"... In PAGE 76: ... We give the maximum possible value dmax, in addition the used values of d for the associated values of n, m and t. Table6 :1 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 1- resilient n m S-boxes using maximum possible value of dmax. Note that dused lt; dmax is the actual value of the parameter d, which is used in the construction whenever dmax does not yield a proper S-box.... In PAGE 76: ...1: Highest Achieved Nonlinearity and d Values (nl(S)=dused=dmax) for 1-resilient n m S-boxes n = 9 n = 10 n = 11 m Ours 1st Con 2nd Con Ours 1st Con 2nd Con Ours 1st Con 2nd Con 2 224=3=4 240 224 448=3=4 480 480 896=3=5 992 960 3 224=3=4 224 224 448=3=4 480 480 896=3=5 992 960 4 224=3=4 224 224 448=3=4 448 480 960=4=5 960 960 5 224=3=3 224 224 480=4=4 448 480 960=4=5 960 960 6 192=2=2 192 192 448=3=3 448 448 960=4=5 960 960 (res.1,3) Table6 :2 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 2-resilient n m S-boxes. We have found some results better than the others, such as the nonlinearity of the 2-resilient 9 3 S-box as 224, whereas the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] is 192.... In PAGE 77: ...ot exist 6 nonintersecting [5; 2; 3] linear codes. Then d can not be taken as 4. By decreasing d by 1, its maximum value is 3, hence the nonlinearity is 224. Table6 :3 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 3-resilient n m S-boxes. We have found the nonlinearity of the 3-resilient 9 2 S-box as 224, whereas the results in [Johansson amp; Pasalic, 2000] and in [Pasalic... In PAGE 79: ... Hence, we conclude that the search method in their paper, should be a theoretical assignment in the set of some well-known linear block codes. This theoretical choice seems to work quite well for the case of 1-resilient S-boxes shown in Table6 :4, where only one of our results(shown by bold let- ters) is superior to theirs, whereas 5 of their results(bold) are better than ours. However, in Table 6:6 there are 4 cases(bold) that our S-boxes have higher nonlinearity, and they all correspond to small search spaces.... In PAGE 79: ... This theoretical choice seems to work quite well for the case of 1-resilient S-boxes shown in Table 6:4, where only one of our results(shown by bold let- ters) is superior to theirs, whereas 5 of their results(bold) are better than ours. However, in Table6 :6 there are 4 cases(bold) that our S-boxes have higher nonlinearity, and they all correspond to small search spaces. Table 6.... In PAGE 82: ... We have also shown that the highest possible nonlinearity achievable by Johansson and Pasalic con- struction for 2-resilient 9 2 S-box is 224; therefore, the nonlinearity value of 240 that is claimed to be found in [Johansson amp; Pasalic, 2000] is not pos- sible. As can be observed from Table6 :1, the rst construction [Johansson amp; Pasalic, 2000] seems to be more premising than the second construction [Pasalic amp; Maitra, 2002] in terms of the nonlinearity. Comparing our construction results with those of [Johansson amp; Pasalic, 2000] as shown in Tables 6:4, 6:5 and 6:6, we notice that they have obtained better nonlinearities than ours for some cases, where the cardinality of the set of (n d; m; t + 1) linear block codes is excessively large.... ..."

"... In PAGE 76: ... We give the maximum possible value dmax, in addition the used values of d for the associated values of n, m and t. Table6 :1 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 1- resilient n m S-boxes using maximum possible value of dmax. Note that dused lt; dmax is the actual value of the parameter d, which is used in the construction whenever dmax does not yield a proper S-box.... In PAGE 76: ...1: Highest Achieved Nonlinearity and d Values (nl(S)=dused=dmax) for 1-resilient n m S-boxes n = 9 n = 10 n = 11 m Ours 1st Con 2nd Con Ours 1st Con 2nd Con Ours 1st Con 2nd Con 2 224=3=4 240 224 448=3=4 480 480 896=3=5 992 960 3 224=3=4 224 224 448=3=4 480 480 896=3=5 992 960 4 224=3=4 224 224 448=3=4 448 480 960=4=5 960 960 5 224=3=3 224 224 480=4=4 448 480 960=4=5 960 960 6 192=2=2 192 192 448=3=3 448 448 960=4=5 960 960 (res.1,3) Table6 :2 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 2-resilient n m S-boxes. We have found some results better than the others, such as the nonlinearity of the 2-resilient 9 3 S-box as 224, whereas the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] is 192.... In PAGE 77: ...ot exist 6 nonintersecting [5; 2; 3] linear codes. Then d can not be taken as 4. By decreasing d by 1, its maximum value is 3, hence the nonlinearity is 224. Table6 :3 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 3-resilient n m S-boxes. We have found the nonlinearity of the 3-resilient 9 2 S-box as 224, whereas the results in [Johansson amp; Pasalic, 2000] and in [Pasalic... In PAGE 79: ... Hence, we conclude that the search method in their paper, should be a theoretical assignment in the set of some well-known linear block codes. This theoretical choice seems to work quite well for the case of 1-resilient S-boxes shown in Table6 :4, where only one of our results(shown by bold let- ters) is superior to theirs, whereas 5 of their results(bold) are better than ours. However, in Table 6:6 there are 4 cases(bold) that our S-boxes have higher nonlinearity, and they all correspond to small search spaces.... In PAGE 79: ... This theoretical choice seems to work quite well for the case of 1-resilient S-boxes shown in Table 6:4, where only one of our results(shown by bold let- ters) is superior to theirs, whereas 5 of their results(bold) are better than ours. However, in Table6 :6 there are 4 cases(bold) that our S-boxes have higher nonlinearity, and they all correspond to small search spaces. Table 6.... In PAGE 82: ... We have also shown that the highest possible nonlinearity achievable by Johansson and Pasalic con- struction for 2-resilient 9 2 S-box is 224; therefore, the nonlinearity value of 240 that is claimed to be found in [Johansson amp; Pasalic, 2000] is not pos- sible. As can be observed from Table6 :1, the rst construction [Johansson amp; Pasalic, 2000] seems to be more premising than the second construction [Pasalic amp; Maitra, 2002] in terms of the nonlinearity. Comparing our construction results with those of [Johansson amp; Pasalic, 2000] as shown in Tables 6:4, 6:5 and 6:6, we notice that they have obtained better nonlinearities than ours for some cases, where the cardinality of the set of (n d; m; t + 1) linear block codes is excessively large.... ..."

