Applied Algebra and Number Theory Essays in Honor of Harald Niederreiter on the occasion of his 70th birthday Harald Niederreiter’s pioneering research in the field of applied algebra and number theory has led to important and substantial breakthroughs in many areas This collection of survey articles has been authored by close colleagues and leading experts to mark the occasion of his 70th birthday The book provides a modern overview of different research areas, covering uniform distribution and quasi-Monte Carlo methods as well as finite fields and their applications, in particular cryptography and pseudorandom number generation Many results are published here for the first time The book serves as a useful starting point for graduate students new to these areas, or as a refresher for researchers wanting to follow recent trends G E R H A R D L A R C H E R is Full Professor for Financial Mathematics and Head of the Institute for Financial Mathematics at the Johannes Kepler University Linz F R I E D R I C H P I L L I C H S H A M M E R is Associate Professor in the Institute for Financial Mathematics at the Johannes Kepler University Linz A R N E W I N T E R H O F is Senior Fellow at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) at the Austrian Academy of Sciences, Linz C H A O P I N G X I N G is Full Professor in the Department of Physical and Mathematical Sciences at Nanyang Technological University, Singapore Harald Niederreiter in Marseille, 2013 Applied Algebra and Number Theory Essays in Honor of Harald Niederreiter on the occasion of his 70th birthday Edited by GERHARD LARCHER Johannes Kepler University Linz FRIEDRICH PILLICHSHAMMER Johannes Kepler University Linz ARNE WINTERHOF Austrian Academy of Sciences, Linz C H AO P I N G X I N G Nanyang Technological University, Singapore University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence www.cambridge.org Information on this title: www.cambridge.org/9781107074002 © Cambridge University Press 2014 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2014 Printed in the United Kingdom by Clays, St Ives plc A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Applied Algebra and Number Theory : Essays in Honor of Harald Niederreiter on the occasion of his 70th birthday / edited by Gerhard Larcher, Johannes Kepler Universität Linz, Friedrich Pillichshammer, Johannes Kepler Universität Linz, Arne Winterhof, Austrian Academy of Sciences, Linz, Chaoping Xing, Nanyang Technological University, Singapore pages cm Includes bibliographical references ISBN 978-1-107-07400-2 (hardback) Number theory I Niederreiter, Harald, 1944- honoree II Larcher, Gerhard, editor QA241.A67 2014 512.7–dc23 2014013624 ISBN 978-1-107-07400-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Preface page xi Some highlights of Harald Niederreiter’s work Gerhard Larcher, Friedrich Pillichshammer, Arne Winterhof and Chaoping Xing 1.1 A short biography 1.2 Uniform distribution theory and number theory 1.3 Algebraic curves, function fields and applications 1.4 Polynomials over finite fields and applications 1.5 Quasi-Monte Carlo methods References 10 13 18 Partially bent functions and their properties Ayỗa ầeásmelioglu, Wilfried Meidl and Alev Topuzoglu 22 2.1 2.2 2.3 2.4 2.5 2.6 22 24 28 29 35 Introduction Basic properties Examples and constructions Partially bent functions and difference sets Partially bent functions and Hermitian matrices Relative difference sets revisited: a construction of bent functions References 36 38 Applications of geometric discrepancy in numerical analysis and statistics Josef Dick 39 3.1 3.2 3.3 3.4 39 40 44 47 Introduction Numerical integration in the unit cube Numerical integration over the unit sphere