What is the basis for the difficulty in breaking RSA?

Factoring numbers

https://en.wikipedia.org/wiki/RSA_(cryptosystem)

RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data

transmission. It is also one of the oldest. The acronym RSA comes from the surnames of Ron Rivest,

Adi Shamir, and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent

system was developed secretly, in 1973 at GCHQ (the British signals intelligence agency), by the

English mathematician Clifford Cocks. That system was declassified in 1997.

In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which

is kept secret (private). An RSA user creates and publishes a public key based on two large prime

numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be

encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime

numbers.