Homomorphism of an elementwise sum and dot product

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SUMMARY

The discussion centers on the impracticality of implementing the expression ∑ab in the context of homomorphic functions. Specifically, the user seeks a homomorphic function that allows the expression of ∑ab as ∑a∑b, given the constraints of working with ∑i ai = α and ∑i 10i-|i| separately. A counterexample is provided, illustrating that when all ai and bi equal 1, the equality Σab = Σa = Σb does not hold, confirming the limitations of the proposed approach.

PREREQUISITES
  • Understanding of homomorphic functions in cryptography
  • Familiarity with summation notation and properties
  • Knowledge of polynomial expansions and their implications
  • Basic concepts of elementwise operations in mathematics
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  • Research homomorphic encryption techniques, focusing on additive homomorphism
  • Explore polynomial expansion methods in algebra
  • Study counterexamples in mathematical proofs to understand limitations
  • Investigate the implications of elementwise operations in vector spaces
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Mathematicians, cryptographers, and computer scientists interested in homomorphic encryption and its applications in secure computations.

NotASmurf
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∑ab is needed but is impractical to implement.

Specifically ∑i ai.10i-|i| in any form where I can work with ∑i ai = α and ∑i 10i-|i| separately.

Is there a homomorphic function I can run it through such that ∑ab can be expressed as ∑a∑b? Note: for current problem i cannot simply set it up such that ∑a∑b - ∑ab polynomial expansion saves the day, any help appreciated.
 
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No. A simple counterexample: Let all ai=1 and all bi=1 Then Σab = Σa = Σb.
 

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