Inverse transformation and test sets v vi Contents 3.5 3.6 Acceptance-rejection sampler Markov chain Monte Carlo and completely uniformly distributed sequences 3.7 Uniformly ergodic Markov chains and push-back discrepancy References 48 Discrepancy bounds for low-dimensional point sets Henri Faure and Peter Kritzer 58 4.1 Introduction 4.2 Upper discrepancy bounds for low-dimensional sequences 4.3 Upper discrepancy bounds for low-dimensional nets 4.4 Lower discrepancy bounds for low-dimensional point sets 4.5 Conclusion References 58 66 75 81 87 88 On the linear complexity and lattice test of nonlinear pseudorandom number generators Domingo Gómez-Pérez and Jaime Gutierrez 91 5.1 Introduction 5.2 Lattice test and quasi-linear complexity 5.3 Quasi-linear and linear complexity 5.4 Applications of our results 5.5 An open problem References 91 93 94 97 99 99 A heuristic formula estimating the keystream length for the general combination generator with respect to a correlation attack Rainer Göttfert 51 53 54 102 6.1 The combination generator 6.2 The model 6.3 Preliminaries 6.4 The correlation attack 6.5 The formula References 102 102 103 103 105 108 Point sets of minimal energy Peter J Grabner 109 7.1 7.2 109 111 Introduction Generalized energy and uniform distribution on the sphere Contents 7.3 Hyper-singular energies and uniform distribution 7.4 Discrepancy estimates 7.5 Some remarks on lattices References 116 119 122 123 The cross-correlation measure for families of binary sequences Katalin Gyarmati, Christian Mauduit and András Sárközy 126 8.1 8.2 8.3 8.4 126 129 133 Introduction The definition of the cross-correlation measure The size of the cross-correlation measure A family with small cross-correlation constructed using the Legendre symbol 8.5 Another construction References 10 135 139 141 On an important family of inequalities of Niederreiter involving exponential sums Peter Hellekalek 144 9.1 Introduction 9.2 Concepts 9.3 A hybrid Erd˝os–Turán–Koksma inequality References 144 148 157 161 Controlling the shape of generating matrices in global function field constructions of digital sequences Roswitha Hofer and Isabel Pirsic 164 10.1 10.2 10.3 10.4 Introduction Global function fields Constructions revisited Designing morphological properties of the generating matrices 10.5 Computational results 10.6 Summary and outlook References 11 vii 164 169 170 176 182 185 187 Periodic structure of the exponential pseudorandom number generator Jonas Kaszián, Pieter Moree and Igor E Shparlinski 190 11.1 Introduction 11.2 Preparations 190 194 viii 12 13 14 15 Contents 11.3 Main results 11.4 Numerical results on cycles in the exponential map 11.5 Comments References 195 199 201 202 Construction of a rank-1 lattice sequence based on primitive polynomials Alexander Keller, Nikolaus Binder and Carsten Wächter 204 12.1 Introduction 12.2 Integro-approximation by rank-1 lattice sequences 12.3 Construction 12.4 Applications 12.5 Conclusion References 204 205 206 211 214 214 A quasi-Monte Carlo method for the coagulation equation Christian Lécot and Ali Tarhini 216 13.1 Introduction 13.2 The quasi-Monte Carlo algorithm 13.3 Convergence analysis 13.4 Numerical results 13.5 Conclusion References 216 219 222 229 229 231 Asymptotic formulas for partitions with bounded multiplicity Pierre Liardet and Alain Thomas 235 14.1 Introduction 14.2 Asymptotic expansion of MU,q 14.3 Proof of Theorem 14.2 References 235 239 246 253 A trigonometric approach for Chebyshev polynomials over finite fields Juliano B Lima, Daniel Panario and Ricardo M Campello de Souza 15.1 15.2 15.3 15.4 Introduction Trigonometry in finite fields Chebyshev polynomials over finite fields Periodicity and symmetry properties of Chebyshev