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Example of Pixel-Based Visual Cryptography in Detail Example of Pixel-based Visual Cryptography in Detail All that is not perfect down to the smallest detail is doomed to perish. Gustav Mahler 266 Example of Pixel-based Visual Cryptography Figure 1: Example: Pixel-based Visual Cryptography, Original Picture S. Pape, Authentication in Insecure Environments, DOI 10.1007/978-3-658-07116-5, © Springer Fachmedien Wiesbaden 2014 Example of Pixel-based Visual Cryptography 267 Figure 2: Example: Pixel-based Visual Cryptography, Transparency 1 268 Example of Pixel-based Visual Cryptography Figure 3: Example: Pixel-based Visual Cryptography, Transparency 2 Example of Pixel-based Visual Cryptography 269 Figure 4: Example: Pixel-based Visual Cryptography, Overlay Auxiliary Tables and Proofs Consistency is found in that work whose whole and detail are suitable to the occasion. It arises from circumstance, custom, and nature. Marcus Vitruvius Pollio 272 Auxiliary Tables and Proofs Lemmas ( )= − 2 n Lemma 11.1. The derivative of f n 1 n+1 is positive for all n>1. v v Proof. By applying the chain rule for df f(n)=du du + du dv dn du dn dv dn v v with u = 1 − 2 , v = n, du = vuv−1, and du = log(u)uv it follows: n+1 du dv − df 2 n 1 df 2 f(n)=n 1 − 1 − dn n + 1 dn n + 1 2 n 2 df + 1 − log 1 − (n) n + 1 n + 1 dn − 2 n 1 2 2 2 = 1 − n + 1 − log 1 − n + 1 (n + 1)2 n + 1 n + 1 2 n 2n n − 1 = 1 − + log n + 1 n2 − 1 n + 1 n Since for all n>1it holds that 1 − 2 >0it remains to show that T(n)= n+1 2n + n−1 >0 n = 2 T(2)= n2−1 log n+1 . We start with an evaluation for and get 4 − ( ) ( ) 3 log 3 >0. Next, we regard the derivative of T n : dT dT 2n dT n − 1 T(n)= + log dn dn n2 − 1 dn n + 1 − ( 2 − ) − ( ) (n + 1) dT n 1 = n 1 2 2n 2n + dn n+1 (n2 − 1)2 n − 1 − 2 − ( + )(( + ) −( − ) ) = 2n 2 + n 1 n 1 1 n 1 1 (n2 − 1)2 (n − 1)(n + 1)2 − 2 − ( 2 − ) = 2n 2 + 2 n 1 (n2 − 1)2 (n2 − 1)2 − = 4 (n2 − 1)2 S. Pape, Authentication in Insecure Environments, DOI 10.1007/978-3-658-07116-5, © Springer Fachmedien Wiesbaden 2014 Auxiliary Tables and Proofs 273 This means that T(n) is stricly monotonically decreasing. Next, we consider the second derivative: d2T dT −4 T(n)= dn2 dn (n2 − 1)2 4 dT n4 − 2n2 + 1 = dn (n2 − 1)4 3 − = 16n 16n (n2 − 1)4 = 16n (n2 − 1)3 