Modular multiplication

The basic operation in most of the following schemes is modular multiplication. (Elliptic curve arithmetic, on which some of the schemes could be based instead, is not described here see, e.g., Kobl87].) It is assumed that is represented by the numbers 0,. .., n-1. A modular multiplication can be computed with one multiplication and one division. A multiplication of /-bit numbers on a machine with word length w can easily be performed with fZ/vvl word multiplications. Division takes a small number of word multiplications more. 

Modular inverses can be computed efficiently with the extended Euclidean algorithm. However, this is somewhat slower than modular multiplication and asymptotically 0(P). 

The Chinese remainder algorithm is very efficient after a little precomputation that can be done once and for all in cryptologic situations for two congruences modulo numbers of length 112 it needs one modular and one non-modular multiplication of numbers of length HI. 

Thus the main operation is the computation of y ° ywhich is basically a modular multiplication, and therefore 0(lnl2 ). 

Main key generation Generation of 4 random numbers and two pair exponentiations, which require less than 4k modular multiplications of numbers of length 1. 

Verifying proofs of forgery Either one pair exponentiation to compute gX-x g y-y (jgss tjjgjj 2k modular multiplications of numbers of length /) or, with Remark 9.13, one exponentiation. ... 

Mont85 Peter L. Montgomery Modular Multiplication without trial division Mathematics of Computation 44 (1985) 519-512. 

Modularity problems the instrument includes many different boards which are very well made for standard applications, but it is sometimes diffieult to set up a new type of examination. For example use of multiple probe, or multi-elements probes. 

Multiple applications for resilin-like polypeptides have garnered renewed research interest since the report of the first recombinant resilin in 2005. The excellent mechanical properties of the resilin-like polypeptides has directed investigation toward their use as high-performance materials and in tissue engineering applications. It is widely acknowledged that cells interact and take cues from their microenvironment and, therefore, the development of polymeric scaffolds to mimic the extracellular matrix and drive desired cell or tissue responses has been of wide interest. To this end, our laboratories have developed a modular resilin-like polypeptide (RLP12) (see Fig. 20) that contains not only twelve repeats of the... 

The most complex automated systems are used almost exclusively by centralized HTS operations in large pharmaceutical companies and are referred to as ultra HTS (uHTS) platforms. They typically consist of the same four functional instruments, but have the capacity to process several hundred plates per extended workday. They often incorporate a modular design philosophy with multiple duplicate instruments for enhanced capacity that offer some functional redundancy. The mechanism for moving plates from one instrument module to another is often, but not always, a continuous track-way that resembles an industrial assembly line rather than the robotic arm typically used in a workcell system [5-8],... 

Huynh, C.Q. and Zieler, H. (1999) Construction of modular and versatile plasmid vectors for the high-level expression of single or multiple genes in insects and insect cell lines. Journal of Molecular Biology, 288 (1), 13—20. 

Yoon, Y.J., Beck, B.J., Kim, B.S. et al. (2002) Generation of multiple bioactive macrolides by hybrid modular polyketide synthases in Streptomyces venezuelae. Chemistry Biology, 9, 203-214. 

The older modular simulation mode, on the other hand, is more common in commerical applications. Here process equations are organized within their particular unit operation. Solution methods that apply to a particular unit operation solve the unit model and pass the resulting stream information to the next unit. Thus, the unit operation represents a procedure or module in the overall flowsheet calculation. These calculations continue from unit to unit, with recycle streams in the process updated and converged with new unit information. Consequently, the flow of information in the simulation systems is often analogous to the flow of material in the actual process. Unlike equation-oriented simulators, modular simulators solve smaller sets of equations, and the solution procedure can be tailored for the particular unit operation. However, because the equations are embedded within procedures, it becomes difficult to provide problem specifications where the information flow does not parallel that of the flowsheet. The earliest modular simulators (the sequential modular type) accommodated these specifications, as well as complex recycle loops, through inefficient iterative procedures. The more recent simultaneous modular simulators now have efficient convergence capabilities for handling multiple recycles and nonconventional problem specifications in a coordinated manner. 

Modular design. Synthetic molecules were constructed of multiple, closely related nonsymmetric repeat units that can be covalently attached from either end. This approach allows rational design of various congeners, controlling their physical properties, such as iron-binding capacities and partition coefficients (lipophilic/hydrophilic... 

---