polynomials over finite fields 255 255 257 265 270 328 On the linear complexity of multisequences For m = 2, 3, , M + 1: m−1 − pm p˜ m = p˜ m−1 − p˜M+2−m · (M + − m), dm−1 = b˜m = b˜m−1 − dm−1 , bm := b˜m p˜ m−1 − p˜m M+2−m , For m = M, M − 1, , 2, 1, for i = 0, , p˜ m − pm − do: swap(bm+i , bm+i +1 ) For m = t, t + 1, , M do: swap(bm , bm+1 ) Finally, set d := b M+1 Now (b1 , , b M )} ∈ Z M , and (b1 , , b M , d; T, t) is the state corresponding to the input partition 18.4.2 Correspondence between the NW and VC approaches We will first obtain a connection between the approach of Niederreiter and Wang [18], whose partition of L into ( I˜m ) we will call the “NW-partition,” and the BDM approach taken by Vielhaber and Canales [29], with its corresponding state (b1 , , b M , d; T, t), called the “VC-partition.” Each (nonordered!) NW-partition (I1 , , I M ) is equivalent to the BDM n n state with bm = M+1 − Im , i.e., (b1 , , b M ) = M+1 · (1, , 1) − (I1 , , I M ) Wang and Niederreiter [35] use only the ordered (nonincreasing) NWpartition ( I˜1 , , I˜M ) There are up to M! BDM states subsumed into this NW-partition, all those with {b1 , , b M } = {I1 , , I M } (as a multiset) The corresponding partitions apparently are (no proof yet) those with the highest part not exceeding k (VC-partitions of the BDM states), and those with the sum of the M parts not exceeding k (NW-partitions) The growth rates are accordingly as follows There are exactly n+M = M nM M! + O(n M−1 ) active states or VC-partitions in the BDM up to column n, but M n only PM (n) = M!M! + O(n M−1 ) partitions in the NW approach We obtain the NW-partition from the BDM state and vice versa (see above) and the BDM state from the VC-partition by the previous algorithm We not yet know how to get from a BDM state/NW-partition to the VC-partition, except by walking through the transitions, starting from s0 , and counting inhibitions Some examples for the correspondence of VC-partitions to NW-partitions are as follows From VC-partition (e.g [00]) to NW-partition (e.g 33), for M = 2, n = 9, see Table 18.2a Dropping every (M + 1)st row, and then alternatingly distributing rows and columns onto two parts, we have four submatrices, which now exhibit more structure, see Table 18.2b For M = 3, n = 22, see Table 18.2c Michael Vielhaber 329 Table 18.2 VC-partition to NW-partition, a, b for M = 2, n = 9, c for M = 3, n = 22 a 0] 1] 2] 3] 4] 5] 6] 7] 8] 9] [0 [1 [2 [3 [4 [5 [6 [7 [8 [9 33 43 42 42 44 41 41 54 50 50 43 32 32 53 51 51 54 40 40 52 52 53 31 31 63 60 60 22 62 61 61 63 30 30 62 21 21 72 70 70 71 71 72 20 20 11 81 80 80 81 10 10 90 90 00 b 1] 3] [0 [2/3 [5/6 [8/9 32 51 40 22 61 30 [1 [4 [7 43 53 54 62 63 c 0] 1] 5] 7] 71 20 10 72 9] 00 81 2] 3] 4] [00 111 [10 [11 211 211 211 [20 [21 [22 221 221 221 311 311 311 [30 [31 [32 [33 220 210 210 320 310 310 320 110 410 410 [40 [41 [42 [43 [44 220 210 210 320 300 310 310 320 200 110 410 400 410 100 500 [50 [51 [52 [53 [54 [55 220 210 210 320 300 300 310 310 320 200 200 110 410 400 400 410 100 100 500 500 0] 2] 4] 6] 8] [0 [2/3 [5/6 [8/9 33 42 41 50 52 31 60 21 70 11 80 90 [1 [4 [7 43 44 54 53 63 62 72 81 5] 000 330 On the linear complexity of multisequences As can be observed, for M = and n = 22, the outer VC-partitions are those [abc] with ≥ a ≥ b ≥ c ≥ 0, while the inner NW-partitions are those [abc] with ≥ a ≥ b ≥ c ≥ ∧ ≥ a + b + c ≥ 18.4.3 Summing up (1) The refined LBRMS and mSCFA algorithms are equivalent and are also equivalent to the Feng–Tzeng [6] algorithm (see [36]) concerning the development of degrees/valuations of the approximation (2) The Niederreiter–Wang and Vielhaber–Canales (BDM) approaches are equivalent, the former having less (by a factor M!) states, the