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An introduction to finite fields, their properties, and applications in cryptography. It covers the concepts of groups, rings, fields, modular arithmetic, divisors, and polynomial arithmetic. The document also includes examples and algorithms for finding greatest common divisors and inverses in finite fields.
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, Len Howard
-3^ a= a.a.a
k^ for some
a^ and every
b^ in group
^ a set of “numbers” ^ with two operations (addition and multiplication)which form: ^ an abelian group with addition operation ^ and multiplication:^ ^ has closure^ ^ is associative^ ^ distributive over addition:
a(b+c) = ab + ac ^ if multiplication operation is commutative, itforms a^ commutative ring ^ if multiplication operation has an identity and nozero divisors, it forms an
integral domain
“a mod n”
to be
remainder when a is divided by n use the term
congruence
for:^ a = b mod n ^ when divided by
n,^ a & b have same remainder ^ eg. 100 = 34 mod 11 b is called a
residue^ of a mod n ^ since with integers can always write:
a = qn + b
^ usually chose smallest positive remainder as residue• ie.^0 <= b^ <=^ n- ^ process is known as
modulo reduction
7 =^ -5^ mod
7 =^2 mod^
7 =^9 mod^7
only if^ a^ is relatively prime to
n docsity.com
A^ =^ gcd(a,
b)
3.^ R^ =^ A^ mod
B
4.^ A^ =^ B 5.^ B^ =^ R 6.^ goto^2
gcd(1066, 904) 1066 = 1 x 904 + 162
gcd(904, 162) 904 = 5 x 162 + 94
gcd(162, 94) 162 = 1 x 94 + 68
gcd(94, 68) 94 = 1 x 68 + 26
gcd(68, 26) 68 = 2 x 26 + 16
gcd(26, 16) 26 = 1 x 16 + 10
gcd(16, 10) 16 = 1 x 10 + 6
gcd(10, 6) 10 = 1 x 6 + 4
gcd(6, 4) 6 = 1 x 4 + 2
gcd(4, 2) 4 = 2 x 2 + 0
gcd(2, 0)
n-1^ x + … + a
x +^ a^ =^ ∑ 10
i a x i