Cryptography Reference
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where the coe 7 cients a j are in R for j
0 and a j = 0 for all but a finite
number of those values of j . The set of all such polynomials is denoted by R [ x ].
If a n
= 0, and a j = 0 for j>n , then a n is called the leading coe 7 cient of f ( x ).
If the leading coeGcient a n =1 R , in the case where R is a commutative ring
with identity 1 R , then f ( x ) is said to be monic .
We may add two polynomials from R [ x ], f ( x )= j =0 a j x j and g ( x )=
j =0 b j x j ,by
f ( x )+ g ( x )=
( a j + b j ) x j
R [ x ] ,
j =0
and multiply them by
f ( x ) g ( x )=
c j x j ,
j =0
where
j
c j =
a i b j i .
i =0
Also, f ( x )= g ( x ) if and only if a j = b j for all j =0 , 1 ,... . Under the above
operations R [ x ]is a ring, called the polynomial ring over R in the indeterminant
x . Furthermore, if R is commutative, then so is R [ x ], and if R has identity 1 R ,
then 1 R is the identity for R [ x ]. Notice that with these conventions, we may
write f ( x )= j =0 a j x j , for some n
N
, where a n is the leading coeGcient
since we have tacitly agreed to “ignore” zero terms.
If α
R , we write f ( α ) to represent the element j =0 a j α j
R , called the
substitution of α for x . When f ( α ) = 0, then α is called a root of f ( x ). The
substitution gives rise to a mapping
f : R
R given by f : α
f ( α ) ,
which is determined by f ( x ). Thus, f is called a polynomial function over R .
Definition A.32 ( Degrees of Polynomials ) If f ( x )
R [ x ] , with f ( x )=
j =0 a j x j , and a d
0 is called the degree of f ( x ) over R , denoted
by deg R ( f ) . If no such d exists, we write deg R ( f )=
=0 , then d
, in which case f ( x )
is the zero polynomial in R [ x ] .If F is a field of characteristic zero, then
−∞
deg
( f ) = deg F ( f )
Q
for any f ( x )
Q
[ x ] .If F has characteristic p , and f ( x )
F p [ x ] , then
F p ( f ) = deg F ( f ) .
In either case, we write deg( f ) for deg F ( f ) , without loss of generality, and call
this the degree of f ( x ) .
deg
With respect to roots of polynomials, the following is important.
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