tag:blogger.com,1999:blog-69594678405463718052018-03-05T22:12:18.311-08:00Gish's RepositoryA blog of varied interests, from schooling to work, to gaming and other hobbiesChris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.comBlogger280125tag:blogger.com,1999:blog-6959467840546371805.post-5113417356788315272015-02-21T16:45:00.001-08:002015-02-21T16:45:34.826-08:00Sample Plots<div id="cp_widget_5d8b9265-b1d2-488e-b460-fb7dfb3e692b"><a href="https://www.blogger.com/blogger.g?blogID=6959467840546371805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="453" src="picture2.jpg" style="cursor: move;" width="604" /></a>...</div><script type="text/javascript">var cpo = []; cpo["_object"] ="cp_widget_5d8b9265-b1d2-488e-b460-fb7dfb3e692b"; cpo["_fid"] = "AICAnV8ePpMP"; var _cpmp = _cpmp || []; _cpmp.push(cpo); (function() { var cp = document.createElement("script"); cp.type = "text/javascript"; cp.async = true; cp.src = "//www.cincopa.com/media-platform/runtime/libasync.js"; var c = document.getElementsByTagName("script")[0]; c.parentNode.insertBefore(cp, c); })(); </script><noscript>Powered by Cincopa <a href='http://www.cincopa.com/video-hosting'>Video Hosting for Business</a> solution.</noscript>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-11740287842122178502014-02-05T16:16:00.001-08:002014-02-05T22:31:11.886-08:00Bill Nye vs Ken Ham Debate ResponseI’ve recently seen and heard a lot about the creationist vs evolutionist debate that Bill Nye the science guy had yesterday and wanted to address some points.<br /><br />Disclaimer: I’m not arguing for either side I just want to make sure everyone is talking the same language.<br /><br />The main issue here is the difference in both sides understanding of probability and large numbers. I will now try to explain situations purely using mathematics, and people can draw their own conclusions.<br /><br />Issue 1: Why the law of thermodynamics is a law. <br />This law is obeyed because of how many particles are associated with a given system. Let us talk about a 1 meter cubed box of air, or a 3 foot by 3 foot box of air. Within it we have about 2.1*10^22 particles. If we want to impose order on this system spontaneously, lets say stuffing all the particles into one corner of the box for just 1 second it would be like getting everyone in the world which is only about 6 billion or 6*10^12 people to decide to do nothing but walk in single file for the next 300 years. That’s why this law is obeyed. When a scientist is talking about chances this is the scale of what he’s talking about. Things will most likely happen as predicted unless random occurrences such as what is described above spontaneously happen, which of course is possible, just unlikely.<br /><br />Issue 2: Why the law of thermodynamics is violated. <br />This law is violated all the time because of how similar one situation is to another. Although this box of air behaves the same way most of the time, that odd thing could occur right? It really does happen because of how many boxes of air there are in the world. There are about 5*10^22 boxes of air on the earth from the surface to the atmosphere. This means that every second that passes the concept or situation of the box lives for 1.58*10^14 years. The scientific lifespan of the universe is only 10^10 years which means that every second that passes the idea of the box lives for about 1,000 lifetimes of the universe. In that time you do get lucky enough to see the box of air suddenly compress itself. Therefore the law does get violated all the time. Order can spontaneously happen, and it does happen. That’s the scientist’s explanation for evolution. When they are talking about the “chance” for life to form, the “concept” the situation for that little pond of organic goop existed for 10’s of thousands of lifetimes of the universe for each second that passed until it happened. Not saying that its true, its still a theory, but the waiting period before the first life form on earth was created is a lot longer than most people think, and in that time who knows what could happen?<br /><br />Issue 3: Why the law is still a law even though it is constantly being violated.<br />The second law of thermodynamics is a law of the universe. Out of those 5*10^22 boxes of air, most likely only a few boxes are doing that violation. Not only that those few boxes are not necessarily occurring at the same moment, so in general most of the time the boxes are obeying the second law. Because of that, the entire universe obeys the second law despite localized violations.