>>1339222 (LB)
>https://paillier.daylightingsociety.org/about
I think your on to something here
We believe, however, that the cryptographic research had unnoticeably witnessed
the progressive emergence of a third class of trapdoor techniques: firstly
identified as trapdoors in the discrete log, they actually arise from the common
algebraic setting of high degree residuosity classes. After Goldwasser-Micali’s
scheme [9] based on quadratic residuosity, Benaloh’s homomorphic encryption
function, originally designed for electronic voting and relying on prime residuosity,
prefigured the first attempt to exploit the plain resources of this theory. Later,
Naccache and Stern [16], and independently Okamoto and Uchiyama [19] significantly
extended the encryption rate by investigating two different approaches:
residuosity of smooth degree in Z
∗
pq and residuosity of prime degree p in Z
∗
p
2q
respectively. In the meantime, other schemes like Vanstone-Zuccherato [28] on
elliptic curves or Park-Won [20] explored the use of high degree residues in other
settings.
In this paper, we propose a new trapdoor mechanism belonging to this family.
By contrast to prime residuosity, our technique is based on composite residuosity
classes i.e. of degree set to a hard-to-factor number n = pq where p and q are two
large prime numbers. Easy to understand, we believe that our trapdoor provides
a new cryptographic building-block for conceiving public-key cryptosystems.
In sections 2 and 3, we introduce our number-theoretic framework and investigate
in this context a new computational problem (the Composite Residuosity
Class Problem), which intractability will be our main assumption. Further, we
derive three homomorphic encryption schemes based on this problem, including
a new trapdoor permutation. Probabilistic schemes will be proven semantically
secure under appropriate intractability assumptions. All our polynomial reductions
are simple and stand in the standard model.