Interpolation In Numerical Analysis

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1. INTRODUCTION OF LAGRANGE POLYNOMIAL ITERPOLATION
1.1 Interpolation:

• First of all, we will understand that what the interpolation is.

• Interpolation is important concept in numerical analysis. Quite often functions may not be available explicitly but only the values of the function at a set of points, called nodes, tabular points or pivotal points. Then finding the value of the function at any non-tabular point, is called interpolation.

Definition:

• Suppose that the function f (x) is known at (N+1) points (x0, f0), (x1, f1), . . . , (xN, fN) where the pivotal points xi spread out over the interval [a,b] satisfy a = x0 < x1 < . . . < xN = b and fi = f(xi) then finding the value of the function …show more content…

APPLICATIONS OF LAGRANGE POLYNOMIAL INTERPOLATION

 Lagrange polynomials basis are used in the Newton–Cotes method of numerical integration

 One of its application is in Cryptography such as “ Shamir's secret sharing scheme”.

 It is appropriate for back-of-the-envelope calculations.

 It is uses in Improving the Low Resolution Images Accuracy in Human Face Recognition.

 It is uses in Contrast Based Color Watermarking In Wavelet Domain.

 It is uses in Learning of Neural Network.

 It is uses in Data Compression.

5.APPLICATION OF LAGRANGE POLYNOMIAL INTERPOLATION

Application: Secure Message Transmission using …show more content…

 As any nth degree polynomial is uniquely determined by n+1 points, n +1 points are communicated to the other side, where the polynomial and hence the message is reconstructed.

 Padding of length m is added to the message to overcome the message length issue.

 Huffman coding is used for converting the plaintext into binary form.

5.2.CRYPTO SYSTEM BASED ON LAGRANGE INTERPOLATION

 A random number N is assumed by the sender and the respective N value securely communicated to receiver through the concept of digital enveloping. Security of message is mainly depends on this randomly generated N.

 Encryption: The sender is converting the actual message into points like (xi,yi) by passing the original message using the algorithm given in Fig 1. In Step 4, M is constructed by using the formula.

Decryption: Receiver collects all the interpolation points, retrieves the original message using the algorithm in fig2. Step 2 is used for constructing polynomial function is achieved by using following formula.

5.3.SYSTEM MODEL FOR MESSAGE

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