<br /><br />Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-70976924555391714462014-01-18T00:03:00.001-08:002014-01-18T00:03:52.367-08:00How to Fix No Signal to MonitorSometimes when using an old computer the monitor won't connect to the tower. To resolve this follow these simple steps.<div><ol><li>Check the Power Cable and make sure its firmly connected</li><li>Check the Monitor Cable and make sure its firmly connected</li></ol><div>If the above is true then it is quite possible that the ram might need to be reset in order to do this:</div></div><div><ol><li>Remove the side cover of the tower</li><li>Find the ram which will look something like what is shown below</li><li>Gently push down the two clips holding the ram</li><li>Remove the ram and clean it with a dry cloth</li><li>Replace the ram in its initial position by pushing down firmly, the clips will snap the ram in place if successful</li></ol><div style="text-align: center;"><img height="200" src="http://build-gaming-computer-guide.com/image-files/ddr3-vs-ddr2-ram.jpg" width="200" /></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-35786292419478478712014-01-13T14:02:00.004-08:002014-01-13T14:02:49.480-08:00KP FundamentalsOverview of Kelvin Probe Method<br /><ul><li>Background</li><ul><li>Kelvin method invented by Lord Kelvin over a century ago</li></ul><li>Work Function - energy required to remove an electron from material </li><ul><li>indicates surface conditions such as</li><ul><li>absorbed, evaporated layers, surface reconstruction, surface charging, oxide layer imperfections, surface and bulk recombinations</li></ul></ul><li>Use a reference surface to study a sample electrode</li><li>the two are conductors that form a parallel plate capacitor at the tip</li><li>Capacitance - C = ε A/D</li><ul><li>ε is the permittivity of the material in between plates, A is area connected as capacitor, D is distance between two conductors</li></ul></ul><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li>C = Q/V</li><ul><li>C is capacitance, Q is charge, v is voltage</li></ul><li>Q = V ε A / D</li><ul><li>As long as it is possible to determine voltage difference charge can be calculated, ε = permittivity of free-space if instrument is in vacuum</li><li>Null method - apply external voltage V1 to probe. If V1 is equal to Voltage on probe then V is 0, so no current is flowing to or from the probe</li></ul></ul><div><br /></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li>Current Nullification method</li><ul><li>Probe is vibrated in sinusoid and is perpendicular to the surface</li><li>Distance between plates D = D0 + D1sin(ωt)</li><li>D1 is the amplitude of oscillation</li></ul><li>Find new capacitance</li><ul><li>C = ε A/[D0 + D1sin(ωt)]</li></ul><li>Find Current generated</li><ul><li>I = V dC/dt</li><li>I = V d (ε A/[D0 + D1sin(ωt)]) /dt</li><li>I = V ε A [D1ωsin(ωt)]/[D0 + D1sin(ωt)]2</li><li>Then we nullify this current by bringing Voltage to 0</li></ul></ul><div><br /></div><ul style="margin-bottom: 0pt; margin-top: 0pt;"><li>This is the solution to noise difficulties that come from null field method</li><li>Distance D is maintained due to parallel movement of probe</li><ul><li>Capacitance remains constant as permittivity and area doesn’t change</li><li>Therefore we are looking for the variations in surface potential/charge</li><li>Current I is generated I = (dV/dt) C</li></ul><li>Drawbacks</li><ul><li>This results in an averaged potential measurement</li><li>Given some potential function, the instrument will measure the enclosed charge averaged over the size of the measuring tip</li></ul><li>Resultant Error</li><ul><li>If diameter of measured potential > diameter of tip error results = 2 times the size of the tip</li><li>If diameter of measured potential = diameter of tip error results = 2 times the size of the measured potential</li><li>If diameter of measured potential < diameter of tip error results = 2 times the size of the measured potential</li></ul><li>To get accurate measurements the probe must be both close to the surface and small in comparison to measured charge to minimize edge effects</li></ul><div><br /></div><div><span id="docs-internal-guid-7593a0f4-8d9f-f9ee-fe45-6f2b87015a19"></span></div><div><span id="docs-internal-guid-7593a0ee-8d9d-7ac3-974f-f9649e933ce3"></span></div><div><a href="http://ctluk.blogspot.com/2014/01/kelvin-probe-research-toc.html">Back to ToC</a></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-39706993135718470142014-01-13T12:26:00.001-08:002014-01-13T14:03:19.512-08:00Kelvin Probe Research ToC<h2 style="text-align: center;">Table of Contents</h2><div><ol><li><a href="http://ctluk.blogspot.com/2014/01/kp-fundamentals.html">KP Fundamentals</a></li></ol></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-7299627088685721492013-12-12T15:31:00.003-08:002013-12-12T15:31:59.048-08:00Battery History<span id="docs-internal-guid-2b03776e-e922-b9a3-edc8-83b69083a859"></span><br /><div>The following are notes on battery history</div><ul><li>History of Batteries</li><ul><li>2000 years ago Baghdad battery</li><ul><li>created in Mesopotamia </li><li>Jar of ceramics created with a speculated structure of copper with walls of iron</li><li>if filled with unknown composition of electrolytes meets the requirements for a battery</li><li>Maybe used to plate gold onto silver?