NUMBER THEORY AND ALGEBRA. ℤ set of integers { . . . . -3, -2, -1, 0, 1, 2, 3, . . . } a , b , c , d - integers & belong to set ℤ algebraic operations –: “+”, “-”, and “ ” – valid with set a + b , a – b , a + b + c + d , a b , b d
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set with infinite number of members
other examples of rings, commutative rings, infinite commutative rings?
Algorithm 1.1 Euclidean Algorithm
Input: a, b
Output: gcd (a, b)
r0 ←a
r1 ←b
n ←1
while rn ≠ 0
n ← n – 1
gcd (a, b) ← rn
Algorithm 1.2 Extended Euclidean Algorithm
Input: a, b: Output: gcd (a, b); u, v
r0 ← a; r1 ← b
u0 ← 1; u1 ← 0
v0 ← 0; v1 ← 0
n ← 1
while (rn+1 ≠ 0)
n ← n -1
gcd(a, b) ← rn; u ← un; v ← vn
Add & subtract
(u0 + kb) a + (v0 – k b ) = c
←generalized version
2 ≡ - 10 (mod 12) ←negative numbers
congruence property expressed as
Cryptography starts here
result in ℤ7
See table for general addition of two numbers a and b (mod 7)
Table for ‘mod 7’ multiplication
All non-zero elements of ℤ7 & their respective inverses
Table multiplication table for ℤ6
iffgcd (a, m) = 1
Use multiplicative inverse to carry out equivalent of division in ℤm
gcd (a, b) =1 a & brelatively prime
total number of elements in ℤ*m (m)
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( m1) ( m2) ( m3). . . . ( mk)
Algorithm 1.3 Fast Exponentiation Algorithm
a(mod p), 2a(mod p), 3a(mod p) all non-zero
563 prime a562≡ 1(mod 563) a ℤ563
ai(mod11) values for all a and i values
gi(mod p) takes all values in ℤp as i changes from 1 to p – 1 g is a ‘primitive element’ of ℤp
2, 6, 7, & 8 primitive elements of ℤ11
x: ‘discrete logarithm’ of h to base g
Note apparent lack of order in dependent variable values
Discrete logarithm of ratio of two elements
Find discrete logarithm of (6)(437)-1 in ℤ1319- base 13
h, hf, hf2, . . . hfn.
& ( n + 1) entries
x≡ a2(mod m2), x≡ a3(mod m3), , x≡ at(mod mt)
has a solution.
c2≡c1(mod (m1 m2 m3. . . mt))
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≡ 4(mod 7) ≡ 2(mod 11)
All powers of ≡ 1 (mod 17)
≡ 78 (mod 17)
≡ 16 (mod 17) x0 = 1
x1 = 1
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DL based on the factors of p-1
g, h, & x ℤp: g is a primitive element of ℤp
Factorize p-1
q1, q2, . . qt are primes & e1, e2, . . et respective integer exponents.
Obtain for all i from 1 to t.
Evaluate for all i from 1 to t.
Evaluate for all i from 1 to t.
Use procedure of last algorithm & obtain DL - xi of hi to base gi for all i from 1 to t
Note: definition of hi & gi implies xi exists
Express x as a set of multiple congruences
:k – an integer
implies the congruence x≡x1(mod
x≡x2(mod . . . x≡xt(mod
Use Chinese remainder theorem & solve above congruences & evaluate x
&
x20 = 1
$
Use g2 & h2
x23-digit ternary number:
Take 32 power & simplify using #
Substitute in #, use 17365-1≡ 7406 (mod 18523) & simplify
Take 3rd power & simplify
x21 = 1
$
Leave out PPTs – 35, 36, 37, 54, 82, 83, 87 – 102 : All these have ‘ ’ mark at top right corner
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