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Greatest Common
Divisor:

A positive integer d is called a common divisor of the
integers a and b, if d divides a and b. The greatest possible such d is called
the greatest common divisor of a and b, denoted gcd(a,b).If = 1 gcd(a,b) then
a,b are called relatively prime.

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The Euclidean
Algorithm For Finding GCD:

EUCLID(a,b)

1.    
If b==0

2.    
Return a

3.    
Else return EUCLID(b,a
mod b)

As an example of
the running of EUCLID, consider the computation of  gcd(30,21)

EUCLID(30,21)=
EUCLID(21,9)

= EUCLID(9,3)

= EUCLID(3,0)

=3

This computation calls EUCLID recursively three times.

The algorithm returns a in line 2, if b = 0, so that equation
(31.9) implies that gcd(a,b) = gcd.(a,0) = a. The algorithm cannot recurse
inde?nitely, since the second argument strictly decreases in each recursive
call and is always non negative. Therefore, EUCLID always terminates with the
correct answer.

Running Time Analysis of Euclid’s
Algorithm:

We analyze the worst-case running time of EUCLID as a
function of the size of a and b. We assume with no loss of generality that
a>b>= 0. To justify this assumption, observe that if b>a>= 0, then
EUCLID(a,b) immediately makes the recursive call EUCLID(b,a). That is, if the
?rst argument is less than the second argument, EUCLID spends one recursive
call swapping its arguments and then proceeds. Similarly, if        b = a>0, the

procedure terminates after one recursive call, since a mod b
=0.

The overall
running time of EUCLID is proportional to the number of recursive calls it
makes.

 

The Extended
Euclidean Algorithm For Finding GCD:

We extend the algorithm to compute the integer coef?cients x
and y such that

d =gcd(a,b)= ax
+by

EXTENDED-EUCLID.(a,b)

1.    
If b==0

2.    
Return(a,1,0)

3.    
else
(d’,x’,y’)=EXTENDED-EUCLID(b,a mod b)

4.    
(d,x,y)=(d’,x’,y’-a/by’)

5.    
return(d,x,y)

The
EXTENDED-EUCLID procedure is a variation of the EUCLID procedure. Line 1 is
equivalent to the condition in SIMPLE EUCLIDEAN b == 0 in line 1 of EUCLID. If
b = 0, then EXTENDED-EUCLID returns not only d=a in line 2 but also the coefficients
x=1 and y=0 so that a=ax+by.If b not equal to zero, EXTENDED-EUCLID first
computes(d’,x’,y’) such that d’=gcd(b,a mod b) and d’=bx’+(b,a mod b).

Since the number
of recursive calls made in EUCLID is equal to the number of recursive calls
made in EXTENDED-EUCLID, the running times of EUCLID and EXTENDED-EUCLID are
the same, to within a constant factor. That is, for a>b>0, the number of
recursive calls is O.lgb

 

The RSA
Public-Key Cryptosystem:

RSA (Rivest–
Shamir– Adleman) is one of the main open key cryptosystems and is broadly
utilized for secure information transmission. In such a cryptosystem, the
encryption key is public and it is unique in relation to the decoding key which
is kept private. In RSA, this asymmetry depends on the practically trouble of
the factorization of the result of two extensive prime numbers, the
“factoring issue”.

Public-key
cryptosystem:

Private Key cryptography, or asymmetric
cryptography, is an encryption conspire that utilizations two numerically
related, however not indistinguishable, keys – an open key and a private key.
Not at all like symmetric key calculations that depend on one key to both encode
and decode, each key plays out a remarkable capacity. General society key is
utilized to scramble and the private key is utilized to decode.

It is computationally infeasible to register
the private key depend on public (general) key. With these lines, open keys can
be free of cost shared, permitting clients a simple and advantageous strategy
for encryption content and checking advanced marks, and private keys can be
kept private, guaranteeing just the proprietors of the private keys can
decryption content and make computerized marks.

Since open keys
should be shared however are too huge to be effectively recalled that, they are
put away on computerized declarations for secure transport and sharing. Since
private keys are not shared, they are essentially put away in the product or
working framework you utilize, or on equipment (e.g., USB token, equipment
security module) containing drivers that enable it to be utilized with your
product or working framework.

RSY
Cryptosystem:

Public key algorithm

i- Key1 (public key use for encryption).

ii- Key2 (private key use for decryption).

Encrypting and decrypting use modular exponentiation

Algorithm:

i-                  
Choose two large prime no. P and Q such
that P != Q

ii-               
Calculate N=P*Q

iii-             
Choose E (Public Key) such that E
is not a factor of (P-1)*(Q-1).

iv-             
Choose  D (Private Key) such that (D*E)mod
(P-1)*(Q-1)=1

v-               
Cipher Text (C.T) = (P.T) E mod
N.

vi-             
Plain Text (P.T) = (C.T) D
mod N.

 

Example:

                      A (Sender)                                                                              B(Receiver)

            (Sender A
want to send 5)

P=7, Q=13

            N=7*11=77

            (P-1)(Q-1)=>6*10=60                                    (D*E) mod
60=1

            E=13 (Public
Key)                                         D=37
(Private Key)

            C.T=(5)13
mod 77                                                                   C.T=26

            C.T=26                                                                                   P.T=(26)37
mod 77

                                                                                    (Receiver
receive the value 5 by sender)

                                                                                               

 

 

 

 

Solving Modular
linear equation:

 

As u realizes that this sort of condition is utilized as a
part of Cryptography, so it is issue that how to unravel this condition to
discovering key, utilized. For instance in the event that somebody got an
information having some esteem which is scrambled with some component, now recipient
need to decode it however don’t have a clue about the unscrambling key so it
might be conceivable that key esteem is found by taking modulus with number n
(it is rely upon encryption strategy key if utilizing same procedure for
encryption or it is predefined).

So for this Problem we have answer for fathom this sort of
condition utilizing following advances: assume that a, b, and n are given

Find gcd(a,n)=d i.e d=ax+by

On the off chance that d|b at that point there is arrangement
(at that point there is further strides to settle)

Else no any arrangement                                                              

Algorithm for
Modular equation

For measured Equation there is calculation (Algorithm) known
as MODULAR-LINEAR-EQUATION-SOLVER (a, b, n); MODULAR-LINEAR-EQUATION-SOLVER (a, b, n);

1.
(d, x’, y’) = EXTENDED-EUCLID (a, n)

2.
If d|b

3.           Xo= x’ (b/d) mod n

4.           For i = 0 to d -1

 5.             
Print Xi= (Xo+ i(n/d)) mod n

6.
Else print “no solutions”

Presently I will clarify how these lines functions when we
give some estimation of a, b, n. For instance of the operation of this
strategy, consider the condition 15x ? 40(mod 50) (here, a =14, b =30, and n
=100). Calling EXTENDED-E UCLID in line 1, we register (d, x’, y’) = (5,- 3,
and 1). Since 5 | 50 at that point, lines 3– 5 will execute. Line 3 figures Xo
= (- 3) (8) mod 50 = 26. At that point the circle on lines 4– 5 prints the five
arrangements 26, 36, 46, 6 and 16 by executing line 5

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