"... In PAGE 38: ... We generate a simple example and present it in Section 4:4. To show the restrictions on design parameters, we provide Table4 :1, which shows the highest possible nonlinearity values achievable by this method for n m S-boxes with n = 6, 7 and 8. Later in Chapter 6, we present our construction results for larger values of n, using the Johansson amp; Pasalic method described in the following sections.... In PAGE 49: ... Moreover, t n d m must be satis ed. So, Table4 :1 shows the highest nonlinearity values achievable by this construction for n m S-boxes for n = 6, 7 and 8, choosing the maximum possible value of d for the associated values of n; m and t. Table 4.... ..."

"... In PAGE 76: ... We give the maximum possible value dmax, in addition the used values of d for the associated values of n, m and t. Table6 :1 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 1- resilient n m S-boxes using maximum possible value of dmax. Note that dused lt; dmax is the actual value of the parameter d, which is used in the construction whenever dmax does not yield a proper S-box.... In PAGE 76: ... Table6 :2 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 2-resilient n m S-boxes. We have found some results better than the others, such as the nonlinearity of the 2-resilient 9 3 S-box as 224, whereas the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] is 192.... In PAGE 77: ...ot exist 6 nonintersecting [5; 2; 3] linear codes. Then d can not be taken as 4. By decreasing d by 1, its maximum value is 3, hence the nonlinearity is 224. Table6 :3 compares our highest achieved nonlinearity (nl(S)) results with the results in [Johansson amp; Pasalic, 2000] and in [Pasalic amp; Maitra, 2002] for 3-resilient n m S-boxes. We have found the nonlinearity of the 3-resilient 9 2 S-box as 224, whereas the results in [Johansson amp; Pasalic, 2000] and in [Pasalic... In PAGE 79: ... Hence, we conclude that the search method in their paper, should be a theoretical assignment in the set of some well-known linear block codes. This theoretical choice seems to work quite well for the case of 1-resilient S-boxes shown in Table6 :4, where only one of our results(shown by bold let- ters) is superior to theirs, whereas 5 of their results(bold) are better than ours. However, in Table 6:6 there are 4 cases(bold) that our S-boxes have higher nonlinearity, and they all correspond to small search spaces.... In PAGE 79: ... This theoretical choice seems to work quite well for the case of 1-resilient S-boxes shown in Table 6:4, where only one of our results(shown by bold let- ters) is superior to theirs, whereas 5 of their results(bold) are better than ours. However, in Table6 :6 there are 4 cases(bold) that our S-boxes have higher nonlinearity, and they all correspond to small search spaces. Table 6.... In PAGE 82: ... We have also shown that the highest possible nonlinearity achievable by Johansson and Pasalic con- struction for 2-resilient 9 2 S-box is 224; therefore, the nonlinearity value of 240 that is claimed to be found in [Johansson amp; Pasalic, 2000] is not pos- sible. As can be observed from Table6 :1, the rst construction [Johansson amp; Pasalic, 2000] seems to be more premising than the second construction [Pasalic amp; Maitra, 2002] in terms of the nonlinearity. Comparing our construction results with those of [Johansson amp; Pasalic, 2000] as shown in Tables 6:4, 6:5 and 6:6, we notice that they have obtained better nonlinearities than ours for some cases, where the cardinality of the set of (n d; m; t + 1) linear block codes is excessively large.... ..."

"... In PAGE 18: ... This suggests that p H11005 2 for the C(d) test for nonlinearity. We perform the C(d) test with p H11005 1, 2, d H11349 p and various values of m (the number of start-up observations in the ordered autoregression); the results are given in Table9 . The combination (p, d) H11005 (2, 2) consistently gives the most significant C(d) statistic under different values of m.... ..."