</li></ul><li>200 Years Ago Animal Electricity</li><ul><li>Galvani while doing a dissection a steel scalpel touched a brass hook that was holding a frog leg in place</li><li>Battery composition</li><ul><li>Steel scalpel has iron </li><li>brass hook has copper</li><li>preservatives acted as electrolytes</li></ul></ul></ul><li>First Battery</li><ul><li>Voltaic Pile </li><li>set of individual galvanic cells placed in series</li><li>copper and zinc stacked together with electrolytes sandwiching the two metals</li></ul><li>History of electrochemical energy storage</li><ul><li>Galvani</li><li>Volta - Voltaic Pile </li><li>Daniell - Zn/Cu Cell</li><li>Faraday - Basic Principles of electrochemistry</li><li>Grove - H2/02 cell</li><li>Plante - Lead Acid Cell</li><li>Leclanche - Zn Carbon dry cell</li><li>Edison Zn/Ni and Fe/Ni cells</li></ul><li>Modern Batteries</li><ul><li>modern development started around half a century ago</li><li>Solid Electrolytes</li><ul><li>1966 Yao and Kummer Beta Alumina Solid Electrolyte: Fast Na Ion Conductor</li><li>birthed solid state ionics</li><li>Rubidium silver iodide</li></ul><li>Internal Phenomena</li><ul><li>Whittingham steele insertion reaction electrodes</li><li>created Le/TiS2 cells</li><li>Sony commercialized the first Lithium ion batteries</li></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/11/electrical-engineering-toc.html">Back to ToC</a></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-19874025331325244042013-12-06T14:34:00.002-08:002013-12-06T14:34:31.432-08:00Notes - The ElGamal Public Key CryptosystemThe following are notes from Introduction to Cyrptography with Coding Theory.<br /><div><ul><li>ElGamal Public Key Cryptosystem</li><ul><li>Alice wants to send a message m to Bob</li><li>Bob chooses a large prime p and a primitive root α</li><li>Bob chooses a secret integer a and computes B ≡ α <sup>a </sup>(mod p)</li><ul><li>Make public (p, α, B) and is Bob's public key</li></ul><li>Alice</li><ul><li>downloads (p, α, B)</li><li>Chooses a secret random integer k and computes r ≡ α <sup>k </sup>(mod p)</li><li>Computes t ≡ B <sup>k</sup> m (mod p)</li><li>sends the pair (r,t) to B</li></ul><li>Bob decrypts by computing tr <sup>-a</sup> ≡ m (mod p)</li></ul><li>Security</li><ul><li>depends on a being kept secret which is reliant on the idea that discrete logs are difficult to compute</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-82015138839628450302013-12-06T13:53:00.000-08:002013-12-06T13:53:02.258-08:00Notes - Discrete LogarithmsThe following are notes from Introduction to Cyrptography with Coding Theory.<br /><br /><ul><li>RSA Algorithm relies on the difficulty of factoring, but we can also use the difficulty of discrete logs</li><li>Fix a prime p. Let α and Β be nonzero integers mod p and suppose</li><ul><li>Β ≡ α<sup>x</sup> (mod p)</li></ul><li>It is difficult to find x assuming we choose a large enough prime</li><li>α is taken to be a primitive root mod p so that every Β is a power of α (mod p)</li><li>The size of the largest primes for which discrete logs can be computed is usually about the same size as the largest integers that could be factored</li><li>This means that discrete logs are another example of one-way functions</li><ul><li>where f(x) is easy to compute but given y it is computationally infeasible to find x</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-7530093779146507842013-12-05T17:06:00.000-08:002013-12-05T17:06:04.723-08:00Notes - An Application to Treaty VerificationThe following are notes from Introduction to Cyrptography with Coding Theory.