I was having a discussion with a colleague yesterday whose education and experience exceeded mine. He said something yesterday that sounded incorrect to me intuitively, but I don't have the knowledge to contradict it.
He claimed that the only reason a secure cryptographic hash is better at detecting tampering or corruption is because it has more bits in the output (e.g. SHA-256 has a 256-bit output vs. CRC32's 32-bit output). Right now, we're only talking about detecting corruption/tampering, not recovering data.
My intuition tells me that given the choice of CRC32 or SHA256 (and taking only the low 32 bits), the hash is going to be much more secure as far as detecting changed bits (particularly in the face of a malicious and intelligent attacker, but even due to corruption in "just the wrong way").
Can someone either tell me why I'm wrong & the 2 are essentially equivalent, or support my intuition with math and or proof/reasoning?