The post “Fresh Ideas in Mathematics: Non-Classical Arithmetics and Varieties” suggested to readers the idea of using sets of numerical algebraic operations – arithmetic, functions / equations based on them, now called variety functions / equations, and what are sets / sequences of values of variety functions and sets solutions of the indicated equations – varieties. Specific non-classical arithmetic can be found in the preprint “Arithmetic DR+”, and you can get to know her informally at video examples.
Further, KA is an abbreviation for classical arithmetic, NKA is for non-classical, -ih.
One of the declared advantages of the idea was the possibility of transferring (reusing) the usual constructions of classical arithmetic to non-classical ones in order to save thinking. Here we designate one theoretical example, beginning with the questions: “In the given arithmetic A there is an analogue of a prime number with respect to its multiplication or not? Arithmetic of remainders in A exists? The RSA algorithm is implemented in A?”
Let us outline the dotted arithmetic of interest to us A as follows: its multiplication is an algorithm that takes numeric operands as input, the final multiplication table i and final addition table j for numbers, like classical operations. It is mathematically correct to state that in A as many multiplications as possible pairs “multiplication table – addition table”. Same with additions, subtractions, divisions.
So, we have a lot of multiplications, and it is colossal. Once the arithmetic BUT is such that it makes sense to talk about prime (with respect to a given multiplication) numbers, then we have the following “simple entanglement of a cracker”: if he does not know the operation of multiplication, i.e., does not know the multiplication and addition tables, then it is not known how the given number. Indeed, in one multiplication it factorizes into primes a, bin the other – to simple c, d, e, fin the third it is itself simple.
So, RSA implemented in the set ABOUTt arithmetic operations BUT, will work with secret tables and will have a public key. The closedness of the tables of decryption operations entails the need for them to be closed for encryption, because, unlike RSA with classical arithmetic, Bob’s message will be decrypted by Alice if both of them know the operations: Alice’s public key is known to everyone, anyone can encrypt it, but which table the recipient has is unknown . On the other hand, for the same reasons as in RSA, we do not need to change tables in a new encryption session.
Suppose RSA is implemented in ABOUTt . It is reasonable to assume that, with equally “long” numbers, RSA, equipped with ABOUTt, has greater cryptographic stability due to the secrecy of the tables than the one provided with the classical one. but BUT may be slower in the machine sense. Then you can try to work with shorter numbers. This may slightly reduce resistance, but, nevertheless, keep it higher than RSA with KA. Or you can simplify the encryption algorithm, still remaining in the high reliability interval.
PS The author is not an encryption specialist. The DR+ Arithmetic preprint does not contain any details on the given topic as of February 25, 2022.