Since the second derivative is non-negative, T(n) is strictly convex. Furthermore, since the the limit of both summands as n approaches infinity is zero, the limit of T(n) as n approaches infinity is also zero: 2n n − 1 lim + log = 0 n→∞ n2 − 1 n + 1 This means that T(n) decreases slower and slower, but is always positive until − 2 n n approaches infinity. Hence, the derivative of 1 n+1 is also positive and n therefore we showed that df 1 − 2 >0for all n>1. dn n+1 274 Auxiliary Tables and Proofs Tables Table 1: Possible Pairs of Dice Codings with n = 9 for a Given Number of Different Dots difference possible pairs 0 (0,0); (1,1); (2,2); (3,3); (4,4); (5,5); (6,6); (7,7); (8,8); (9,9) 1 (0,1); (1,0); (1,2); (2,1); (2,3); (3,2); (3,4); (4,3); (4,5); (5,4); (5,6); (6,5); (6,7); (7,6); (7,8); (8,7); (8,9); (9,8) 2 (0,2); (1,1); (1,3); (2,0); (2,2); (2,4); (3,1); (3,3); (3,5); (4,2); (4,4); (4,6); (5,3); (5,5); (5,7); (6,4); (6,6); (6,8); (7,5); (7,7); (7,9); (8,6); (8,8); (9,7) 3 (0,3); (1,2); (1,4); (2,1); (2,3); (2,5); (3,0); (3,2); (3,4); (3,6); (4,1); (4,3); (4,5); (4,7); (5,2); (5,4); (5,6); (5,8); (6,3); (6,5); (6,7); (6,9); (7,4); (7,6); (7,8); (8,5); (8,7); (9,6) 4 (0,4); (1,3); (1,5); (2,2); (2,4); (2,6); (3,1); (3,3); (3,5); (3,7); (4,0); (4,2); (4,4); (4,6); (4,8); (5,1); (5,3); (5,5); (5,7); (5,9); (6,2); (6,4); (6,6); (6,8); (7,3); (7,5); (7,7); (8,4); (8,6); (9,5) 5 (0,5); (1,4); (1,6); (2,3); (2,5); (2,7); (3,2); (3,4); (3,6); (3,8); (4,1); (4,3); (4,5); (4,7); (4,9); (5,0); (5,2); (5,4); (5,6); (5,8); (6,1); (6,3); (6,5); (6,7); (7,2); (7,4); (7,6); (8,3); (8,5); (9,4) 6 (0,6); (1,5); (1,7); (2,4); (2,6); (2,8); (3,3); (3,5); (3,7); (3,9); (4,2); (4,4); (4,6); (4,8); (5,1); (5,3); (5,5); (5,7); (6,0); (6,2); (6,4); (6,6); (7,1); (7,3); (7,5); (8,2); (8,4); (9,3) 7 (0,7); (1,6); (1,8); (2,5); (2,7); (2,9); (3,4); (3,6); (3,8); (4,3); (4,5); (4,7); (5,2); (5,4); (5,6); (6,1); (6,3); (6,5); (7,0); (7,2); (7,4); (8,1); (8,3); (9,2) 8 (0,8); (1,7); (1,9); (2,6); (2,8); (3,5); (3,7); (4,4); (4,6); (5,3); (5,5); (6,2); (6,4); (7,1); (7,3); (8,0); (8,2); (9,1) 9 (0,9); (1,8); (2,7); (3,6); (4,5); (5,4); (6,3); (7,2); (8,1); (9,0) Auxiliary Tables and Proofs 275 Table 2: Upper Bounds of Probabilities (in %) that k Matching Ciphertexts Occur Given ∗ N1∗ Ciphertexts of 1 and n = 9 (cf. Term 5.8) ⎧ ⎪0, if N ∗ <k ⎪ 1 ⎨⎪ 1, if N1∗ > (k − 1)n P (∃c.