (0, 10, 7)},
C7 = {(0, 1, 5), (0, 2, 10), (0, 3, 4), (0, 4, 9), (0, 5, 3), (0, 6, 8), (0, 7, 2),
(0, 8, 7), (0, 9, 1), (0, 10, 6)},
C8 = {(0, 1, 6), (0, 2, 1), (0, 3, 7), (0, 4, 2), (0, 5, 8), (0, 6, 3), (0, 7, 9),
(0, 8, 4), (0, 9, 10), (0, 10, 5)},
C9 = {(0, 1, 7), (0, 2, 3), (0, 3, 10), (0, 4, 6), (0, 5, 2), (0, 6, 9), (0, 7, 5),
(0, 8, 1), (0, 9, 8), (0, 10, 4)},
C10 = {(0, 1, 8), (0, 2, 5), (0, 3, 2), (0, 4, 10), (0, 5, 7), (0, 6, 4), (0, 7, 1),
(0, 8, 9), (0, 9, 6), (0, 10, 3)},
C11 = {(0, 1, 9), (0, 2, 7), (0, 3, 5), (0, 4, 3), (0, 5, 1), (0, 6, 10), (0, 7, 8),
(0, 8, 6), (0, 9, 4), (0, 10, 2)},
C12 = {(0, 1, 10), (0, 2, 9), (0, 3, 8), (0, 4, 7), (0, 5, 6), (0, 6, 5), (0, 7, 4),
(0, 8, 3), (0, 9, 2), (0, 10, 1)},
C13 = {(1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0), (5, 0, 0), (6, 0, 0), (7, 0, 0),
(8, 0, 0), (9, 0, 0), (10, 0, 0)},
C14 = {(1, 0, 1), (2, 0, 2), (3, 0, 3), (4, 0, 4), (5, 0, 5), (6, 0, 6), (7, 0, 7),
(8, 0, 8), (9, 0, 9), (10, 0, 10)},
C15 = {(1, 0, 2), (2, 0, 4), (3, 0, 6), (4, 0, 8), (5, 0, 10), (6, 0, 1), (7, 0, 3),
(8, 0, 5), (9, 0, 7), (10, 0, 9)},
C16 = {(1, 0, 3), (2, 0, 6), (3, 0, 9), (4, 0, 1), (5, 0, 4), (6, 0, 7), (7, 0, 10),
(8, 0, 2), (9, 0, 5), (10, 0, 8)},
16
C17 = {(1, 0, 4), (2, 0, 8), (3, 0, 1), (4, 0, 5), (5, 0, 9), (6, 0, 2), (7, 0, 6),
(8, 0, 10), (9, 0, 3), (10, 0, 7)},
C18 = {(1, 0, 5), (2, 0, 10), (3, 0, 4), (4, 0, 9), (5, 0, 3), (6, 0, 8), (7, 0, 2),
(8, 0, 7), (9, 0, 1), (10, 0, 6)},
C19 = {(1, 0, 6), (2, 0, 1), (3, 0, 7), (4, 0, 2), (5, 0, 8), (6, 0, 3), (7, 0, 9),
(8, 0, 4), (9, 0, 10), (10, 0, 5)},
C20 = {(1, 0, 7), (2, 0, 3), (3, 0, 10), (4, 0, 6), (5, 0, 2), (6, 0, 9), (7, 0, 5),
(8, 0, 1), (9, 0, 8), (10, 0, 4)},
C21 = {(1, 0, 8), (2, 0, 5), (3, 0, 2), (4, 0, 10), (5, 0, 7), (6, 0, 4), (7, 0, 1),
(8, 0, 9), (9, 0, 6), (10, 0, 3)},
C22 = {(1, 0, 9), (2, 0, 7), (3, 0, 5), (4, 0, 3), (5, 0, 1), (6, 0, 10), (7, 0, 8),
(8, 0, 6), (9, 0, 4), (10, 0, 2)},
C23 = {(1, 0, 10), (2, 0, 9), (3, 0, 8), (4, 0, 7), (5, 0, 6), (6, 0, 5), (7, 0, 4),
(8, 0, 3), (9, 0, 2), (10, 0, 1)},
C24 = {(1, 1, 0), (2, 2, 0), (3, 3, 0), (4, 4, 0), (5, 5, 0), (6, 6, 0), (7, 7, 0),
