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Mathematics of Cryptography
Prepared By: Shaikh Amrin
2.1 Integer arithmeticSet of integers
Z={….,-2,-1,0,1,2,…}In cryptography we are working in binary operations applied to set of integers like addition, subtraction, multiplication, division.
Integer division
n(divisor) 23 q(quotient)11 255 a(divident) 22 35 33 2 r(remainder)
Restrictions:1. Divisor be a positive number2. Reminder be a positive number
255=(11*23)+2-255=(11*-23)+-2Here in 2nd example r=-2How can we make it positive?
Add(-1) to q and add the value n to r.
-255=(11*-24)+9
divisibilityIf a is not zero and r=0 in division
relationSo, we can say that
n divides a or n is divisor of a ora is divisible by n.Denoted by n|aEx: 13|78,7|98,-6|24
Properties of divisibility
1. If a|1, then a=+-12. If a|b and b|c, then a=+-b3. If a|b and b|c, then a|c4. If a|b and a|c, then a|(m*b+n*c)
where m and n are arbitrary integers
If 3|15 and 15|45 then 3|45If 3|15 and 3|9 then 3|
(15*1+9*1)
Greatest Common Divisor
Euclidean AlgorithmGcd(a,0)=aGcd(a,b)=gcd(b,r) where r=a%b
Ex:1 Ex:2Gcd(36,12) Gcd(50,3)=Gcd(12,0) =Gcd(3,2)=12 =Gcd(2,1)
=1
Find GCD(2740,1760)
q r1 r2 r
1 2740 1760 980
1 1760 980 780
1 980 780 200
3 780 200 180
1 200 180 20
9 180 20 0
20 0
Find GCD(25,60)
Calculate GCD(17,0),GCD(0,45)
Extended Euclidean AlgorithmGiven two integers a and b we
often need to find s & t such that(s*a)+(t*b)=gcd(a,b)
It is useful when a and b are co-prime.
compute multiplicative inverse.
q r1 r2 r s1 s2 s t1 t2 t
1 0 0 1
s=s1-(q*s2)t=t1-(q*t2)
q r1 r2 r s1 s2
s t1 t2 t
5 161
28
21
1 0 1-(5*0)=1
0 1 0-(5*1)=-5
Find gcd(161,28) and also calculate s and t
q r1 r2 r s1 s2
s t1 t2 t
5 161
28
21
1 0 1-(5*0)=1
0 1 0-(5*1)=-5
28 21
0 1 1 -5
Find gcd(161,28) and also calculate s and t
q r1 r2 r s1 s2
s t1 t2 t
5 161
28
21
1 0 1-(5*0)=1
0 1 0-(5*1)=-5
1 28 21
7 0 1 0-(1*1)=-1
1 -5 1-(1*-5)=6
Find gcd(161,28) and also calculate s and t
q r1 r2 r s1 s2
s t1 t2 t
5 161
28
21
1 0 1-(5*0)=1
0 1 0-(5*1)=-5
1 28 21
7 0 1 0-(1*1)=-1
1 -5 1-(1*-5)=6
3 21 7 0 1 -1 1-(3*-1)=4
-5 6 -5-(3*6)=-23
Find gcd(161,28) and also calculate s and t
q r1 r2 r s1 s2
s t1 t2 t
5 161
28
21
1 0 1-(5*0)=1
0 1 0-(5*1)=-5
1 28 21
7 0 1 0-(1*1)=-1
1 -5 1-(1*-5)=6
3 21 7 0 1 -1 1-(3*-1)=4
-5 6 -5-(3*6)=-23
7 0 -1 4 6 -23
Find gcd(161,28) and also calculate s and t
gcd(161,28)=r1=7 and also s=-1 and t=6
(s*a)+(t*b)=gcd(a,b)=> (-1*161)+(6*28)=gcd(161,28)=7
Modular Arithmetic
a=q*n+rn is called modulusr is called as residuezn is called as set of residues {0,1…n-1}
Ex:n=11,a=3535=3*11+2Z11={0,1,2,3,4,5,6,7,8,9,10}
CongruenceMapping from Z to Zn is not one-one.
For ex: 2%10=2,12%10=2…
so,{2,12,22} are called congruent to n.
is called as congruent operator.
a is congruent to b mod m if m divides b-a.
a b mod m
-2 19 mod 21
Any integer is congruent to itself modulo m (reflexivity) a a mod m.
a b mod m implies that b a mod m (Symmetry)
a b mod m and b c mod m implies that a c mod m (transitivity)
Residue classes of a%m m=4 1 % 4={1,1+-4,1+-(2*4),1+-
(3*4)..}2 % 4={2,2+-4,2+-(2*4),2+-
(3*4)..}3 % 4={3,3+-4,3+-(2*4),3+-
(3*4)..}0 % 4={0,0+-4,0+-(2*4),0+-
(3*4)..}
Inverse
Additive inverse:
In Zn two numbers a,b are additive inverse of each other if a+b 0 mod n
Ex: additive inverse of 4 in Z10 =10-4=6
Multiplicative InverseIn Zn two numbers a,b are multiplicative inverse of each other if a*b 1 mod n
Ex: multiplicative inverse of 3 in Z10
=3*7%10=1
q r1 r2 r t1 t2 t
3 10 3 1 0 1 -3
3 3 1 0 1 -3 10
1 0 -3=7 10=0
Z26={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
Z26*={1,3,5,7,9,11,15,17,19,21,23}
Matrix Inverse
M-1 = 1/det(M)*MT
