Furthermore after
Conversely, if I∉U,this implies that the complement of I
the slash is used for the setminus operation. I think using \setminus there (which generates a backslash) would be a more standard notation less likely to be mistaken for quotient structures.
This would assume that either Bob is unaware that an attack might be happening, or that he can’t be bothered to do statistical analysis on his vector.
If the original value (without the noise chosen by Alice) is non-obvious to the attacker, but obvious to Bob (e.g., they use redundancy and encryption—it is a well known fact that Alice and Bob like cryptography), and the magnitude of the noise is common knowledge, then all attempts to modify the message will, on average, increase the standard deviation of the noise as measured by Bob. If my math is right, the attacker could modify each value by about half the width of the noise and end up with an expected χ2 sum of 112 instead of 100, which will probably not be suspicious to Bob.
If Bob has an idea of the attackers objective, detecting tampering will get much easier. If Bob suspects that the attacker wants a huge sum, he can just calculate the sum of the noise terms and compare that to the expected distribution. Then any deniable tampering would have to be within expected random fluctuations. (Of course, for every vector, there is some base in which it looks very suspicious.)
Often, we have an idea what the objective of an adversary using a sum-threshold attack might be. There is more utility in influencing who get’s to be president of a country than in influencing who will become the tenth-ranking janitor in their residence. Some bosses would like to pressure their employees into having sex, few if any want to condition them to speak sentences with a prime number of syllables.