(8, 8, 0), (9, 9, 0), (10, 10, 0)},
C25 = {(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4), (5, 5, 5), (6, 6, 6), (7, 7, 7),
(8, 8, 8), (9, 9, 9), (10, 10, 10)},
C26 = {(1, 1, 2), (2, 2, 4), (3, 3, 6), (4, 4, 8), (5, 5, 10), (6, 6, 1), (7, 7, 3),
(8, 8, 5), (9, 9, 7), (10, 10, 9)},
C27 = {(1, 1, 3), (2, 2, 6), (3, 3, 9), (4, 4, 1), (5, 5, 4), (6, 6, 7), (7, 7, 10),
(8, 8, 2), (9, 9, 5), (10, 10, 8)},
C28 = {(1, 1, 4), (2, 2, 8), (3, 3, 1), (4, 4, 5), (5, 5, 9), (6, 6, 2), (7, 7, 6),
(8, 8, 10), (9, 9, 3), (10, 10, 7)},
C29 = {(1, 1, 5), (2, 2, 10), (3, 3, 4), (4, 4, 9), (5, 5, 3), (6, 6, 8), (7, 7, 2),
(8, 8, 7), (9, 9, 1), (10, 10, 6)},
C30 = {(1, 1, 6), (2, 2, 1), (3, 3, 7), (4, 4, 2), (5, 5, 8), (6, 6, 3), (7, 7, 9),
(8, 8, 4), (9, 9, 10), (10, 10, 5)},
C31 = {(1, 1, 7), (2, 2, 3), (3, 3, 10), (4, 4, 6), (5, 5, 2), (6, 6, 9), (7, 7, 5),
(8, 8, 1), (9, 9, 8), (10, 10, 4)},
C32 = {(1, 1, 8), (2, 2, 5), (3, 3, 2), (4, 4, 10), (5, 5, 7), (6, 6, 4), (7, 7, 1),
(8, 8, 9), (9, 9, 6), (10, 10, 3)},
C33 = {(1, 1, 9), (2, 2, 7), (3, 3, 5), (4, 4, 3), (5, 5, 1), (6, 6, 10), (7, 7, 8),
(8, 8, 6), (9, 9, 4), (10, 10, 2)},
C34 = {(1, 1, 10), (2, 2, 9), (3, 3, 8), (4, 4, 7), (5, 5, 6), (6, 6, 5), (7, 7, 4),
(8, 8, 3), (9, 9, 2), (10, 10, 1)},
C35 = {(1, 2, 0), (2, 4, 0), (3, 6, 0), (4, 8, 0), (5, 10, 0), (6, 1, 0), (7, 3, 0),
(8, 5, 0), (9, 7, 0), (10, 9, 0)},
17
C36 = {(1, 2, 1), (2, 4, 2), (3, 6, 3), (4, 8, 4), (5, 10, 5), (6, 1, 6), (7, 3, 7),
(8, 5, 8), (9, 7, 9), (10, 9, 10)},
C37 = {(1, 2, 2), (2, 4, 4), (3, 6, 6), (4, 8, 8), (5, 10, 10), (6, 1, 1), (7, 3, 3),
(8, 5, 5), (9, 7, 7), (10, 9, 9)},
C38 = {(1, 2, 3), (2, 4, 6), (3, 6, 9), (4, 8, 1), (5, 10, 4), (6, 1, 7), (7, 3, 10),
(8, 5, 2), (9, 7, 5), (10, 9, 8)},
C39 = {(1, 2, 4), (2, 4, 8), (3, 6, 1), (4, 8, 5), (5, 10, 9), (6, 1, 2), (7, 3, 6),
(8, 5, 10), (9, 7, 3), (10, 9, 7)},
C40 = {(1, 2, 5), (2, 4, 10), (3, 6, 4), (4, 8, 9), (5, 10, 3), (6, 1, 8), (7, 3, 2),
(8, 5, 7), (9, 7, 1), (10, 9, 6)},
C41 = {(1, 2, 6), (2, 4, 1), (3, 6, 7), (4, 8, 2), (5, 10, 8), (6, 1, 3), (7, 3, 9),
(8, 5, 4), (9, 7, 10), (10, 9, 5)},
