第二章 密码学的数学基础.pdf
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1
•
A(0),B(1),…,Z(25)
•
2
• b ma, a b
2 6,3 18,−5 25,3 8?
•
−a b ∧b c ⇒a c
a b b a a b
− ∧ ⇒
−a b ∧a c ⇒a bx ±cy
proof ?
3
• p+p,
-p,+1,-1
2,3,5,7,11,13,17,…,101,…
• P
• n
4,9,187,900,…
4
• Theorem:(Fundamental Theorem of Arithmetic)
e1 e2 ek ei
∀n∈N n= p1 p2 …pk ( or Π p )
p ∈P
i
where ep is the exponent of the prime factor p
• Note: the result of factorization is unique
• Example: 84=22×3×7
5
• Theorem: There are infinitely many primes
• Proof: (by contradiction)
Assume P , build a number N is
max
N P P ...P +1
1 2 max
There N is a new prime.
6
• Definition: the greatest common divisor (GCD) is
the number c
c gcd(a,b) max{d d a ∧d b}
• Properties:
•d n1, d n2 ,..., d nk ⇒d gcd(n1,..., nk )
gcd( , ) gcd( , mod )
• a b b a b
: mod 0
proof
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