The disconcerting case of two private keys with the same public key value
In the cryptographic world, private keys and public keys are essential components that allow safe communication between the parties. However, there is a peculiar case in Bitcoin, a decentralized cryptocurrency, where the private keys to the corresponding public keys have the same X value is theoretically possible.
The SECP256K1 curve is a widely used elliptical curve defined by NIST standards (National Institute of Standards and Technology). It consists of prime numbers of 256 bits as coefficients and can be used to create secure digital signatures and transactions. The curve has two key sizes: SECP384R1 and SECP192R1, which are also part of the NIST standards.
The problem in question is established in a mathematical article entitled “On the existence of points of points in a curve whose value X is the same” (2008). This problem was extended later to include a more general case: to find two different private keys with corresponding public keys that have the same value X.
Find two different private keys with the same public key value
To address this problem, we must understand that each point in a curve corresponds to a unique pair of prime numbers. When we generate a couple of private keys, their corresponding public keys can have different values based on the choice of this prime number.
Let’s denote the two private keys such as (P1, Q1) and (P2, Q2), with the corresponding public keys P1 (X1, Y1) and P2 (X2, Y2). We are interested in the peers of findings (P1, Q1) and (P2, Q2) so that their corresponding X values are the same.
In other words, we need to find two integers in which there are a maximum of two points in the SECP256K1 curve with an X or M value of different pairs (P1, Q1) and (P2, Q2) where its corresponding X values are the same are the same.
The limitation: two points in the curve with X-Value M
To establish that there may be a maximum of two points in the SECP256K1 curve whose value x is M, we need to remember a mathematical fact. The number of different pairs (P1, Q1) and (P2, Q2) so that the corresponding X values are equal are limited by the degree of the curve (m+1), which represent the maximum possible number of points in the curve .
In other words, if there is m cousin number p1, p2, …, pm, then for each pair (p1, p2), we can find a corresponding point P (x1, y1) with the value x m. For the Paloma principle, since we have more than 2 points in the curve (M+1), at least two of the thesis pairs must share an X value.
Theoretical implication
This theoretical implication has significant implications for cryptographic applications in Bitcoin and other blockchain platforms. If it is possible to find two different private keys with corresponding public keys that have the same X value, then it could lead to a new level of security vulnerability.
However, it is essential to keep in mind that this problem remains not proven and requires greater investigation to determine if such pairs can be found. The article of the authors “on the existence of points of points in a curve whose value x is the same” (2008) provides a theoretical framework to understand why finding such pairs could be challenging.
Conclusion
Although we have established that it is theoretically possible to find two different private keys with corresponding public keys that have the same X value, more research is required to determine if this can be achieved in practice. The cryptographic community continues to explore new methods and challenges, ensuring the safety and resistance of Blockchain platforms such as Bitcoin.
