Ethereum: Is there a point on the secp256k1 curve for any given X coordinate?

Ethereum: Understanding point generation on the SECP256K1 curve

Ethereum blockchain relies heavily on cryptographic algorithms, including elliptical curve cryptography (ECC), to provide transactions and control access to the network. One of these algorithms is the SECP256K1, which is a popular choice on many Ethereum-based intelligent contract platforms. However, in this article, we are immersed in the characteristics of generating points on the Secp256k1 curve, and that there is a point where the curve algorithm returns invalid or meaningless value.

The Secp256k1 curve

The SECP256K1 curve is a widely used elliptical curve defined by the elliptical curve discreet logarithm problem (ECDLP). This curve is chosen for efficiency, scalability and safety properties. It consists of 256 bit numbers, each of which represents one element on this curve.

Generation of points on the SECP256K1 curve

To create the point of the SECP256K1 curve, you need to choose two random points for $ g $, $ p $ and $ a $ (or $ q $, as they are usually used for signature and control purposes) on the curve. The resulting point is then generated by the following steps:

1.

• Calculate $ g = a^{(a^2)/8} \ mod 2^n $

• Calculate $ y = g^x \ mod n $

where:

– $ \ cdot $ represents the multiplication modulo $ n $

– $ (\) $ denotes exponence

– $ g $, $ p $ and $ the $ randomly chosen elements on the curve (usually using a safe pseudonym number -generator)

-$ a $ is a whole number of parameters for ECDLP, usually set to 65537

Point generation returns a valid group item

The SECP256K1 curve ensures that at the points on this curve, all operations will result in valid group elements. This means that two points cannot be mapped to the same point of the curve (unless they are collected and are exactly one of the four points defined by the equation of the curve).

Is there a point where Secp256k1 returns invalid value?

Theoretically, there is no set of finite $ x $ value that returns a meaningless or invalid value for the SECP256K1 curve. In practice, however, there are cases where certain mathematical manipulations can produce unexpected results:

* Computing complexity : Some calculations can be exponentially slower than the finite fields (FVF), due to the difficulty of solving ECDLP. This can lead to efficiency in certain scenarios, but does not mean that such operations return invalid values.

* Curve Properties : The SECP256K1 curve is designed for specific uses and has optimized properties for performance and safety. Although these properties do not guarantee validity, they ensure that the curved points remain valid.

In summary

Generation of the points on the SECP256K1 curve includes the careful selection of $ g $, $ p $ and $ a $ (or $ q $) with a careful choice of $, $ g $ and $ $ parameters. The resulting point shall then be calculated by a series of modular arithmetic operations. The SECP256K1 curve ensures that all calculations at this curve result in valid group elements, making it suitable for use in various cryptographic applications.

Although there are no theoretical cases where the Secp256K1 curve returns an invalid value, calculation and mathematical complexities can lead to efficiency or unexpected results. However, these issues do not affect the validity of the points obtained on the curve obtained, which remain valid in accordance with the properties of the elliptical curve cryptography (ECC).