Homomorphic encryption as an elegant manner to save privacy

Sabine Hossenfelder talked about homomorphic encryption, which is an elegant and extremely general algebraic manner to guarantee data privacy (see this). The idea is that the encryption respects the algebraic operations: sums go to sums and products go to products. The processing can be done for the encrypted data without decryption. The outcome is then communicated to the user and decrypted only at this stage. This saves a huge amount of time.

What comes first in mind is Boolean algebra (see this). In this case the homomorphism is truth preserving. The Boolean statement formed as a Boolean algebra element is mapped to the same statement but with images of the statements replacing the original statements. In the set theoretic realization of Boolean algebra this means that unions are mapped to unions and intersections to intersections. In Boolean algebra, the elements are representable as bit sequences and sum and product are done element-wise: one has x²=1 and x+x=0. Ordinary computations can be done by representing integers as bit sequences.

In any computation one must perform a cutoff and the use of finite fields is the neat way to do it. Frobenius homomorphism x→x^p in a field of characteristic p maps products to products and, what is non-trivial, also sums to sums since one has (x+y)^p= x^p+y^p. For finite fields F_p the Frobenius homomorphism is trivial but for F_p^e, e>1, this is not the case. The inverse is in this case x→x ^p_e-1. These finite fields are induced by algebraic extensions of rational numbers. e corresponds to the dimension of the extension induced by the roots of a polynomial

Frobenius homomorphism extends also to the algebraic extensions of p-adic number fields induced by the extensions of rationals. This would make it possible to perform calculations in extensions and only at the end to perform the approximation replaces the algebraic numbers defining the basis for the extension with rationals. To guess the encryption one must guess the prime that is used and the use of large primes and extensions of p-adic numbers induced by large extensions of rationals could keep the secrecy.

p-Adic number fields are highly suggestive as a computational tool as became clear in p-adic thermodynamics used to calculate elementary particle masses: for p= M₁₂₇= 2¹²⁷-1 assignable to electron, the two lowest orders give practically exact result since the higher order corrections are of order 10^-76. For p-adic number fields with very large prime p the approximation of p-adic integers as a finite field becomes possible and Frobenius homomorphism could be used. This supports the idea that p-adic physics is ideal for the description of cognition.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.