No abstract

Visual cryptography and (k; n)-visual secret sharing schemes were introduced by Naor and Shamir (Advances in Cryptology -Eurocrypt 94, Springer, Berlin, 1995, pp. 1-12). A sender wishing to transmit a secret message distributes n transparencies amongst n recipients, where the transparencies contain seemingly random pictures. A (k; n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k − 1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message. The important parameters of a scheme are its contrast, i.e., the clarity with which the message becomes visible, and the number of subpixels needed to encode one pixel of the original picture. Naor and Shamir constructed (k; k)-schemes with contrast 2 −(k−1) . By an intricate result from Linial (Combinatorica 10 (1990) 349 -365), they were also able to prove the optimality of these schemes. They also proved that for all ÿxed k6n, there are (k; n)-schemes with contrast (2e) −k = √ 2 k. For k = 2; 3; 4 the contrast is approximately 1 105 ; 1 698 and 1 4380 . In this paper, we show that by solving a simple linear program, one is able to compute exactly the best contrast achievable in any (k; n)-scheme. The solution of the linear program also provides a representation of a corresponding scheme. For small k as well as for k = n, we are able to analytically solve the linear program. For k = 2; 3; 4, we obtain that the optimal contrast is at least 1 4 ; 1 16 and 1 64 . For k = n, we obtain a very simple proof of the optimality of Naor's and Shamir's (k; k)-schemes. In the case k = 2, we are able to use a di erent approach via coding theory which allows us to prove an optimal tradeo between the contrast and the number of subpixels.

Abstruct-An artificial neural network (ANN) is commonly modeled by a threshold circuit, a network of interconnected processing units called linear threshold gates. The depth of a circuit represents the number of unit delays or the time for parallel computation. The size of a circuit is the number of gates and measures the amount of hardware. It was known that traditional logic circuits consisting of only unbounded fanin AND, OR, NOT gates would require at least R(log nllog log n) depth to compute common arithmetic functions such as the product or the quotient of two n-bit numbers, if the circuit size is polynomially bounded (in n). It is shown that ANN'S can be much more powerful than traditional logic circuits, assuming that each threshold gate can be built with a cost that is comparable to that of AND/OR logic gates. In particular, the main results show that powering and division can be computed by polynomial-size ANN'S of depth 4, and multiple product can be computed by polynomial-size ANN'S of depth 5. Moreover, using the techniques developed here, a previous result can be improved by showing that the sorting of n n-bit numbers can be carried out in a depth-3 polynomial size ANN. Furthermore, it is shown that the sorting network is optimal in depth.

Visual cryptography and (k; n)-visual secret sharing schemes were introduced by Naor and Shamir (Advances in Cryptology -Eurocrypt 94, Springer, Berlin, 1995, pp. 1-12). A sender wishing to transmit a secret message distributes n transparencies amongst n recipients, where the transparencies contain seemingly random pictures. A (k; n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k − 1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message. The important parameters of a scheme are its contrast, i.e., the clarity with which the message becomes visible, and the number of subpixels needed to encode one pixel of the original picture. Naor and Shamir constructed (k; k)-schemes with contrast 2 −(k−1) . By an intricate result from Linial (Combinatorica 10 (1990) 349 -365), they were also able to prove the optimality of these schemes. They also proved that for all ÿxed k6n, there are (k; n)-schemes with contrast (2e) −k = √ 2 k. For k = 2; 3; 4 the contrast is approximately 1 105 ; 1 698 and 1 4380 . In this paper, we show that by solving a simple linear program, one is able to compute exactly the best contrast achievable in any (k; n)-scheme. The solution of the linear program also provides a representation of a corresponding scheme. For small k as well as for k = n, we are able to analytically solve the linear program. For k = 2; 3; 4, we obtain that the optimal contrast is at least 1 4 ; 1 16 and 1 64 . For k = n, we obtain a very simple proof of the optimality of Naor's and Shamir's (k; k)-schemes. In the case k = 2, we are able to use a di erent approach via coding theory which allows us to prove an optimal tradeo between the contrast and the number of subpixels.

Monoterpenoids, such as the plant metabolite geraniol, are of high industrial relevance since they are important fragrance materials for perfumes, cosmetics, and household products. Chemical synthesis or extraction from plant material for industry purposes are complex, environmentally harmful or expensive and depend on seasonal variations. Heterologous microbial production offers a cost-efficient and sustainable alternative but suffers from low metabolic flux of the precursors and toxicity of the monoterpenoid to the cells. In this study, we evaluated two approaches to counteract both issues by compartmentalizing the biosynthetic enzymes for geraniol to the peroxisomes of Saccharomyces cerevisiae as production sites and by improving the geraniol tolerance of the yeast cells. The combination of both approaches led to an 80% increase in the geraniol titers. In the future, the inclusion of product tolerance and peroxisomal compartmentalization into the general chassis engineering toolbox for monoterpenoids or other host-damaging, industrially relevant metabolites may lead to an efficient, low-cost, and eco-friendly microbial production for industrial purposes.