I Homomorphism of an elementwise sum and dot product

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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