# Implementation of Diffie-Hellman Algorithm

Background

Elliptic Curve Cryptography (ECC) is an approach to public-key cryptography, based on the algebraic structure of elliptic curves over finite fields. ECC requires a smaller key as compared to non-ECC cryptography to provide equivalent security (a 256-bit ECC security have an equivalent security attained by 3072-bit RSA cryptography).

For a better understanding of Elliptic Curve Cryptography, it is very important to understand the basics of Elliptic Curve. An elliptic curve is a planar algebraic curve defined by an equation of the form where ‘a’ is the co-efficient of x and ‘b’ is the constant of the equation

The curve is non-singular; that is its graph has no cusps or self-intersections (when the characteristic of the
co-efficient field is equal to 2 or 3).
In general, an elliptic curve looks like as shown below. Elliptic curves could intersect atmost 3 points when a straight line is drawn intersecting the curve. As we can see that elliptic curve is symmetric about the x-axis, this property plays a key role in the algorithm. Diffie-Hellman algorithm

The Diffie-Hellman algorithm is being used to establish a shared secret that can be used for secret
communications while exchanging data over a public network using the elliptic curve to generate points and get the secret key using the parameters.

• For the sake of simplicity and practical implementation of the algorithm, we will consider only 4 variables one prime P and G (a primitive root of P) and two private values a and b.
• P and G are both publicly available numbers. Users (say Alice and Bob) pick private values a and b and they generate a key and exchange it publicly, the opposite person received the key and from that generates a secret key after which they have the same secret key to encrypt.

Step by Step Explanation

Alice Bob
Public Keys available = P, G Public Keys available = P, G
Private Key Selected = a Private Key Selected = b
Key generated = Key generated = Exchange of generated keys takes place
Generated Secret Key = Generated Secret Key = Algebraically it can be shown that Users now have a symmetric secret key to encrypt

Example

```Step 1: Alice and Bob get public numbers P = 23, G = 9

Step 2: Alice selected a private key a = 4 and
Bob selected a private key b = 3

Step 3: Alice and Bob compute public values
Alice:    x =(9^4 mod 23) = (6561 mod 23) = 6
Bob:    y = (9^3 mod 23) = (729 mod 23)  = 16

Step 4: Alice and Bob exchange public numbers

Step 5: Alice receives public key y =16 and
Bob receives public key x = 6

Step 6: Alice and Bob compute symmetric keys
Alice:  ka = y^a mod p = 65536 mod 23 = 9
Bob:    kb = x^b mod p = 216 mod 23 = 9

Step 7: 9 is the shared secret.
```

Implementation:

 `/* This program calculates the Key for two persons  ` `using the Diffie-Hellman Key exchange algorithm */` `#include ` `#include ` ` `  `// Power function to return value of a ^ b mod P ` `long` `long` `int` `power(``long` `long` `int` `a, ``long` `long` `int` `b, ` `                                     ``long` `long` `int` `P) ` `{  ` `    ``if` `(b == 1) ` `        ``return` `a; ` ` `  `    ``else` `        ``return` `(((``long` `long` `int``)``pow``(a, b)) % P); ` `} ` ` `  `//Driver program ` `int` `main() ` `{ ` `    ``long` `long` `int` `P, G, x, a, y, b, ka, kb;  ` `     `  `    ``// Both the persons will be agreed upon the  ` `        ``// public keys G and P  ` `    ``P = 23; ``// A prime number P is taken ` `    ``printf``(``"The value of P : %lld\n"``, P);  ` ` `  `    ``G = 9; ``// A primitve root for P, G is taken ` `    ``printf``(``"The value of G : %lld\n\n"``, G);  ` ` `  `    ``// Alice will choose the private key a  ` `    ``a = 4; ``// a is the chosen private key  ` `    ``printf``(``"The private key a for Alice : %lld\n"``, a); ` `    ``x = power(G, a, P); ``// gets the generated key ` `     `  `    ``// Bob will choose the private key b ` `    ``b = 3; ``// b is the chosen private key ` `    ``printf``(``"The private key b for Bob : %lld\n\n"``, b); ` `    ``y = power(G, b, P); ``// gets the generated key ` ` `  `    ``// Generating the secret key after the exchange ` `        ``// of keys ` `    ``ka = power(y, a, P); ``// Secret key for Alice ` `    ``kb = power(x, b, P); ``// Secret key for Bob ` `     `  `    ``printf``(``"Secret key for the Alice is : %lld\n"``, ka); ` `    ``printf``(``"Secret Key for the Bob is : %lld\n"``, kb); ` `     `  `    ``return` `0; ` `} `

Output

```The value of P : 23
The value of G : 9

The private key a for Alice : 4
The private key b for Bob : 3

Secret key for the Alice is : 9
Secret Key for the Bob is : 9
```

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