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4.1 The sound of a single secret key
Figure 12 (adjunta abajo) depicts the spectrogram of two identical RSA signing operations in sequence, using the same 4096 bit key and message. Each signing operation is preceded by a short delay during which the CPU is in a sleep state. Figure 12 contains several interesting effects. The delays, where the computer is idle, are manifested as bright horizontal strips. Between these strips, the two signing operations can be clearly distinguished. Halfway through each signing operation there is a transition at several frequency bands (marked with yellow arrows). This transition corresponds to a detail in the RSA implementation of GnuPG. For public key n = pq, the RSA signature s = md mod n is computed by evaluating md mod (p−1) mod p and md mod (q−1) mod q, and combining these via the Chinese Remainder Theorem. The first half of the signing operation corresponds to the exponentiation modulo p, and the secondto the exponentiation modulo q. Note that the transition between these secret modules is clearly visible. Moreover, this effect is consistent and reproducible (in various frequency ranges) not only on the Evo N200, but on many modern machines made by various manufacturers. For example, Figure 13 contains the recording of RSA signatures executed on a Lenovo ThinkPad T61 using the same key as in Figure 12. Note that while not as prominent as in the case of the Evo N200, the RSA operations are clearly visible
at 34–36 kHz.
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