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There are many parallels between the discrete log algorithms discussed in this section and the factoring algorithms in Section 7.2. In particular, Dixon's factoring algorithm and the index calculus discrete log algorithm have many similarities. Both algorithms have an equation finding phase and a linear algebra phase. For properly chosen parameters, the costliest part of the index calculus algorithm is solving for the logarithms of the elements in the factor base. As with Dixon's Algorithm, the equation finding part of the index calculus algorithm can be distributed among multiple processors, but the linear algebra part cannot. There exists another class of algorithms for factoring and discrete logarithm which are known as collision search techniques. The best general collision search method is Pollard's Rho Algorithm [32]. However, collision search algorithms have an exponential work factor, while the quadratic sieve and index calculus are subexponential (see Section 7.2.4 for a discussion of the work factor for the quadratic sieve). Finally, it is worth noting that there is no analog of the quadratic sieve or index calculus for elliptic curve cryptosystems since, for elliptic curves, there is no analog of a factor base. Consequently, collision search is the best available technique for breaking systems based on elliptic curves, and this is why smaller parameters sizes can be used with elliptic curve cryptosystems without sacrificing security.

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where p (t) are the values of (P - Pc)/P along the paths. Close to transition line and near t = 0, we can take them to be the same linear function of t: p (t) "" ct, where c is a constant. Show how the latent heat varies with t.





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8. Indicate whether you want the system to automatically enter a default amount for most employees. If the number varies with employees or is computed based on a rate computation for each employee, leave this eld blank. Also, indicate whether the tax is subject to an annual limit, which is the maximum amount to be paid by any employee. Enter the limit in the appropriate eld, as illustrated in Figure 21-6. After this limit is entered, QuickBooks displays a warning if you attempt to withhold in excess of that amount. Click Next. 9. QuickBooks prompts you to specify which Items are subject to the rate. Click Finish when you are done.

16.4 The accompanying graph shows the magnetization M vs. magnetic field H for nickel, as measured by Weiss and Forrer in 1926. Some numerical values are listed in Table 16.6. Given that T,. = 627.2 K, f3 = 0.368, and 8 = 4.22, select data from a few temperatures above and below T, to verify the scaling form

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If factoring and discrete log algorithms represent classical attacks on public key systems, then the attacks in this section are strictly punk rock. The attacks discussed here do not follow the usual cryptanalytic approach of analyzing the underlying crypto algorithm for weaknesses. Instead, the cryptanalyst looks for any weak link in the overall implementation that might allow information about the private key to leak. First, we discuss three different timing attacks. By exploiting small timing differences that occur in specific met hods of modular exponentiation, the attacker can gain information about the private key. Then we consider a glitching attack, where the attacker induces an error in a computation (for example, by abusing a smartcard). For a particular method of modular exponentiation, a single induced error can enable an attacker to easily recover the private key. Both timing and glitching attacks are practical and it is therefore critical that cryptosystems are built to resist such attacks. It is important to emphasize that the attacks discussed in this section do not exploit any inherent weakness in the RSA algorithm itself. Instead, these attacks exploit specific implementation issues that allow information to leak out -with potentially devastating consequences.

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At one time, it was widely believed that if RSA was implemented correctly, thc only realistic way to attack it was by factoring the modulus. In 1996 Paul Kocher [85]surprised the crypto world when he demonstrated a practical sidechannel attack on RSA. A side channel is an unintended source of information. In the crypto cow tcxt , side channels sometimes leak information about a computation, which in turn reveals information about the key. For example, careful measurements of the amount of current used by smartcards have been used to recover keys. Kocher s side-channel attack on RSA is based on a careful timing of various cryptographic operations. Using selected inputs, he was able to recover the private keys from smartcards which used a relatively simple method of modular exponentiation. Kocher conjectured that his technique could also be used in settings where the modular exponentiation was computed using more efficient means, but other researchers soon discovered this was not the case. Schindler [129] was able to develop a timing attack that succeeds when modular exponentiation is computed in a more optimized fashion than the

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