Beyond Bitcoin
The digital security of banks, 500 Fortune companies, governments and many IoT devices depend on RSA — a cryptographic system whose security is provided by the difficulty of factoring integers into their prime factors, and in particular, the difficulty of factoring integers that only have two prime factors where both have exactly the same size in number of digits. These numbers are called strong semiprimes, and factoring them is the RSA problem.
It seems to me, after speaking with mathematicians, cryptographers, and random users on the internet, the reason a blockchain based on the RSA factoring problem has not been created until now is because no one could figure out how the blockchain could generate strong semiprimes for miners to factor without first knowing what the prime factors were.
My solution to this problem is simple: instead of generating strong semiprimes without knowing their factors a priori — which no one can figure out how to do — create conditions under which miners can find these strong semiprimes by way of factoring and reward them for finding them. In the process, tie the blockheader data to this process to secure the blockchain.
The essence of PoW is as follows: generate a random number by hashing the data in the block header of the block to be validated, give miners a range around this generated integer, and allow miners to factor all these integers. If they find a strong semiprime reward them accordingly. If they do not find a strong semiprime they can change the nonce and try again. The miners can generate as many random numbers as they want using nonces, but the search range allowed will always be about the same.