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Because 2 2Cjj is the standard error of the regression coefficient j, we would also write the CI formula as j t 2,n p se1 j 2 t 2,n p se1 j 2. j j t

(12-34)

(i) Compute MA = MO AM, and M i = M I AM1. The precise values of AM0 and AM1 are specified by the input differential. For Wang s differential, these values are given in the next section.

EXAMPLE 12-6

We will construct a 95% confidence interval on the parameter 1 in the wire bond pull strength problem. The point estimate of 1 is 1 2.74427 and the diagonal element of 1X X2 1 corresponding to 1 is C11 0.001671. The estimate of 2 is 2 5.2352, and t0.025,22 2.074. Therefore, the 95% CI on 1 is computed from Equation 12-34 as 2.74427 12.0742 215.235221.0016712

(j) The MD5 hash of message M = ( M 0 , M l ) is equal to the MD5 hash of the message M = (MA,M i ) .

12.0742 215.23522 1.0016712

438 12-3.2

We may also obtain a confidence interval on the mean response at a particular point, say, x01, x02, p , x0k. To estimate the mean response at this point, define the vector 1 x01 Ex02U o x0k

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See Problem 25 for one simple improvement to the computational phase of the attack as described here. For Wang s differential, the work factor for the computational part of the attack is dominated by finding Mo. The work factor for finding MO is on the order of 2n MD5 hashes, where n is the number of conditions for MO that are not satisfied by the modification techniques mentioned above (and described in more detail, below). As originally implemented by Wang, the computational phase had a work factor significantly greater than 240. Subsequent improvements have steadily lowered the work factor and, to date, the best claimed work factor is on the order of about 232.25 MD5 hashes [144]. It is possible that this will be reduced further by incremental improvements to Wang s attack. Below, we discuss each part of the attack, with the emphasis on the computational aspects. But before diving into the details, we mention an interesting insight due to Daum [34]. Suppose we have an MD-like hash function that only has three rounds--such as MD4 but not MD5. Then we would expect to find a collision using Wang s technique, since, roughly speaking, the input differential can be selected so that the third round conditions hold, singlestep modifications can then be used to ensures that the first round conditions hold, and, finally, multi-step modifications can ensure that the second round conditions hold. However, MD5 has four rounds, so some special property of MD5 must be exploited to make Wang s attack succeed. We briefly consider this special feature after discussing the attack outlined above.

(12-36)

The PowerShot G11 not only performs better in relation to noise levels than previous G-series cameras, but it now includes a new Low Light mode. This mode sets the ISO Speed setting automatically between 320 and 12,800 to attain optimal settings for handholding the camera in low light. Try this mode when you need to go beyond ISO 3200 yet still don t want to use ash. Low Light mode is indicated on the Mode dial by a candle icon and enables you to shoot up to an expanded ISO of 12,800 in a reduced 2.5MP resolution at a quick 2.4fps (frames per second) capturing natural indoor shots without the need for a ash. Low Light mode is set at 1824 x 1368 resolution but this is more than suf cient for 4x6-inch prints and, in some cases, 5x7-inch prints as well.

(12-37)

The objective of a time-memory trade-off (TMTO) is to do some one-time work so that each time the algorithm is executed it is more efficient. A TMTO is a general technique that can be applied to improve the performance of many types of algorithms. In this section, we present Hellman s cryptanalytic TMTO, which was developed to attack the Data Encryption Standard (DES), but the approach will work against any block cipher. Our discussion here closely follows that given in [142].

For the multiple linear regression model, a 100(1 mean response at the point x01, x02, . . . , x0k is

x 1 2 0

(12-38)

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