latter having a smaller (almost trivial) recurrence equation (state transition) (3) The results of Niederreiter and Wang are mathematically proven and give closed formulas for M = 1, 2, (at the moment), while the result of Vielhaber and Canales gives a single closed formula valid for every M ∈ N, but this has only been verified numerically for M ≤ and M = 16 18.4.4 From Number to Zahlen tuples: bijections between N0M and Z M This section describes experimentally obtained findings, verified only up to the precision stated in the previous paragraphs The remarkable fact is that the BDM states/NW-partitions and the corresponding VC-partitions apparently induce a 1:1 correspondence between the sets Z M and N0M , for all dimensions M While such bijections abound, it is nevertheless surprising to get such an algorithm out of the seemingly unrelated field of multisequence complexity or Diophantine approximation There is a conjectured bijection from Z M to N0M : for all ≤ T ≤ M, ≤ t ≤ M, there are M(M + 1) different functions from Z M to N0M , using the Z M -tuple as battery values, setting the drain according to the invariant d + T + m bm = 0, and obtaining the (nonincreasing) partition ( p1 , , p M ) The N0M -tuple then is ( p1 − p2 , p2 − p3 , , p M−1 − p M , p M ) Theorem 18.22 For ≤ M ≤ 6, for n ≤ 400 − 50M, we have that the function is injective for all BDM states with class up to n If the all-zero vectors of N0M and Z M map onto each other, only the parameters (T, t) = (0, t) and (1, 1) are permitted Proof By simulation over the mentioned ranges Michael Vielhaber 331 Conjecture 18.23 The preceeding theorem is valid for all M ∈ N, n ∈ N Furthermore, the bijections behave well in the following sense Let (b1 , , b M , d; 0, M + 1) be a state with corresponding set {Im |1 ≤ m ≤ M} of inhibitions counts Let ( I˜m ) be the ordered nonincreasing tuple, and pm = I˜m − I˜m−1 , ≤ m < M, p M = I˜M Then b = (b1 , , b M ) ∈ Z M and p = ( p1 , , p M ) ∈ N0M , and with the three standard norms M b = M |bm |, b = m=1 M 2, bm b ∞ = max |bm |, 2, pm p ∞ = max | pm |, m=1 m=1 and M p = M | pm |, p = m=1 m=1 M m=1 we have the following conjecture Conjecture 18.24 For all M ≥ ∈ N, for all b ∈ Z M (BDM state) and the corresponding p = p(b) ∈ N0M (VC-partition according to the algorithm): 2M b M + b M +1 b M + O(1) ≤ p ≤ 2M b + O(1) ≤ p ≤ (3M − 1.8) b ∞ + O(1) ≤ p ∞ ≤ 2M b + O(1) ∞ + O(1) + O(1) and all bounds are sharp with O(1) dependent only on M 18.5 Open questions and further research (q) (1) Derive a closed form for N4 (L , n), starting from Niederreiter and Wang’s recursions (2) Repeat this for all M ≥ (3) Give a proof for the parsimonious formula γ ( , d) from [29, Theorem 22] for all M, d, q, (4) Prove that the mapping from states to partitions is a bijection onto the set of all partitions into at most M parts 332 On the linear complexity of multisequences (5) Give an algorithm for this mapping (6) Prove that a state s with partition ( I˜1 , , I˜M ) acquires positive mass at the I˜1 th column, for all s ∈ S, all M, and give an algorithm for I˜1 using (b1 , , b M , d), not I˜1 (Question (5) would include this.) 18.6 Conclusion The field of modeling linear complexity, for sequences and multisequences, is both mature and active, with precise results for the expectation and variance in cases M up to 3, and two approaches for higher M: a conjectured closed form and a general recurrence relation, both employing partitions Our birthday celebrant Harald Niederreiter has been shaping this field from the beginning and continues to so, on starting his eighth decade, by recently giving us closed formulas for the numbers N2 (n, L) and N3 (n, L) of sequences with prescribed length and linear