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Mandelbrot sets are very popular infinitely complex fractals. Mandelbrot sets have been used for generating colorful images, melodious music etc. In this paper, authors propose the use of Mandelbrot sets for public key cryptography. In the past, RSA and Elliptic curve cryptography have been used successfully. RSA uses the modular exponentiation concept from Number theory to generate the public key and private key pairs. The proposed system generates public key and private key pairs using Mandelbrot sets. It is also demonstrated using examples on how it is infeasible to compute private key by using the public key. It is well known fact that colorful Mandelbrot plots can be generated by Mandelbrot equations. Since Mandelbrot plots are infinitely complex, the public key cryptographic system can be built using this principle.