Why can't someone create a valid digital signature without knowing your private key?
Bitcoin uses something called the Elliptic Curve Digital Signature Algorithm (ECDSA) for signing and verifying transactions. Read the linked Wikipedia article for exact process of how ECDSA works. If you understand how the modulo operator works, it should make sense as to why you can't make a signature without the private key.



It's kind of hard for me to explain this as it does rely on some complex mathematics, but I will try.

To produce a valid ECDSA signature, you need the private key, which is a large integer. It is explicitly used in the signature creation algorithm, but the public key is not. Furthermore, the public key is not an integer like the private key; rather it is a point on an elliptic curve. So even if you know the public key, you can't create a signature with it because it is not the private key and you can't recover the private key from the public key.

The private key cannot be recovered from the public key because of a problem called the Elliptic Curve Discrete Logarithm Problem. From Wikipedia:


  it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" (ECDLP).


The private key is the discrete logarithm. Although this is an assumption, it has been shown in practice that ECDLP is a hard problem, i.e. it is hard to find the discrete log given a base point and public key point. So you can't get the private key that is required in ECDSA to produce the signature.