complexity for multisequences of width and Acknowledgements Harald Niederreiter has been my “academic father” ever since accepting me as an external doctoral student in 1994 Hence, for all my publications on stream ciphers and complexity, I would consider him as a “spiritual co-author” (of course without any responsibility for potential typos, something unheard of and unseen in his own papers; in this regard many thanks to the anonymous referee for pointing out not only typos, but numerous stylistic improvements of this paper) Mónica del Pilar Canales Chacón, my wife, co-investigator, and proofreader, has again managed to point out the weak spots in earlier versions Rainer Göttfert, whom I first met in 1993 at Fq2 in Las Vegas, has since then been a good friend in many ways and aspects I am really lucky to know all three of them References [1] M del P Canales Chacón and M Vielhaber, Structural and computational complexity of isometries and their shift commutators Electronic Colloq on Computational Complexity, ECCC, TR04–057, 2004 Michael Vielhaber 333 [2] Z Dai, X Feng and J Yang, Multi-continued fraction algorithm and generalized B–M algorithm over F2 In: T Helleseth, D Sarwate and H.-Y Song (eds.), Proc SETA 2004, Int Conf on Sequences and their Applications, October 24–28, 2004, Seoul, Korea Lecture Notes in Computer Science, volume 3486 Springer, Berlin, 2005 [3] J L Dornstetter, On the equivalence of Berlekamp’s and Euclid’s algorithms IEEE Trans Inf Theory 33(3), 428–431, 1987 [4] P Erd˝os and P Révész, On the length of the longest head-run Colloq Math Soc J Bolyai, Keszthely, 1975 Top Inf Theory 16, 219–228, 1975 [5] X Feng and Z Dai, Expected value of the linear complexity of twodimensional binary sequences In: T Helleseth, D Sarwate and H.-Y Song (eds.), SETA 2004, Int Conf on Sequences and their Applications, October 24–28, 2004, Seoul, South Korea Lecture Notes in Computer Science, volume 3486, pp 113–128 Springer, Berlin, 2005 [6] G L Feng and K K Tzeng, A generalized Euclidean algorithm for multisequence shift-register synthesis IEEE Trans Inf Theory 35, 584–594, 1989 [7] F G Gustavson, Analysis of the Berlekamp–Massey linear feedback shift-register synthesis algorithm IBM J Res Develop 20, 204–212, 1976 [8] N Moshchevitin and M Vielhaber, On an improvement of a result by Niederreiter and Wang concerning the expected linear complexity of multisequences, arXiv:math/0703655v2, 2007 [9] H Niederreiter, Sequences with almost perfect linear complexity profile Proc EUROCRYPT 1987 Lecture Notes in Computer Science, volume 304, pp 37–51 Springer, Berlin, 1988 [10] H Niederreiter, The probabilistic theory of linear complexity Proc EUROCRYPT 1988 Lecture Notes in Computer Science, volume 330, pp 191–209 Springer, Berlin, 1988 [11] H Niederreiter, Keystream sequences with a good linear complexity profile for every starting point Proc EUROCRYPT 1989 Lecture Notes in Computer Science, volume 434, pp 523–532 Springer, Berlin, 1990 [12] H Niederreiter, A combinatorial approach to probabilistic results on the linearcomplexity profile of random sequences J Cryptol 2, 105–112, 1990 [13] H Niederreiter, The linear complexity profile and the jump complexity of keystream sequences Proc EUROCRYPT 1990, Lecture Notes in Computer Science, volume 473, pp 174–188 Springer, Berlin, 1991 [14] H Niederreiter, Linear complexity and related complexity measures for sequences INDOCRYPT 2003 Lecture Notes in Computer Science, volume 2904, pp 1–17 Springer, Berlin, 2003 [15] H Niederreiter and M Vielhaber, Linear complexity profiles: hausdorff dimensions for almost perfect profiles and measures for general profiles J Complexity 13(3), 353–383, 1997 [16] H Niederreiter and M Vielhaber, Simultaneous shifted linear complexity profiles in quadratic time Appl Algebra Eng Commun Comput 9(2), 125–138, 1998 334 On the linear complexity of multisequences [17] H Niederreiter and M Vielhaber, An algorithm for shifted continued fraction expansions in parallel linear time Theor Comput Sci 226, 93–114, 1999 [18] H Niederreiter and L.