C42 = {(1, 2, 7), (2, 4, 3), (3, 6, 10), (4, 8, 6), (5, 10, 2), (6, 1, 9), (7, 3, 5),
(8, 5, 1), (9, 7, 8), (10, 9, 4)},
C43 = {(1, 2, 8), (2, 4, 5), (3, 6, 2), (4, 8, 10), (5, 10, 7), (6, 1, 4), (7, 3, 1),
(8, 5, 9), (9, 7, 6), (10, 9, 3)},
C44 = {(1, 2, 9), (2, 4, 7), (3, 6, 5), (4, 8, 3), (5, 10, 1), (6, 1, 10), (7, 3, 8),
(8, 5, 6), (9, 7, 4), (10, 9, 2)},
C45 = {(1, 2, 10), (2, 4, 9), (3, 6, 8), (4, 8, 7), (5, 10, 6), (6, 1, 5), (7, 3, 4),
(8, 5, 3), (9, 7, 2), (10, 9, 1)},
C46 = {(1, 3, 0), (2, 6, 0), (3, 9, 0), (4, 1, 0), (5, 4, 0), (6, 7, 0), (7, 10, 0),
(8, 2, 0), (9, 5, 0), (10, 8, 0)},
C47 = {(1, 3, 1), (2, 6, 2), (3, 9, 3), (4, 1, 4), (5, 4, 5), (6, 7, 6), (7, 10, 7),
(8, 2, 8), (9, 5, 9), (10, 8, 10)},
C48 = {(1, 3, 2), (2, 6, 4), (3, 9, 6), (4, 1, 8), (5, 4, 10), (6, 7, 1), (7, 10, 3),
(8, 2, 5), (9, 5, 7), (10, 8, 9)},
C49 = {(1, 3, 3), (2, 6, 6), (3, 9, 9), (4, 1, 1), (5, 4, 4), (6, 7, 7), (7, 10, 10),
(8, 2, 2), (9, 5, 5), (10, 8, 8)},
C50 = {(1, 3, 4), (2, 6, 8), (3, 9, 1), (4, 1, 5), (5, 4, 9), (6, 7, 2), (7, 10, 6),
(8, 2, 10), (9, 5, 3), (10, 8, 7)},
C51 = {(1, 3, 5), (2, 6, 10), (3, 9, 4), (4, 1, 9), (5, 4, 3), (6, 7, 8), (7, 10, 2),
(8, 2, 7), (9, 5, 1), (10, 8, 6)},
C52 = {(1, 3, 6), (2, 6, 1), (3, 9, 7), (4, 1, 2), (5, 4, 8), (6, 7, 3), (7, 10, 9),
(8, 2, 4), (9, 5, 10), (10, 8, 5)},
C53 = {(1, 3, 7), (2, 6, 3), (3, 9, 10), (4, 1, 6), (5, 4, 2), (6, 7, 9), (7, 10, 5),
(8, 2, 1), (9, 5, 8), (10, 8, 4)},
C54 = {(1, 3, 8), (2, 6, 5), (3, 9, 2), (4, 1, 10), (5, 4, 7), (6, 7, 4), (7, 10, 1),
(8, 2, 9), (9, 5, 6), (10, 8, 3)},
18
C55 = {(1, 3, 9), (2, 6, 7), (3, 9, 5), (4, 1, 3), (5, 4, 1), (6, 7, 10), (7, 10, 8),
(8, 2, 6), (9, 5, 4), (10, 8, 2)},
C56 = {(1, 3, 10), (2, 6, 9), (3, 9, 8), (4, 1, 7), (5, 4, 6), (6, 7, 5), (7, 10, 4),
(8, 2, 3), (9, 5, 2), (10, 8, 1)},
C57 = {(1, 4, 0), (2, 8, 0), (3, 1, 0), (4, 5, 0), (5, 9, 0), (6, 2, 0), (7, 6, 0),
(8, 10, 0), (9, 3, 0), (10, 7, 0)},
C58 = {(1, 4, 1), (2, 8, 2), (3, 1, 3), (4, 5, 4), (5, 9, 5), (6, 2, 6), (7, 6, 7),
(8, 10, 8), (9, 3, 9), (10, 7, 10)},
C59 = {(1, 4, 2), (2, 8, 4), (3, 1, 6), (4, 5, 8), (5, 9, 10), (6, 2, 1), (7, 6, 3),
