You're thinking of a very limited subset of cryptography (I assume: ciphers).
As a counter-example of having to trust the receiving end, consider zero knowledge proofs [1]. Even if you don't trust the prover, you become convinced of what they're trying to prove to you (if it's true).
The basic trick to proper cryptographic voting machines is zero-knowledge-proving that they are working correctly. Any tampering, whether it be in the source code or at the hardware level, will either not affect the tally or break the zero knowledge proofs.
Most protocols I saw did let you verify your own vote, but how can you check other's votes not knowing what they voted? You can't, and >90% of people wouldn't even know how to check their votes.
Everybody understands paper voting and there are several (more or less) trustworthy protocols for paper-based vote count.
Maybe I'm pessimistic, but I can't picture a near future where the populace understands computers enough to be able to trust e-voting.
But how can you check that their encrypted vote is not tampered with even if it's well formed? I can craft "virtual individuals" which replace the real people voting and you wouldn't even notice. The vote is cast and well-formed, it's just not their vote.
If vote is secret you just know that someone voted X. Who is that someone? Is he even real?
Since we need secrecy we need to know X, but not who is "someone", so we're blind there.
Every encrypted vote is linked to a voter registration. As with the current system, you can determine if someone has voted but not who they voted for.
If you can't prevent fake voters from registering, you've failed before the election even begins.
(Yes, this does allow you to coerce people with known preferences into not voting in order to affect the result. This issue, and many others, are discussed in the video.)
As a counter-example of having to trust the receiving end, consider zero knowledge proofs [1]. Even if you don't trust the prover, you become convinced of what they're trying to prove to you (if it's true).
The basic trick to proper cryptographic voting machines is zero-knowledge-proving that they are working correctly. Any tampering, whether it be in the source code or at the hardware level, will either not affect the tally or break the zero knowledge proofs.
1: http://en.wikipedia.org/wiki/Zero-knowledge_proof