Number Theory 2 요약 - Book Test
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12장: 소수의 무한성과 분포

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abcd

efg

13장: ㅆ

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ㅁㄴㅇㄹㅁㄴㅇㄹ
ㅁㄴㅇㄹㅁㄴㅇㄹ

as

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a

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asd

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ear on low computing power devices such as mobile phones, or chips, it has become essential to create protocols for which we can reach the same level of security without spending considerable computing power setting up the system in the first place. Elliptic curve cryptography (ECC) provides an exciting alternative to RSA, and has shown to be a lot more efficient in terms of key size. In this paper, we provide a description of how elliptic curves are used in modern cryptography, as well as their current limitations and future prospects. Because quantum computers pose a serious threat to the currently in use public-key systems, we also describe the recent progress on super singu- lar elliptic curves isogenies, which may offer a quantum resistant cryptosystem and a viable alternative for the future of elliptic curve based cryptography.
1. Introduction and History
Up until the 1970’s, all the encryption in use around the world was based on sym- metric ciphers, which means that in order for two parties to communicate securely, they must have had to previously meet and agree on a shared secret. While such methods provided all the security needed at the start of the 20th century, the second World War and the rise of the internet and online information, motivated the idea of a cryptographic protocol where two parties could create a secure communication channel without any kind of previous communication.
This kind of cryptography is known as asymmetric, or public-key cryptography. The first to demonstrate the existence of such methods were Diffie, Hellman and Merkle in 1976 [9]. While their protocol is usually referred to as the Diffie-Helllman Key Exchange, in 2002, Hellman suggested that it should be called the Diffie- Helllman-Merkle Key Exchange, including Merkle in recognition of his contribution to the invention of public key cryptography [14]. Because asymmetric protocols tend to require relatively large amounts of computation, it is common to use them to send a small key which can then be used to establish a secure symmetric channel.
Key words and phrases. Elliptic curve cryptography, quantum computing, super singular elliptic curve isogeny.
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2 JEREMY WOHLWEND
Definition 1.1. We denote the discrete logarithm problem on the multiplicative group of the integers modulo p, Z/Zp , as follows. Given g, a ∈ Z/Zp , where a is a member of the cyclic subgroup generated by g, find an integer k such that:
(1.1) gk ≡ amodp
The security of the Diffie-Hellman-Merkle protocol relies on the assumption that while ga can be computed easily using repeated squaring, its counterpart, the dis- crete logarithm problem, is computationally hard. While there is no formal proof of the hardness of the problem, this assumption is widely conjectured to be true. In particular, there are groups under which the logarithm is easy but also groups where it is believed to be particularly hard. Elliptic curves are believed to be part of the latter, as is discussed see later in this paper, and to be considerably more resistant than Diffie-Hellman-Merkle and prime groups, for which efficient methods in practice exist, for instance, index calculus [4].
In 1977, Ron Rivest, Adi Shamir, and Leonard Adleman proposed another asym- metric encryption scheme, known as RSA [24]. The security of RSA is based on an similar problem to Diffie-Hellman-Merkle, namely, the discrete factoring problem.

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Thank you