#c = k|N1∗ ) = n! 1 ⎪1 − N , if k N ∗ (k − 1)n,k = 2 ⎪ (n−N1∗ )! 1 ⎪ n1∗ ⎪ N1∗ −1 ⎩ N1∗ N1∗ −k ≈ 1 ( − ) ∗ ( − ) k n n 1 , if k N1 k 1 n,k > 2 k 23456 N1∗ 2 11.1 0 0.0 0.0 0.0 3 30.9 1.2 0.0 0.0 0.0 4 53.9 4.4 0.1 0.0 0.0 5 74.4 9.8 0.6 1.5 · 10−2 0.0 6 88.6 17.3 1.6 8.1 · 10−2 1.7 · 10−3 7 96.2 27.0 3.4 0.3 1.1 · 10−2 8 99.2 38.4 6.0 0.6 3.7 · 10−2 9 99.9 51.2 9.6 1.2 0.1 10 100.0 65.0 14.2 2.1 0.2 k 78 910 N1∗ 2 0.0 0.0 0.0 0.0 3 0.0 0.0 0.0 0.0 4 0.0 0.0 0.0 0.0 5 0.0 0.0 0.0 0.0 6 0.0 0.0 0.0 0.0 7 1.9·10−4 0.0 0.0 0.0 8 1.3·10−3 2.1 · 10−5 0.0 0.0 9 5.4·10−3 1.6 · 10−4 2.3 · 10−6 0.0 10 1.6·10−2 7.4 · 10−4 2.1 · 10−5 2.6 · 10−7 276 Auxiliary Tables and Proofs ∗ Table 3: Probabilities (in %) that Ni∗ Ciphertexts of i out of N Ciphertexts Occur with n = 9 if i Follows a Uniform Distribution − N 2 Ni∗ 2 N Ni∗ P (#i = Ni∗ |N) = + 1 − + Ni∗ n 1 n 1 N ∗ i 012345 N 1 80.0 20.0 0.0 0.0 0.0 0.0 2 64.0 32.0 4.0 0.0 0.0 0.0 3 51.2 38.4 9.6 0.8 0.0 0.0 4 41.0 41.0 15.3 2.6 0.2 0.0 5 32.8 41.0 20.5 5.1 0.6 3.2·10−2 6 26.2 39.3 24.5 8.2 1.5 0.2 7 21.0 36.7 27.5 11.5 2.9 0.4 8 16.8 33.6 29.4 14.7 4.6 0.9 9 13.4 30.2 30.2 17.6 6.6 1.6 10 10.7 26.8 30.2 20.1 8.8 2.6 11 8.6 23.6 29.5 22.2 11.1 3.9 12 6.9 20.6 28.4 23.6 13.3 5.3 13 5.5 17.9 26.8 24.6 15.4 6.9 14 4.3 15.4 25.0 25.0 17.2 8.6 15 3.5 13.2 23.1 25.0 18.8 10.3 16 2.8 11.2 21.1 24.6 20.0 12.0 17 2.3 9.6 19.1 23.9 20.9 13.6 18 1.8 8.1 17.2 23.0 21.5 15.1 19 1.4 6.8 15.4 21.8 21.8 16.4 20 1.2 5.8 13.7 20.5 21.8 17.5 N ∗ i 678910 N 1 to 5 0.0 0.0 0.0 0.0 0.0 6 6.4·10−3 0.0 0.0 0.0 0.0 7 3.5·10−2 1.2·10−3 0.0 0.0 0.0 8 0.1 8.2·10−3 2.6·10−4 0.0 0.0 9 0.3 2.9·10−2 1.8·10−3 5.1·10−5 0.0 10 0.6 7.8·10−2 7.3·10−3 4.1·10−4 1.0·10−5 11 1.0 0.2 2.1·10−2 1.8·10−3 9.0·10−5 12 1.6 0.3 5.2·10−2 5.8·10−3 4.3·10−4 13 2.3 0.6 0.1 1.5·10−2 1.5·10−3 14 3.2 0.9 0.2 3.3·10−2 4.2·10−3 15 4.3 1.4 0.3 6.7·10−2 1.0·10−2 16 5.5 2.0 0.6 0.1 2.1·10−2 17 6.8 2.7 0.8 0.2 4.2·10−2 18 8.2 3.5 1.2 0.3 7.5·10−2 19 9.5 4.4 1.7 0.5 0.1 20 10.9 5.5 2.2 0.7 0.2 Auxiliary Tables and Proofs 277 ∗ Table 4: Probabilities (in %) that N1∗ Ciphertexts of 1 out of N Ciphertexts Occur with N0∗ = 2 and n = 9 if i Follows a Uniform Distribution (cf.
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