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Orson Hardwick
We report a metamaterial which simultaneously possesses a negative bulk modulus and mass density. This metamaterial is a zinc blende structure consisting of one fcc array of bubble-contained-water spheres (BWSs) and another relatively shifted fcc array of rubber-coated-gold spheres (RGSs) in epoxy matrix. The negative bulk modulus and mass density are simultaneously derived from the coexistent monopolar resonances from the embedded BWSs and dipolar resonances from the embedded RGSs. The Poisson ratio of the metamaterial also turns negative near the resonance frequency.
In this case I have replaced three mods with three subtractions and three multiplications. Thus although I have replaced a single operator (%) with two operations (- *) I still expect an increase in speed because the modulus operator is actually three operators in one (- * /). Thus effectively I have eliminated three divisions and so I expected a significant improvement in speed. The results however were a little surprising:
Not only did replacing the modulus operator with alternative operations result in faster code, it also opened up the possibility for further optimizations. As a result with the AVR & MSP430 I was able to more than halve the execution time.
A similar problem (with a similar solution) occurs when one wants to display integers on a display. For example if you are using a custom LCD panel with say a 3 digit numeric field, then the problem arises as to how to determine the value of each digit. The obvious way, using the modulus operator is as follows:
The problem is that C provides to access to the underlying instructions. All hardware integer dividers produce the modulus as a side effect of doing the division. This is why most instruction sets have an instruction that returns both div and mod. But this efficiency is not captured in most languages operators, including C, perhaps because built in language operators generally return an atom of data not a dual. However, you can get it through a function call, div() or ldiv() which are in stdlib.h. ANd that is the answer! Not this crazy long essay.
modulus sometimes come for free if the two operators (% and /) are in the same block in the source and use the same operands, . I looked at some code generated by GCC and it appears that the divide routine produces the modulus simultaneously.
I think this is only true for some 32 bit assemblers. For example most (all?) 8 /16 bit architectures lack a divide instruction. However your point that the remainder (modulus) inherently falls out of the division operator is valid. An earlier commenter had implied this in mentioning the standard div() function.
No offense taken. This is a blog about embedded systems and the modulus operator crops up a lot. To give you examples from my work unfortunately requires you to know a fair amount about embedded systems. Anyone out there have some simple real world examples for Lucas?
For those who are more curious, this article examines the fastest way to get the remainder between using modulus or an alternative with addition in C#, so there would definitely be a C alternative:
-sharp/use-the-modulus-operator-or-alternative
I'm guessing the problem could be small subgroup attacks. But this can be as much of a problem as when the modulus is prime. Or better: a non-prime group could have a big prime subgroup while a prime group could have small prime subgroups.
Here the modulus $p$ is prime, so we can naturally compute the number of public keys (elements) in our group: $p-1$. By factoring this number you can also get the possible subgroups. If you have enough small subgroups $p_i$ then you can use the Chinese Remainder Theorem to stitch together the many partial private keys you found into the real private key.
Here the prime $n$ is not so much a prime anymore. We instead use a RSA modulus $n = p \times q$. Since $n$ is not a prime anymore, to compute the number of possible public keys in our new DH group, we need to compute $(p-1)(q-1)$ (the number of elements co-prime to $n$). This is a bit tricky and only us, with the knowledge of $p$ and $q$ should be able to compute that.This way, under the assumptions of RSA, we know that no-one will be able to factor the number of elements ($(p-1)(q-1)$) to find out what subgroups there are. And now our small subgroups are well hidden for us, and only us, to perform Pohlig-Hellman.
We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.
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