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The earliest public key cryptosystems using number theory were based on the structure either of the multiplicative group (Z/NZ)* or the multiplicative group of a finite field G¥(q), q = p". The subsequent construction of analogous systems based on other finite Abelian groups, together with H. W. Lenstra's success in using elliptic curves for integer factorization, make it natural to study the possibility of public key cryptography based on the structure of the group of points of an elliptic curve over a large finite field. We first briefly recall the facts we need about such elliptic curves (for more details, see [4] or [5]). We then describe elliptic curve analogs of the Massey-Omura and ElGamal systems. We give some concrete examples, discuss the question of primitive points, and conclude with a theorem concerning the probability that the order of a cyclic subgroup is nonsmooth. I would like to thank A. Odlyzko for valuable discussions and correspondence, and for sending me a preprint by V. S. Miller, who independently arrived at some similar ideas about elliptic curves and cryptograph
The earliest public key cryptosystems using number theory were based on the structure either of the multiplicative group (Z/NZ)* or the multiplicative group of a finite field G¥(q), q = p". The subsequent construction of analogous systems based on other finite Abelian groups, together with H. W. Lenstra's success in using elliptic curves for integer factorization, make it natural to study the possibility of public key cryptography based on the structure of the group of points of an elliptic curve over a large finite field. We first briefly recall the facts we need about such elliptic curves (for more details, see [4] or [5]). We then describe elliptic curve analogs of the Massey-Omura and ElGamal systems. We give some concrete examples, discuss the question of primitive points, and conclude with a theorem concerning the probability that the order of a cyclic subgroup is nonsmooth. I would like to thank A. Odlyzko for valuable discussions and correspondence, and for sending me a preprint by V. S. Miller, who independently arrived at some similar ideas about elliptic curves and cryptograph
Elliptic Curve CryptoElliptic Curve Cryptography (ECC) is one of the most powerful but least understood types of cryptography in wide use today. At CloudFlare, we make extensive use of ECC to secure everything from our customers HTTPS connections to how we pass data between our data centers.Fundamentally, we believe its important to be able to understand the technology behind any security system in order to trust it. To that end, we looked around to find a good, relatively easy-to-understand primer on ECC in order to share with our users. Finding none, we decided to write one ourselves. That is what follows. Be warned, this is a complicated subject and its not possible to boil down to a pithy blog post. In other words, settle in for a bit of an epic because theres a lot to cover. If you just want the gist, the TL;DR is: ECC is the next generation of public key cryptography and, based on currently understood mathematics, provides a significantly more secure foundation than first generation public key cryptography systems like RSA. If youre worried about ensuring the highest level of security while maintaining performance, ECC makes sense to adopt. If youre interested in the details, read on.
The earliest public key cryptosystems using number theory were based on the structure either of the multiplicative group (Z/NZ)* or the multiplicative group of a finite field G¥(q), q = p". The subsequent construction of analogous systems based on other finite Abelian groups, together with H. W. Lenstra's success in using elliptic curves for integer factorization, make it natural to study the possibility of public key cryptography based on the structure of the group of points of an elliptic curve over a large finite field. We first briefly recall the facts we need about such elliptic curves (for more details, see [4] or [5]). We then describe elliptic curve analogs of the Massey-Omura and ElGamal systems. We give some concrete examples, discuss the question of primitive points, and conclude with a theorem concerning the probability that the order of a cyclic subgroup is nonsmooth. I would like to thank A. Odlyzko for valuable discussions and correspondence, and for sending me a preprint by V. S. Miller, who independently arrived at some similar ideas about elliptic curves and cryptograph
The earliest public key cryptosystems using number theory were based on the structure either of the multiplicative group (Z/NZ)* or the multiplicative group of a finite field G¥(q), q = p". The subsequent construction of analogous systems based on other finite Abelian groups, together with H. W. Lenstra's success in using elliptic curves for integer factorization, make it natural to study the possibility of public key cryptography based on the structure of the group of points of an elliptic curve over a large finite field. We first briefly recall the facts we need about such elliptic curves (for more details, see [4] or [5]). We then describe elliptic curve analogs of the Massey-Omura and ElGamal systems. We give some concrete examples, discuss the question of primitive points, and conclude with a theorem concerning the probability that the order of a cyclic subgroup is nonsmooth. I would like to thank A. Odlyzko for valuable discussions and correspondence, and for sending me a preprint by V. S. Miller, who independently arrived at some similar ideas about elliptic curves and cryptograph
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