dissertacao.pdf

If we want to solve a system of linear modular equations there is one efficient
theorem which provides the correct answer quickly:
Theorem 1. (Chinese Remainder Theorem): Let a1, ..., .an be n integers
and p1, ..., pn be relatively prime positive integers. Set P = n
i=1 pi and, for
i = 1, ..., n define yi such that:
Then, one solution of the system
is given by:
⎧
⎪⎨
⎪⎩
P
yi
∼= 1 (mod pi) (9)
pi
x ∼ = a1 (mod p1)
x ∼ = a2 (mod p2)
.
x ∼ = an (mod pn)
x0 =
n
i=1
P
aiyi
pi
Any other integer solution, x, of the system of congruences satisfies:
1.5.4 Useful Algorithms and Results
(10)
x ∼ = x0 (mod P ) (11)
We now include some useful definitions and results used in the proofs of the
attacks presented in chapter 3.
Definition 12. Let
p(x) = anx n + an−1x n−1 + ... + a1x + a0
be a polynomial of degree n with roots αi for i = 1, ..., n and
q(x) = bmx m + bm−1x m−1 + ... + b1x + b0
be a polynomial of degree m with roots βi for i = 1, ..., n.
The Resultant of p and q, denoted Resultant(p, q), is defined by
n
Resultant(p, q) = a m n b n m
i=1 j=1
10
(12)
(13)
m
(αi − βi) (14)