### Table 1.1 shows the comparison of linear-, polynomial-, and exponential-time algorithms (Garey amp; Johnson 1979).

### Table 2.1, Schneier [24] illustrates the various classes of algorithms and examples of their running times. From this table, it is apparent that if you have a cryptographic algorithm that requires exponential time to break, you can rest assured that, barring any fantastic leaps in the computer industry, it will not be broken.

1997

### Table 6: Performance of modular exponentiation algorithms for jnj = 512 (in msec)

"... In PAGE 10: ... Except for the classical algorithm, all other reduction algorithms require more or less precom- putations based on the modulus. The running times for modular reduction shown in Table 2 Table 5 do not include such precomputation time, while the running times for modular exponenti- ation shown in Table6 Table 9 do include the precomputation time. For exponentiation we used the window algorithm.... ..."

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### Table 9: Performance of modular exponentiation algorithms for jnj = 2048 (in msec)

"... In PAGE 9: ... Partial assembly language implementations are also done for PCs. Table 1 Table9 show our implementation results. The following notations are used in the tables: Machines and languages: { S20/60/C: implementation by C on SPARC20/60MHz { US/167/C: implementation by C on ULTRASPARC/167MHz... In PAGE 10: ... Except for the classical algorithm, all other reduction algorithms require more or less precom- putations based on the modulus. The running times for modular reduction shown in Table 2 Table 5 do not include such precomputation time, while the running times for modular exponenti- ation shown in Table 6 Table9 do include the precomputation time. For exponentiation we used the window algorithm.... ..."

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### Table 1 distinction thus shown between these three fault models is (to

### Table 7. Polynomials found by the LLL algorithm

"... In PAGE 13: ...This means that LLL must be performed on a basis of size 2n n,whichleadstoan exponential time algorithm. Some results are tabulated in Table7 , for Pisot numbers, Salem numbers, and non-Pisot non-Salem numbers. 7.... ..."

### Table 1: Approximation ratios for scheduling with con icts. The rst two algorithms have running time exponential in m while the third has running time polynomial in m. Lines above the double rule refer to results based on previous work.

2007

"... In PAGE 4: ...3 Our results: the offline model. The approximation ratios in the of ine model are summarized in Table1 . Given the hardness results mentioned previously, we focus on the basic case of m = 2.... ..."

### Table 1: Approximation ratios for scheduling with conflicts. The first two algorithms have running time exponential in m while the third has running time polynomial in m. Lines above the double rule refer to results based on previous work.

2007

"... In PAGE 4: ...3 Our results: the offline model. The approximation ratios in the offline model are summarized in Table1 . Given the hardness results mentioned previously, we focus on the basic case of m = 2.... ..."

### Table 1. Typical results of optimal coloring algorithm on rings of 7 or 8 nodes. Note that execution time varies greatly due to the exponential size of the set of possible solutions.

2004

### Table 1: Several discrete-time gradient-based algorithms: EGU|Unnormalized Exponentiated Gradient [KW97b], EG|Exponentiated Gradient [KW97b], and BEG|Bounded Exponentiated Gradient [Byl97]. Here 5t;i is short hand for @Lt(!t) @!t[i]

1997

"... In PAGE 3: ... The Euler-discretization of the dual update (2) gives !t+h := !t ? h 5 Lt(!t) or t+h := f(f?1( t) ? h 5 Lt(!t)) : (4) For example if f is the identity function then both the main update and its dual collapse to the conventional gradient descent update. However if f(x) = ln(x) then the discretized version (3) of the main update with h = 1 gives the Unnormalized Exponentiated Gradient Update (EGU) of [KW97b] (See the Table1 for more examples).In the next section we discuss the purpose and desired properties of link functions.... ..."

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