### Table 1: The primes of special forms

"... In PAGE 6: ... First we consider the curves de ned over Q. See Table1 for the list of integers D, the corresponding j-invariants of the curves whose complex multiplications are the maximal order in Q(p D), and the forms of the primes p such that at least one of the isomorphic classes of the curves has exactly p Fp-points. If p has one of the special forms in Table 1, we can easily construct an elliptic curve E=Fp with exactly p Fp-points.... In PAGE 6: ... See Table 1 for the list of integers D, the corresponding j-invariants of the curves whose complex multiplications are the maximal order in Q(p D), and the forms of the primes p such that at least one of the isomorphic classes of the curves has exactly p Fp-points. If p has one of the special forms in Table1 , we can easily construct an elliptic curve E=Fp with exactly p Fp-points. See [14] for the algorithm to decide the right isomorphic classes.... ..."

### Table 1: The primes of special forms

"... In PAGE 4: ... If jE(Fp)j = p, then its quadratic twist has p + 2 Fp-points. See Table1 for the list of integers D, the corresponding j-invariants of the curves whose complex multiplications are the maximal order in Q(p D), and the forms of the primes p such that at least one of the isomorphic classes of the curves has exactly p Fp-points. If p has one of the special forms in Table 1, we can easily construct an elliptic curve E=Fp with exactly p Fp-points.... In PAGE 4: ... See Table 1 for the list of integers D, the corresponding j-invariants of the curves whose complex multiplications are the maximal order in Q(p D), and the forms of the primes p such that at least one of the isomorphic classes of the curves has exactly p Fp-points. If p has one of the special forms in Table1 , we can easily construct an elliptic curve E=Fp with exactly p Fp-points. See [10] for the algorithm to decide the right isomorphic classes.... ..."

### Table 4: Twin primes: estimates and counts.

2001

"... In PAGE 15: ... Thus, Conjecture B of [8] then predicts the number of primes up to M (up to choosing twists) that give prime order elliptic curves. Table4 gives a comparison between the Koblitz predicted value, the Gross- Smith twin primes value, and actual counts of twin primes and of anomolous primes. The anomolous values are primes naturally paired with themselves in our construction.... ..."

Cited by 2

### Table 5. Mixed walks for elliptic curve subgroups of prime group order

"... In PAGE 12: ... Here, we worked with 200 di erent instances of the DLP and performed 10 DL computations for each instance. We did this for several combinations of r and q,asshownin Table5 . A graphic interpretation of these results is given in Figure 5.... ..."

### Table 4. DL-computation in elliptic curve subgroups of prime order, with iterating function (3.2)

"... In PAGE 10: .... Actions of the functions in (3.1) for G = (Z=11Z) , g = 2, h = 2. When conducting the analogous experiment in elliptic curve subgroups of prime order, we get the results shown in Table4 . They roughly agree with those for prime-order subgroups of (Z=pZ) .... ..."

### Table 4. DL-computation in elliptic curve subgroups of prime order, 3-adding walk

2001

"... In PAGE 10: ... DL-computation in elliptic curve subgroups of prime order, r-adding walk function of r decreases fast until a close-to-random performance is achieved, and then remains rather constant. 1 On the other hand, for r = 3 we do not even obtain a constant average value for L for di erent ranges of group orders: As the results in Table4 show, we have 1Blackburnand Murphy [BM] suggestunder certainheuristicassumptionsthat the relationship between L and r follows the rule L = c q r r?1. Their reasoning matchesBrent and Pollard [BP81], who conjectured that what matters most for the performance of random walks is the variance of... ..."

Cited by 15

### Table 4. DL-computation in elliptic curve subgroups of prime order, 3-adding walk

"... In PAGE 10: ... DL-computation in elliptic curve subgroups of prime order, r-adding walk function of r decreases fast until a close-to-random performance is achieved, and then remains rather constant. 1 On the other hand, for r = 3 we do not even obtain a constant average value for L for di erent ranges of group orders: As the results in Table4 show, we have 1Blackburn and Murphy [BM] suggest under certain heuristic assumptions that the relationship between L and r follows the rule L = c q r r 1. Their reasoning matches Brent and Pollard [BP81], who conjectured that what matters most for the performance of random walks is the variance of... ..."

### Table 3. DL-computation in elliptic curve subgroups of prime order, r-adding walkaverage average r

2001

"... In PAGE 9: ... For r 4 the average value for L (taken over the ratios for a certain range [10n?1; 10n]) appeared to be independent of the size of the group order. The results for r 4, in terms of the average values for L taken over all 11000 ratios, are shown in Table3... ..."

Cited by 15

### Table 5. DL-computation in elliptic curve subgroups of prime order (20-adding walk)

"... In PAGE 12: ... The corresponding results are given in Table 5. The average values for L shown in Table5 are convincingly stable for dif- ferent sizes of the group order, and very close to the random case value L0 = 1:41. This does not yet guarantee that 20-linear walks suit for sim- ulating random random walks for any size of group orders.... ..."

### Table 3. DL-computation in elliptic curve subgroups of prime order, r-adding walk

"... In PAGE 9: ... For r 4 the average value for L (taken over the ratios for a certain range [10n 1; 10n]) appeared to be independent of the size of the group order. The results for r 4, in terms of the average values for L taken over all 11000 ratios, are shown in Table3... ..."