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Thanh Toan
 
myetherwallet private key ky thuat tu

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Bitcoin Daily News – 2019-04-01Thanh Toan
To understand the motivation for elliptic curve cryptography, we must first understand the purpose of public key cryptography as a whole. To do this, we introduce a hypothetical situation involving two old friends of cryptographers everywhere, Alice and Bob. Example 1. Alice and Bob Suppose Alice and Bob would like to communicate secret messages to each other. The only problem is, everyone knows that an evil eavesdropper appropriately named Eve has access to all communication between Alice and Bob. How can they tell their secrets without Eve hearing or at least without Eve hearing anything of significance , and keep her from tampering with the information on its way from one person to another? This is where the idea of public keys comes in. Alice and Bob each have a key, some number or mathematical procedure that can be applied to messages, composed of a public piece and a private piece. The private pieces of these keys are never transmitted, while the public pieces are accessible to everyone, including Eve. Then Bob uses the private part of his key to decrypt the information. Because Bob is the only one who has the private part of his key, he is the only one who can decrypt it. For additional security, Alice and Bob may also have public and private signatures, which work similarly to the keys. If Eve tampers with the signature, it will return garbled, and Bob will know it is corrupted. Alice and Bob both publicly distribute copies and copies of each of their locks, but always keep the key safely with them. He has the single key to all of his locks, so he is only one who can open it. Notice that the security of this system does not rely at all on Alice and Bob finding a secure way to transmit information, but it relies very heavily on Alice and Bob each having private keys that are very, very difficult to retrieve using only their public keys. Eve can only be thwarted if the information that she can intercept is totally useless. This brings us to the elliptic curve discrete logarithm problem, which we will see can be made sufficiently difficult to give us a useful pair of keys. First we must explain elliptic curves. Definition 2. So long as k does not have characteristic 2 or 3, this will be a smooth plane cubic curve with the point at infinity, and we can describe the curve as points satisfying the equation. The group law on an elliptic curve. The operation exploited for key selection in elliptic curve cryptography comes from considering the elliptic curve as an abelian group with points as elements. The resulting point, R, will be the sum of P and Q. The formal properties of the addition law are described below. Theorem 2. In short, the addition law gives us the group properties that we desire. Additionally, we will note that the subset of points in this group whose both coordinates belong to a given field k, along with the point at infinity, will form a subgroup of the curve group C. This will be important, because the curves used in elliptic curve cryptography are defined over a finite field, and we need that set to be closed under point addition. Because our goal now is not to construct elliptic curve cryptography, but rather to understand how it works, we will omit the formal proof, but notice that most of the properties above follow directly from the geometric description of point addition. Now that we understand the properties of elliptic curves as groups, we can approach the elliptic curve discrete logarithm problem, from which elliptic curve cryptosystems draw their strength. Definition 3. It can be understood on a very elementary level why this problem might be difficult to solve. Imagine going through several iterations of the point adding process described above on a curve that has many, many points, then erasing all of the intermediate steps. It is not immediately apparent how to proceed when trying to recreate the process you have just made invisible. In fact, nobody knows exactly how difficult this problem is to solve, because no one has come up with an efficient algorithm to solve it. It is, however, believed to be more difficult to solve than the general discrete logarithm problem, and the various factorization problems that are used in other cryptosystems and the best methods for cracking these problems do not seem to adapt easily to elliptic curve problems , which suggests that elliptic curve cryptography is the strongest of all the available cryptographic systems. There are multiple ways to construct cryptosystems that operate this way, so we will provide two as examples. Both are elliptic curve analogues of preexisting cryptosystems that were created to use the general discrete logarithm problem; adaptation is easy since the structure of the ECDLP is so similar to that of the original DLP. Both also assume some existing system of embedding messages into points on the elliptic curve. There are a number of ways to do this, none of which are specifically attached to the given cryptosystems, so we just assume that we have chosen some embedding of message m into point Pm, and that this embedding is publicly known so that Bob can retrieve the embedded message once he obtains Pm. Our first example is an adaptation of the ElGamal public key cryptosystem:. Example 4. When Alice wants to communicate secretly with Bob, they proceed thus:. This is a successful cryptosystem because every operation that Alice and Bob have to carry out addition and subtraction on the curve is relatively easy, while the operation that Eve would have to perform to crack the system is extremely difficult or for real-life villains without the proper resources, perhaps impossible. Our next example, an analogue of the Massey-Omura public key cryptosystem, operates on a similar back-and-forth series of easy problems for Alice and Bob that produces the ECDLP for Eve. We did not need the number of points on C in the previous example, but this information is never relied on to be secret, because it can be calculated with relative efficiency. We must also remember that not every choice of curve, finite field, and point is created equal. In the ElGamal system, Alice and Bob are operating not on the entire curve, but on the cyclic group generated by G, and in the Massey-Omura system, that generated by Pm. These decisions must be made well, and it would take much more space than this to explain how to make them well. However, with the appropriate choices, we do get to see the power of something that is quite simple, in the elliptic curve group law. A geometric idea as simple as connecting dots though one dot does have to lie at infinity gives us a secret messaging system that has yet to be cracked. Guide to Elliptic Curve Cryptography. Springer, Elliptic curve cryptosystems. Mathematics of Computation, — , Elliptic curve cryptography. An ECC research project , Undergraduate Algebraic Geometry. Cambridge University Press, Image source. Save my name, email, and website in this browser for the next time I comment. Sign in. Log into your account. Forgot your password? Password recovery. Recover your password. Sunday, August 22, Get help. Please enter your comment! Please enter your name here. You have entered an incorrect email address! Mr Good - April 20, Load more. 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