(8, 10, 5), (9, 3, 7), (10, 7, 9)},
C60 = {(1, 4, 3), (2, 8, 6), (3, 1, 9), (4, 5, 1), (5, 9, 4), (6, 2, 7), (7, 6, 10),
(8, 10, 2), (9, 3, 5), (10, 7, 8)},
C61 = {(1, 4, 4), (2, 8, 8), (3, 1, 1), (4, 5, 5), (5, 9, 9), (6, 2, 2), (7, 6, 6),
(8, 10, 10), (9, 3, 3), (10, 7, 7)},
C62 = {(1, 4, 5), (2, 8, 10), (3, 1, 4), (4, 5, 9), (5, 9, 3), (6, 2, 8), (7, 6, 2),
(8, 10, 7), (9, 3, 1), (10, 7, 6)},
C63 = {(1, 4, 6), (2, 8, 1), (3, 1, 7), (4, 5, 2), (5, 9, 8), (6, 2, 3), (7, 6, 9),
(8, 10, 4), (9, 3, 10), (10, 7, 5)},
C64 = {(1, 4, 7), (2, 8, 3), (3, 1, 10), (4, 5, 6), (5, 9, 2), (6, 2, 9), (7, 6, 5),
(8, 10, 1), (9, 3, 8), (10, 7, 4)},
C65 = {(1, 4, 8), (2, 8, 5), (3, 1, 2), (4, 5, 10), (5, 9, 7), (6, 2, 4), (7, 6, 1),
(8, 10, 9), (9, 3, 6), (10, 7, 3)},
C66 = {(1, 4, 9), (2, 8, 7), (3, 1, 5), (4, 5, 3), (5, 9, 1), (6, 2, 10), (7, 6, 8),
(8, 10, 6), (9, 3, 4), (10, 7, 2)},
C67 = {(1, 4, 10), (2, 8, 9), (3, 1, 8), (4, 5, 7), (5, 9, 6), (6, 2, 5), (7, 6, 4),
(8, 10, 3), (9, 3, 2), (10, 7, 1)},
C68 = {(1, 5, 0), (2, 10, 0), (3, 4, 0), (4, 9, 0), (5, 3, 0), (6, 8, 0), (7, 2, 0),
(8, 7, 0), (9, 1, 0), (10, 6, 0)},
C69 = {(1, 5, 1), (2, 10, 2), (3, 4, 3), (4, 9, 4), (5, 3, 5), (6, 8, 6), (7, 2, 7),
(8, 7, 8), (9, 1, 9), (10, 6, 10)},
C70 = {(1, 5, 2), (2, 10, 4), (3, 4, 6), (4, 9, 8), (5, 3, 10), (6, 8, 1), (7, 2, 3),
(8, 7, 5), (9, 1, 7), (10, 6, 9)},
C71 = {(1, 5, 3), (2, 10, 6), (3, 4, 9), (4, 9, 1), (5, 3, 4), (6, 8, 7), (7, 2, 10),
(8, 7, 2), (9, 1, 5), (10, 6, 8)},
C72 = {(1, 5, 4), (2, 10, 8), (3, 4, 1), (4, 9, 5), (5, 3, 9), (6, 8, 2), (7, 2, 6),
(8, 7, 10), (9, 1, 3), (10, 6, 7)},
C73 = {(1, 5, 5), (2, 10, 10), (3, 4, 4), (4, 9, 9), (5, 3, 3), (6, 8, 8), (7, 2, 2),
(8, 7, 7), (9, 1, 1), (10, 6, 6)},
19
C74 = {(1, 5, 6), (2, 10, 1), (3, 4, 7), (4, 9, 2), (5, 3, 8), (6, 8, 3), (7, 2, 9),
(8, 7, 4), (9, 1, 10), (10, 6, 5)},
C75 = {(1, 5, 7), (2, 10, 3), (3, 4, 10), (4, 9, 6), (5, 3, 2), (6, 8, 9), (7, 2, 5),
(8, 7, 1), (9, 1, 8), (10, 6, 4)},
C76 = {(1, 5, 8), (2, 10, 5), (3, 4, 2), (4, 9, 10), (5, 3, 7), (6, 8, 4), (7, 2, 1),
(8, 7, 9), (9, 1, 6), (10, 6, 3)},