-P Wang, Proof of a conjecture on the joint linear complexity profile of multisequences INDOCRYPT 2005 Lecture Notes in Computer Science, volume 3797, pp 13–22 Springer, Berlin, 2005 [19] H Niederreiter and L.-P Wang, The asymptotic behavior of the joint linear complexity profile of multisequences Monatsh Math 150, 141–155, 2007 [20] H Niederreiter, M Vielhaber and L.-P Wang, Improved results on the probabilistic theory of joint linear complexity of multisequences Sci China Inf Sci 55(1), 165–170, 2012 [21] F Piper, Stream ciphers Elektrotech Machinen 104, 564–568, 1987 [22] P Révész, Strong theorems on coin tossing Proceedings of the International Congress Mathematicians, Helsinki, pp 749–754, 1978 [23] P Révész, Random Walk in Random and Non-Random Environment World Scientific, Singapore, 1990 [24] R A Rueppel, Analysis and Design of Stream Ciphers Springer, Berlin, 1986 [25] M Vielhaber, Kettenbrüche, Komplexitätsmaße und die Zufälligkeit von Symbolfolgen Dissertation, University of Vienna, 1997 (PhD Advisor Harald Niederreiter) [26] M Vielhaber, A unified view on sequence complexity measures as isometries In: T Helleseth, D Sarwate and H.-Y Song (eds.), SETA 2004, Proc Int Conf on Sequences and their Applications, October 24–28, 2004, Seoul, Korea Lecture Notes in Computer Science, volume 3488 Springer, Berlin, 2005 [27] M Vielhaber, Continued fraction expansion as isometry – the law of the iterated logarithm for linear, jump, and 2-adic complexity IEEE Trans Inf Theory 53(11), 4383–4391, 2007 [28] M Vielhaber and M del P Canales, The Hausdorff dimension of the set of d-perfect M-multisequences SETA 2006, pp 259–270 Springer, Berlin, 2006 [29] M Vielhaber and M del P Canales, Towards a general theory of simultaneous Diophantine approximation of formal power series: linear complexity of multisequences, arXiv.org/abs/cs.IT/0607030, 2006 [30] M Vielhaber and M del P Canales, On a class of bijective binary transducers with finitary description despite infinite state set CIAA 2005 Lecture Notes Computer Science, volume 3854, pp 356–357 Springer, Berlin, 2006 [31] M Vielhaber and M del P Canales, Simultaneous Diophantine approximation of formal power series: the asymptotic distribution of the linear complexity of multisequences Project FONDECYT, 1040975 manuscript 8, 2006 [32] M Vielhaber and M del P Canales, The asymptotic normalized linear complexity of multisequences J Complexity 24(3), 410–422, 2008 [33] M Vielhaber and M del P Canales, The linear complexity deviation of multisequences: formulae for finite lengths and asymptotic distributions SETA 2012, Int Conf on Sequences and their Applications Lecture Notes in Computer Science, volume 7280, pp 168–180 Springer, Berlin, 2012 [34] M Z Wang and J L Massey, The characterization of all binary sequences with perfect linear complexity profile Presented at EUROCRYPT’86, Linköping, 1986 www.isiweb.ee.ethz.ch/archive/massey_pub/pdf/BI959.pdf Michael Vielhaber 335 [35] L.-P Wang and H Niederreiter, Enumeration results on the joint linear complexity of multisequences Finite Fields Appl 12, 613–637, 2006 [36] L.-P Wang, Y.-F Zhu and D.