Mathematical Relativity / Similar Mathematics and Public Key Encryption

The Mathematical Relativity / Similar Mathematics papers indirectly suggest a way to make reverse factoring of public keys back into their "primes" much more difficult. In these papers, I have demonstrated that there are an infinite number of fields (R_n,+_n,*_n,0_n,1_n) that are isomorphic to the Real field (R,+,*,0,1). Because of the isomorphism, you can take any mathematical proposition of the Real Field (R), or its subsets (S) and demonstrate a similar proposition in the R_n field or its corresponding S_n subsets . It is also possible to standardly numerate R_n or S_n (i.e., reverse the map of the isomorphism with respect to the numeric symbols but NOT the values [unless 0_n = 0 and 1_n = 1]):

0 ~ 0_n
1 ~ 1_n
2 ~ 2_n
etc.

Consequently, all of the fields that I have proposed have a subset that can be considered to be "prime" with respect to the "positive integers" of that field when they are standardly numerated. Standard numeration of these fields provides notational or symbolical correspondence, but not similar value correspondence (unless 0_n = 0 and 1_n = 1).

Specifically, an element, A_n, of the meta-field (R_n,+_n,*_n,0_n,1_n) is considered to be a "positive integer", if, when standardly numerated, it is an element of the meta-set {1_n,2_n,3_n,4_n,...}. An element, A_n, of the meta-field (R_n,+_n,*_n,0_n,1_n) is considered to be "prime", if, when standardly numerated, it is an element of the meta-set {2_n,3_n,5_n,7_n,...}.

For example, in the field (R₀,+₀,*₀,0₀,1₀) where 0₀ is equal to 0 and 1₀ is equal to -1, the values in this field that would be considered to be "positive integers", when standardly numerated, would be {-1 (1_n), -2 (2_n), -3 (3_n), -4 (4_n)} and those that would be considered "prime", when standardly numerated, would be {-2 (2_n), -3 (3_n), -5 (5_n), -7 (7_n), ...}. In the field (R₀,+₀,*₀,0₀,1₀) where 0₀ is equal to 0 and 1₀ is equal to 1+i, the values in this field that would be considered to be "positive integers", when standardly numerated, would be {1+i,2+2i,3+3i,4+4i} and those that would be considered "prime", when standardly numerated, would be {2+2i,3+3i,5+5i,7+7i, ...}. In the field (R₁,+₁,*₁,0₁,1₁) where 0₁ is equal to 1and 1₁ is equal to 2, the values in this field that would be considered to be "positive integers", when standardly numerated, would be {2,4,8,16,...} and those that would be considered "prime", when standardly numerated, would be {4,8,32,128,...}.

Now, instead of having just the usual prime subset of the Real field from which to generate public keys, you have an infinite number of "prime" subsets that can also be used to generate public keys. Each of these subsets of "primes" has a unique actual value that is difficult to reverse factor unless the eavesdropper knows what the original map was (the map can be exchanged between the sender and receiver using a previously, strongly encrypted message).

As a very simple illustration, consider the field (R₀,+₀,*₀,0₀,1₀), where 0₀ = -57,814,193 and 1₀ = -57,814,192. It is relatively easy to generate a public key with the actual value of 0 using the "primes" -57,807,114 (7079_n) and -57,806,026 (8167_n). Please note: when you "multiply" the two "primes" together to get 0 you need to use the definition of multiplication found on the bottom of this page Sim Math Ver.2 p.1.

Without knowing what 0₀ and 1₀ are, it is extremely difficult for an eavesdropper to even begin to reverse factor the public key of 0, back into its primes of -57,807,114 and -57,806,026. And this public key generation can be done for any 0₀,1₀ and "prime" (2₀,3₀,5₀,7₀,...) subset of (R₀,+₀,*₀,0₀,1₀).