Any ideas to make an Electronic Voter Machine more secure? EVMs are not secure they say. So how can we make it more secure tham the existing one using cryptography?
We can't make satisfactory Electronic Voting Machines. Their design face conflicting goals that are impossible to reconcile, even in the simplest conceivable use case: a yes/no vote, a single machine.


Count votes (or at least: determine if there was more yes than no) with the result public.
Limit voting to one per registered voter.
Keep individual votes secret, even from organizers or/and if a person casting vote is actively trying to prove how s/he voted (that requirement helps freedom of vote despite attempted bribery/duress), within the limits inherent to what gets published of the result.
Resist denial of service.
Convince reasonable observers with ordinary skills that the above goals are met, even if observers do not trust the organizers and designers of the machine, understandably so [*].


Among the few non-electronic approaches that work is one that evolved over time: paper ballot freely available to all, put in opaque envelope mandatorily in a private booth, with the envelope publicly inserted in a transparent urn (with mechanical interlock preventing unauthorized insertion), check of the voter's identity and that the voting role is unsigned right before that insertion, and signing the voting role right afterwards, with the urn and envelopes publicly opened in the end and counted, under public scrutiny all along.

Alternatives have been tried:


Mechanical counters, with interlocks preventing multiple voting. There have been jams (perhaps intentional). Only people understanding mechanical machinery (similar to watchmaking) can observe and confirm that counting work as intended before and after voting. And it is to fear that various side channels (lifting a cover hidding the value, sound, ...) can compromise vote secrecy. On the positive side, it can be me made slow and noisy to covertly alter the counters. 
Electromechanical counters: reportedly more reliable, but side channels are rather worse, altering the counters might be faster and easier, and (because wires and air gaps can be hair-thin) an observer (needing basic understanding of electric circuit) could miss something redirecting counting to the wrong counter. While it would be conceivable and useful to make counters that the voter (only) can see moving when casting vote, without being able to tell the count, I have not heard that it was used.


The more we go towards modern electronics and complex cryptography, the worse the "convince reasonable observers with ordinary skills" goal is met. Finding backdoors in silicon and software is extremely hard, and entirely impossible at the voting location. For most reasonable observers, a finite field is a bounded piece of land.



[*] Voting machines in use in (few and mid-sized) French cities are purchased, stored, serviced and operated (with supervision from the ministry of home affairs) under the authority of the Mayor, yet are used to (re-)elect the Mayor. Their specification and type approval is under the authority of the ministry of home affairs, which head is chosen by the prime minister, which is chosen by the Président de la République, which the machines contribute to (re-)elect. In 2007 that election was won by the former head of the ministry of home affairs that gave delegation for establishing the specifications as law, and was again head of that ministry weeks before his own election and days before a software change was made to the most common type of machines. BTW that software is secret, and it's integrity is publicly demonstrated by a checksum that the software computes and displays. Descartes reportedly turned in his grave.