-F Pei, On the lattice basis reduction multisequence synthesis algorithm IEEE Trans Inf Theory 50, 2905–2910, 2004 30 25 0.8 20 0.6 15 0.4 10 0.2 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 40 0.8 30 0.6 20 0.4 10 0.2 0 0.2 0.4 0.6 0.8 Figure 3.2 Illustration of the acceptance-rejection algorithm Points below the curved line are accepted and then projected onto the x-axis The top row shows the acceptance-rejection algorithm using a deterministic point set PM,s+1 , whereas the bottom row shows the acceptance-rejection sampler using random samples PM,s+1 In both cases the number of points is M = 27 (See page 50.) Discrepancy 0.875 N−0.785 0.05 0.01 0.001 0.0001 100 101 102 103 104 105 Figure 3.3 Numerical result of the acceptance-rejection algorithm using low-discrepancy point sets The convergence rate is approximately of order N −0.8 , which is better than the rate one would expect when using random samples (which is N −0.5 ) (See page 51.) Figure 12.1 Convergence test using a more challenging test scene (top row) for light transport simulation Inside each of the two symmetric rooms one polyhedral light source illuminates the inner court through the slit between the blockers in the door frame The two images in the bottom row have been rendered using the same number of 218 samples per pixel, where the left image used the Sobol’ sequence, while the right image used the new rank-1 lattice construction to sample light transport paths The images cannot be distinguished with respect to quality, however, sampling using the rank-1 lattice sequence algorithm is simpler and much more efficient (See page 212.) Figure 12.2 A visual comparison of the new methodology (top) and the Sobol’ sequence (bottom) in NVIDIA iray® While the Sobol’ sequence exposes the typical structured artifacts in the form of rectangular patterns (for example on the open book), the rank-1 lattice sequence exposes more noise (for example in the highlight on the glass of the petrol lamp) Both images have been rendered using only 64 path space samples per pixel (See page 213.) Class K(s) 2 3 4 4 T,t [1,0] [2,0] [1,1] [3,0] [2,1] [4,0] [3,1] [2,2] [,] T,t Part [0,0] 2,3 ( 1, 1,0) ( 1,0, 1) (0, 1, 1) ( 2,0,0) ( 2, 1,1) (0, 2,0) ( 2,1, 1) ( 1, 2,1) (0,0, 2) (,,) 2,3 0,1 (0,0,0) (0,1, 1) (1,0, 1) ( 1,1,0) ( 1,0,1) (1, 1,0) ( 1,2, 1) (0, 1,1) (1,1, 2) (,,) 0,1 0,2 (0,0,0) ( 1,1,0) ( 1,0,1) (0,1, 1) (1,0, 1) (0, 1,1) ( 1,2, 1) (1, 1,0) ( 2,1,1) (,,) 0,2 0,3 (0,0,0) ( 1,0,1) ( 1,1,0) (0, 1,1) (1, 1,0) (0,1, 1) ( 1, 1,2) (1,0, 1) ( 2,1,1) (,,) 0,3 1,1 (0,0, 1) ( 1,0,0) ( 1,1, 1) (0, 1,0) (1, 1, 1) (0,1, 2) ( 1, 1,1) (1,0, 2) ( 2,1,0) (,,) 1,1 1,2 ( 1,0,0) (0,0, 1) ( 1,1, 1) (0, 1,0) ( 1, 1,1) ( 2,1,0) (1, 1, 1) ( 2,0,1) (0,1, 2) (,,) 1,2 1,3 ( 1,0,0) (0, 1,0) ( 1, 1,1) (0,0, 1) ( 1,1, 1) ( 2,0,1) (1, 1, 1) ( 2,1,0) (0, 2,1) (,,) 1,3 2,1 ( 1,0, 1) (0, 1, 1) ( 1, 1,0) (0,0, 2) ( 1,1, 2) ( 2,0,0) (1, 1, 2) ( 2,1, 1) (0, 2,0) (,,) 2,1 2,2 ( 1,0, 1) ( 1, 1,0) (0, 1, 1) ( 2,0,0) ( 2,1, 1) (0,0, 2) ( 2, 1,1) ( 1,1, 2) (0, 2,0) (,,) 2,2 2,3 ( 1, 1,0) ( 1,0, 1) (0, 1, 1) ( 2,0,0) ( 2, 1,1) (0, 2,0) ((,,) 2,1, 1) ( 1, 2,1) (0,0, 2) (,,) 2,3 0.7968 0.9437 0.8437 0.9657 0.8671 0.9730 0.8906 0.9803 0.9140 0.9876 1.0 1.0 Accumulated Mass up to this column for q2 q3 0.375 0.5925 0.5625 0.7901 0.6562 0.8559 0.75 0.9218 D I N ,N d ,b+ Legend: Figure 18.4 BDM states for M = with class up to (See page 323.) ... subjects such as algebraic number theory and algebraic geometry and even coding theory The method that Harald Niederreiter employed is class field theory in algebraic number theory He found many... pseudorandom number generation, quasi-Monte Carlo methods, cryptology, finite fields, applied algebra, algorithms, number theory and coding theory He has published more than 350 research papers and. .. Applied Algebra and Number Theory Essays in Honor of Harald Niederreiter on the occasion of his 70th birthday Harald Niederreiter’s pioneering research in the field of applied algebra and number