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Elliptic curves and use of their endomorphism rings in cryptography

Sülçe, Ali Mert
Although elliptic curves have been studied for hundreds of years, the inception of elliptic curve cryptography is 1985 by Koblitz’s and Miller’s independent proposals that is based on the discrete logarithm problem on an elliptic curve defined over a finite field. After that date, there are a lot of advances and studies in elliptic curve cryptography(ECC) which provide high security with relatively small block sizes and high speed compared to the other public key cryptosystems. For instance, 160-bit elliptic curve key provides the same level of security as a 1024-bit RSA key. Meantime, quantum computers, which provide efficient and very fast parallel computation, are developed. In the near feature, widely used public key cryptosystems, including ECC, are vulnerable to quantum algorithms which means not only ECC but also almost all public key cryptosystems will be dead or seriously wounded in the near future. Therefore, efficient public key systems should be designed for post-quantum world. In this world, elliptic curves with some properties do not lose their popularity. In this work, we shall study the mathematical backgrounds of elliptic curves and isogenies on elliptic curves which are the essential concept in post-quantum cryptography(PQC).