C77 = {(1, 5, 9), (2, 10, 7), (3, 4, 5), (4, 9, 3), (5, 3, 1), (6, 8, 10), (7, 2, 8),
(8, 7, 6), (9, 1, 4), (10, 6, 2)},
C78 = {(1, 5, 10), (2, 10, 9), (3, 4, 8), (4, 9, 7), (5, 3, 6), (6, 8, 5), (7, 2, 4),
(8, 7, 3), (9, 1, 2), (10, 6, 1)},
C79 = {(1, 6, 0), (2, 1, 0), (3, 7, 0), (4, 2, 0), (5, 8, 0), (6, 3, 0), (7, 9, 0),
(8, 4, 0), (9, 10, 0), (10, 5, 0)},
C80 = {(1, 6, 1), (2, 1, 2), (3, 7, 3), (4, 2, 4), (5, 8, 5), (6, 3, 6), (7, 9, 7),
(8, 4, 8), (9, 10, 9), (10, 5, 10)},
C81 = {(1, 6, 2), (2, 1, 4), (3, 7, 6), (4, 2, 8), (5, 8, 10), (6, 3, 1), (7, 9, 3),
(8, 4, 5), (9, 10, 7), (10, 5, 9)},
C82 = {(1, 6, 3), (2, 1, 6), (3, 7, 9), (4, 2, 1), (5, 8, 4), (6, 3, 7), (7, 9, 10),
(8, 4, 2), (9, 10, 5), (10, 5, 8)},
C83 = {(1, 6, 4), (2, 1, 8), (3, 7, 1), (4, 2, 5), (5, 8, 9), (6, 3, 2), (7, 9, 6),
(8, 4, 10), (9, 10, 3), (10, 5, 7)},
C84 = {(1, 6, 5), (2, 1, 10), (3, 7, 4), (4, 2, 9), (5, 8, 3), (6, 3, 8), (7, 9, 2),
(8, 4, 7), (9, 10, 1), (10, 5, 6)},
C85 = {(1, 6, 6), (2, 1, 1), (3, 7, 7), (4, 2, 2), (5, 8, 8), (6, 3, 3), (7, 9, 9),
(8, 4, 4), (9, 10, 10), (10, 5, 5)},
C86 = {(1, 6, 7), (2, 1, 3), (3, 7, 10), (4, 2, 6), (5, 8, 2), (6, 3, 9), (7, 9, 5),
(8, 4, 1), (9, 10, 8), (10, 5, 4)},
C87 = {(1, 6, 8), (2, 1, 5), (3, 7, 2), (4, 2, 10), (5, 8, 7), (6, 3, 4), (7, 9, 1),
(8, 4, 9), (9, 10, 6), (10, 5, 3)},
C88 = {(1, 6, 9), (2, 1, 7), (3, 7, 5), (4, 2, 3), (5, 8, 1), (6, 3, 10), (7, 9, 8),
(8, 4, 6), (9, 10, 4), (10, 5, 2)},
C89 = {(1, 6, 10), (2, 1, 9), (3, 7, 8), (4, 2, 7), (5, 8, 6), (6, 3, 5), (7, 9, 4),
(8, 4, 3), (9, 10, 2), (10, 5, 1)},
C90 = {(1, 7, 0), (2, 3, 0), (3, 10, 0), (4, 6, 0), (5, 2, 0), (6, 9, 0), (7, 5, 0),
(8, 1, 0), (9, 8, 0), (10, 4, 0)},
C91 = {(1, 7, 1), (2, 3, 2), (3, 10, 3), (4, 6, 4), (5, 2, 5), (6, 9, 6), (7, 5, 7),
(8, 1, 8), (9, 8, 9), (10, 4, 10)},
C92 = {(1, 7, 2), (2, 3, 4), (3, 10, 6), (4, 6, 8), (5, 2, 10), (6, 9, 1), (7, 5, 3),
(8, 1, 5), (9, 8, 7), (10, 4, 9)},
20
C93 = {(1, 7, 3), (2, 3, 6), (3, 10, 9), (4, 6, 1), (5, 2, 4), (6, 9, 7), (7, 5, 10),
(8, 1, 2), (9, 8, 5), (10, 4, 8)},
C94 = {(1, 7, 4), (2, 3, 8), (3, 10, 1), (4, 6, 5), (5, 2, 9), (6, 9, 2), (7, 5, 6),
