In WWII the primary way of encrypting radio broadcasts was to have the broadcaster transmit a signal with noise added from record, while the receiver would have an identical copy of the record subtracting the noise.
Assuming that the noise is totally random, and the record is never replayed. How would one attack it?
While this isn't strictly a cryptography question, do modern encrypted radios that work in frequencies that are too low to transmit digital data like this still use this principle?
In WWII the primary way of encrypting radio broadcasts was to have the broadcaster transmit a signal with noise added from record, while the receiver would have an identical copy of the record subtracting the noise.
This is referred to as "scrambling", which is "encryption" in the analogue domain.
Assuming that the noise is totally random, and the record is never replayed. How would one attack it?
Randomness of noise is quite a vague concept. You seem to be hinting at some kind of "one time pad" here. I'll first hint you why this isn't a thing for a scrambler, then give you some hints at possible attacks.
One time pads are carried out using a xor operation. A key property of the XOR operator is that when the distribution of $s$ is equiprobable (50% 1's, 50% 0's, which hold for the pad of a one time pad), the resulting ciphertext $e=m\oplus s$ will also have this probability, independently of the distribution of $m$. This is what makes the one time pad unbreakable.
Compare this to a scrambler. A scrambler adds the message to the pad: $e=m + s$. The plus operator does not have this equiprobability; it even, when $m$ and $s$ are independent, mixes the frequency spectrum! (To see this: the Fourier transform is a linear operator)
Let's break some analogue scramblers, using digital technology! Let's assume our message is speech, which feels like a pretty broad assumption. Speech has some cool properties; one of which is that it has a lot of repetition it vowels, and that most information is below 3000 Hz.
You can, for example, try a windowed cross-correlation in windows of around 30 milliseconds. The peaks of the cross-correlation give you an idea of the repetition frequency of certain vowels (out of which you even might extract the vowels). You may now attempt averaging out over different shifts, which should give you your vowels back.
A bunch of other statistical attacks can be carried out here. You can try yourself easily in MATLAB: sample a sine wave, add white noise, and try the above trick. Speech is a bit more difficult, but should be doable.
As always, give me a ping if you want me to go more in depth. I have no real references here, as I have this in my head from my signal processing and statistics courses.
do modern encrypted radios that work in frequencies that are too low to transmit digital data like this still use this principle?
I cannot really answer that question, sadly. e-sushi made a comment that points to this question though, which seems to answer your question partly.
I very much doubt the question's assertion that
In WWII the primary way of encrypting radio broadcasts was to have the broadcaster transmit a signal with noise added from record, while the receiver would have an identical copy of the record subtracting the noise.
It's attested that synchronized phone records used as one-time key have been used in the voice encryption system SIGSALY. However:
Addition of noise directly to a voice signal can't work as an encryption mean, because
Tentative: In SIGSALY, the voice was split into 10 frequency bands by filters, the amplitude in each band digitized to one of six level (an A/D converter of a little less than 3 bits) at a 50 Hz sampling frequency. There was also two channels for "pitch". Each of the 12 channels's data was encrypted by addition modulo 6 (or similar group operation in a group of order 6) of the pad somewhat encoded in the records serving as one-time-pad, and transmitted. Decryption inverted this process. For details see the cryptomuseum's entry on SIGSALY aka project X or the Green Hornet. I want to thank the author of this answer.
Because SIGSALY essentially used a One Time Pad (assuming no reuse of the records), it was very secure. Contrary to A.3, there is no indication that it was ever broken.
