Ethereum: Understanding a point generation on the SecP256K1
curve **
Blockchain Ethereum is largely relied on cryptographic algorithms, including the cryptography of the elliptical curve (ECC) to ensure transactions and manage access to the network. One of these algorithms is SecP256K1, a popular choice for many smart platforms based on Ethereum. In this article, however, we dive into the specificities of generating points on the SECP256K1 curve and whether there is sometimes a point at which the curve algorithm returns an invalid or nonsensical value.
Curve SECP256K1
The SecP256K1 curve is a widely used elliptical curve defined by the problem of the discrete logarithm of the elliptical curve (ECDLP). This curve is selected for its efficiency, scalability and safety properties. It consists of 256-bit numbers, each of which represents an element of this curve.
Points generating on the SECP256K1 curve **
If you want to generate a point on the SecP256K1 curve, you need to choose two random points $ g $, $ P $, $ and $ (or $ q $, as they are commonly used to sign and verify the curve. The resulting point is then generated using the following steps:
- Select $ and \ in \ Mathbb {from} _2^*$
- Calculate $ g = a^{(a^2)/8} \ mod 2^n $
- Calculate $ y = g^x \ mod n $
where:
– $ \ Cdot $ represents a multiplication module $ n $
– $ (\) $ indicates exponence
– $ G $, $ P $ and $ and $ randomly selected elements on the curve (usually using a safe pseudorandom generator)
– $ A $ is an integer parameter for ECDLP, usually set to 65537
Poin generation returns a valid group element
The SecP256K1 curve ensures that all operations performed in the points on this curve lead to valid group elements. This means that no two points can be mapped to the same point on the curve (unless they are colinearly and lie exactly on one of the four points defined by the equation curve).
Is there a point where SecP256K1 returns invalid value?
Theoretically, there is no final set of $ x $ x values to return a senseless or invalid value for the SecP256K1 curve. In practice, however, there are cases where certain mathematical manipulations can produce unexpected results:
* Computing complexity : Some points calculations on the SECP256K1 curve may be exponentially slower than calculations performed in the final fields (FVF) due to difficulty in solving ECDLP. This could lead to inefficiency in some scenarios, but this does not mean that such operations return invalid values.
* Curve properties : SECP256K1 curve is designed for specific cases of use and has its own features that are optimized for performance and safety. Although these features do not guarantee validity, they ensure that the resulting points of the curves remain valid.
at the end
Generating points on the SECP256K1 curve includes carefully selection of random elements $ g $, $ P $ and $ (or $ q $) based on parameters $ and $, $ G $ and $ n $. The resulting point is then calculated using a series of modular arithmetic operations. The SecP256K1 curve ensures that all calculations made in the points on this curve lead to the applicable elements of the group, making it suitable for use in various cryptographic applications.
Although there are no theoretical cases where the SecP256K1 curve returns an invalid value, computing and mathematical complexities can lead to inefficiency or unexpected results. However, these problems do not affect the validity of the resulting points on the curve, which remain valid according to the properties of the cryptography of the elliptical curve (ECC).
