Spreading - Disappearing Cryptography - page 290

8

1.5

1

6

0.5

4

50 100 150 200 250

-0.5

2

-1

50 100 150 200 250

-1.5

8

1

6

0.5

4

50 100 150 200 250

2

-0.5

50 100 150 200 250 -1

Figure 14.10: The four steps in smoothing noisy data with the FFT.
The first graph shows the data; the second, the result of computing
the FFT; the third, data after small values are set to zero; and fourth,
the final result.

y 251 .Themore
difficult challenge is doing this in a way that is hard to detect and re-
sistant to changes to the file that are either malicious or incidental.
Here's an example. Figure 14.11 shows the absolute value of the
first 600 coefficients from the Fourier transform of the voice signal
shown in Figure 14.3. The major frequencies are easy to identify and
change if necessary.
A simple watermark or signal can be inserted by changing setting
y 300 = 100000 . The result after taking the inverse transform looks
identical to Figure 14.3 at this level of detail. The numbers still range
from

can be added by just increasing the values of y 4 and

000 . The difference, though small, can be seen
by subtracting the original signal from the watermarked one. Figure
14.12 shows that the difference oscilliates between 750 and

− 30 , 000 to 30 ,

− 750 with

the correct frequency.

14.7.1 Tweaking a Number of Coefficients

Ingemar Cox, Joe Kilian, Tom Leighton and Talal Shamoon [CKLS96]
offer a novel way to hide information in an image or sound file by
tweaking the k largest coefficients of an FFT or a DCT of the data.

Call these {y 1 ,y 2 ,...y k−1 } . The largest coefficients correspond to the