## encrypt Using Abel-Ruffini theorem on encryptions

2. If the attacker has the resultant polynomial equation of degree 512, its roots must be found, but there’s no direct formula to do so analytically due to Abel-Ruffini theorem, and factoring such a polynomial with 156 different named constants efficiently is very, very complicated and convoluted 3. If the attacker has direct access to the server, the expected polynomial equation can be retrieved, but the attacker still has to solve that polynomial equation of degree 512 to find all its roots with the right ordering among them and all their correct literal expressions 4. If the attacker has direct access to the client software source codes, the pepper for hashing the password in plaintext, the pepper used on the polynomial equation roots, and the highly secret and obfuscated algorithm for using them with the salt counterparts can be retrieved, but it’s still far from being able to find all the roots of the expected polynomial equation of degree 512 5. If the attacker has all those roots, the right ordering among them and all their correct literal expressions still have to be figured out, and the salts and peppers for those roots has to be properly removed as well 6. If the attacker has all those roots with the right ordering among them, all their correct literal expressions, and salts and peppers on them removed, the attacker has effectively recovered the hashed password, which is mixed with salts and peppers in plaintext 7. The attacker then has to figure out the password in plaintext even with the hashing function, salt, pepper, and the highly secret and obfuscated algorithm that combines them known 8. Unless there are really efficient algorithms for every step involved, the time complexity of the whole process can be as high as factorial time 9. As users are still inputting passwords in plaintexts, dictionary attacks still work to some extent, but if the users are careless with their password strengths, then no amount of cryptography will be safe enough 10. Using numerical methods to find all the roots won’t work in most cases, because such methods are unlikely to find those roots analytically, let alone with the right ordering among them and all their right literal expressions, which are needed to produce the resultant polynomial equation with literal expressions exactly matching the expected one 11. Using rainbow tables won’t work well either, because such table would be way too large to be used in practice, due to the number of polynomial equations with degree 512 being unlimited in theory 12. Strictly speaking, the whole password encryption scheme isn’t a one-way function, but the time complexity needed for encryption compared to that for decryption is so trivial that this scheme can act like such a function Areas demanding further researches: 1. The time complexity for factoring a polynomial of degree n with named constants into n factors analytically 2. Possibilities of collisions from the ordering among all roots and all their different literal expressions 3. Existence of efficient algorithms on finding the right ordering among all roots and all their right literal expressions 4. Strategies on setting up the fixed peppers and generating random salts to form roots with maximum encryption strength Essentially, the whole approach on using polynomial equations for encryptions is to exploit equations that are easily formed by their analytical solution sets but very hard to solve analytically, especially when exact literal matches, rather than just mathematical identity, are needed to match the expected equations. So it’s not strictly restricted to polynomial equations with a very high degree, but maybe very high order partial differential equations with many variables, complex coefficients and functions accepting complex numbers can also work, because there are no known analytical algorithm on solving such equations yet, but analytical solutions are demanded to reproduce the same partial differential equations with exact literal matches, as long as performing partial differentiations analytically can be efficient enough.

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