(8, 1, 10), (9, 8, 3), (10, 4, 7)},
C95 = {(1, 7, 5), (2, 3, 10), (3, 10, 4), (4, 6, 9), (5, 2, 3), (6, 9, 8), (7, 5, 2),
(8, 1, 7), (9, 8, 1), (10, 4, 6)},
C96 = {(1, 7, 6), (2, 3, 1), (3, 10, 7), (4, 6, 2), (5, 2, 8), (6, 9, 3), (7, 5, 9),
(8, 1, 4), (9, 8, 10), (10, 4, 5)},
C97 = {(1, 7, 7), (2, 3, 3), (3, 10, 10), (4, 6, 6), (5, 2, 2), (6, 9, 9), (7, 5, 5),
(8, 1, 1), (9, 8, 8), (10, 4, 4)},
C98 = {(1, 7, 8), (2, 3, 5), (3, 10, 2), (4, 6, 10), (5, 2, 7), (6, 9, 4), (7, 5, 1),
(8, 1, 9), (9, 8, 6), (10, 4, 3)},
C99 = {(1, 7, 9), (2, 3, 7), (3, 10, 5), (4, 6, 3), (5, 2, 1), (6, 9, 10), (7, 5, 8),
(8, 1, 6), (9, 8, 4), (10, 4, 2)},
C
100
= {(1, 7, 10), (2, 3, 9), (3, 10, 8), (4, 6, 7), (5, 2, 6), (6, 9, 5), (7, 5, 4),
(8, 1, 3), (9, 8, 2), (10, 4, 1)},
C
101
= {(1, 8, 0), (2, 5, 0), (3, 2, 0), (4, 10, 0), (5, 7, 0), (6, 4, 0), (7, 1, 0),
(8, 9, 0), (9, 6, 0), (10, 3, 0)},
C
102
= {(1, 8, 1), (2, 5, 2), (3, 2, 3), (4, 10, 4), (5, 7, 5), (6, 4, 6), (7, 1, 7),
(8, 9, 8), (9, 6, 9), (10, 3, 10)},
C
103
= {(1, 8, 2), (2, 5, 4), (3, 2, 6), (4, 10, 8), (5, 7, 10), (6, 4, 1), (7, 1, 3),
(8, 9, 5), (9, 6, 7), (10, 3, 9)},
C
104
= {(1, 8, 3), (2, 5, 6), (3, 2, 9), (4, 10, 1), (5, 7, 4), (6, 4, 7), (7, 1, 10),
(8, 9, 2), (9, 6, 5), (10, 3, 8)},
C
105
= {(1, 8, 4), (2, 5, 8), (3, 2, 1), (4, 10, 5), (5, 7, 9), (6, 4, 2), (7, 1, 6),
(8, 9, 10), (9, 6, 3), (10, 3, 7)},
C
106
= {(1, 8, 5), (2, 5, 10), (3, 2, 4), (4, 10, 9), (5, 7, 3), (6, 4, 8), (7, 1, 2),
(8, 9, 7), (9, 6, 1), (10, 3, 6)},
C
107
= {(1, 8, 6), (2, 5, 1), (3, 2, 7), (4, 10, 2), (5, 7, 8), (6, 4, 3), (7, 1, 9),
(8, 9, 4), (9, 6, 10), (10, 3, 5)},
C
108
= {(1, 8, 7), (2, 5, 3), (3, 2, 10), (4, 10, 6), (5, 7, 2), (6, 4, 9), (7, 1, 5),
(8, 9, 1), (9, 6, 8), (10, 3, 4)},
C
109
= {(1, 8, 8), (2, 5, 5), (3, 2, 2), (4, 10, 10), (5, 7, 7), (6, 4, 4), (7, 1, 1),
(8, 9, 9), (9, 6, 6), (10, 3, 3)},
C
110
= {(1, 8, 9), (2, 5, 7), (3, 2, 5), (4, 10, 3), (5, 7, 1), (6, 4, 10), (7, 1, 8),
(8, 9, 6), (9, 6, 4), (10, 3, 2)},
C
111
= {(1, 8, 10), (2, 5, 9), (3, 2, 8), (4, 10, 7), (5, 7, 6), (6, 4, 5), (7, 1, 4),
(8, 9, 3), (9, 6, 2), (10, 3, 1)},