<br /><ul><li>This method will essentially describe the RSA signature Scheme</li><li>Example</li><ul><li>Countries A and B have signed a nuclear test ban treaty</li><li>Country A wants to verify the treaty by placing sensors in B</li><ul><li>Country A wants to ensure sensors are sending correct data without tampering</li><li>Country B wants to look at the message before its being sent to ensure that espionage data is not being transmitted</li></ul><li>A chooses n = pq to be the product of two large primes and chooses encryption/decryption exponents e and d</li><li> n and e are given to B but p q and d are kept secret</li><li>sensor collects data x and uses d to encrypt x to y ≡ x<sup>d</sup> (mod n)</li><li>both x and y are sent to country B to check that y<sup>e</sup> ≡ x (mod n) and then to Country A to ensure that the number x has not been modified. If x was modified it would be the same difficulty as decrypting the rsa message x</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-6198187161012786302013-12-05T16:52:00.003-08:002013-12-05T16:52:47.291-08:00Notes - The RSA AlgorithmThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>Public Key Cryptosystem</li><ul><li>Presented by Diffie-Hellman</li><li>Goal - allow Alice to send a message to Bob without previous contact and without the use of a courier to exchange a key but protect the message from Eve a potential attacker</li></ul><li>The RSA Algorithm</li><ul><li>Bob chooses secret primes p and q and computes n = pq</li><li>Bob chooses e with gcd(e, (p-1)(q-1)) = 1</li><li>Bob computes d with de ≡ 1 (mod(p-1)(q-1))</li><li>Bob makes n and e public, and keeps p, q, d secret</li><li>Alice encrypts m as c ≡ m<sup>e</sup> (mod n) and sends c to Bob</li><li>Bob decrypts by computing m ≡ c<sup>d</sup> (mod n)</li></ul><li>Example of the RSA Algorithm</li><ul><li>p = 885320963, q = 238855417</li><li>n = p * q = 211463707796206571</li><li>Let the encryption exponent be e = 9007</li><li>Let the sample message m be 30120</li><li>c ≡ m<sup>e</sup> ≡ 30120<sup>9007</sup> ≡ 113535859035722866 (mod n)</li><ul><li>Where c stands for the ciphertext Alice is sending to Bob</li></ul><li>Bob knows p and q so he knows (p-1)(q-1) then he uses the extended euclidean algorithm to compute d such that de ≡ 1 mod (p-1)(q-1)</li><li>d = 116402471153538991</li><li>Bob computes c<sup>d</sup> ≡ 113535859035722866<sup><span style="font-size: small;">116402471153538991 </span></sup>≡ 30120 (mod n) to obtain the original message</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-56111897818057694312013-11-26T23:48:00.001-08:002013-11-26T23:48:06.584-08:00Battery Applications and ParametersThe following are notes on Batteries for their common applications and parameters<span id="docs-internal-guid-2b03776e-987c-375e-6334-8fcf9bbdd084"></span><br /><ul><li>Battery Applications</li><ul><li>Portable Applications</li><ul><li>Portable electronics, laptops, iphones, toys, medical devices</li><li>battery used to start engine in cars is the largest market</li><ul><li>lead acid and lithium ion</li></ul><li>High energy batteries, highest energy for given size</li></ul><li>Stationary Applications</li><ul><li>Pairing with with intermittent sources of electricity</li><li>high energy is less important</li></ul></ul><li>Common Forms Factors of Batteries</li><ul><li>Button Cells in Watches</li><li>Cylinder Cells</li><li>Lithium Ion</li><li>Prismatic Cells</li></ul><li>Sizes of Batteries</li><ul><li>4.5-volt (3R12) battery </li><li>D Cell</li><li>C cell</li><li>AA cell</li><li>AAA cell</li><li>AAAA cell</li><li>A23 battery</li><li>9-volt PP3 </li><li>button cells</li></ul><li>Types of Batteries</li><ul><li>Chargeable or non-chargeable</li><li>Battery Chemistry</li><ul><li>Alkaline </li><li>Lead acid </li><li>NiCd nickel cadmium</li><li>NiMH nickel metal hydride</li><li>Lithium ion</li></ul></ul><li>Important Parameters of Batteries</li><ul><li>Energy (Watt Hour)</li><ul><li>Energy Density - Energy Per Unit Volume</li><li>Specific Energy - Energy Per unit Weight</li><li>Characterize the total amount of energy it carries, lifetime of battery once you charge it up, distance of electric car</li></ul><li>Power (Watt)</li><ul><li>How fast you can take the energy out of a battery</li><li>Power Density Power per unit volume</li><ul><li>Example more important for Laptop, fitting into size constraint for a high usage device</li></ul><li>Specific Power Power per unit weight</li><ul><li>Car, heaviness of battery impacts performance</li></ul><li>Life</li><ul><li>Cycle Life, times we can charge and discharge a battery with its capacity still above 80% of the original</li></ul></ul></ul><ul><ul><ul><li>Calendar Life Time a battery can last if left on shelf</li><ul><li>Electrical car ensure it can last for about 10 years?</li></ul></ul><li>Safety</li><ul><li>Battery intrinsically is in unstable state</li></ul><li>Temperature Performance</li><li>Cost</li><ul><li>Currently its difficult to justify batteries for cars and grid use</li></ul></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/11/electrical-engineering-toc.html"><span id="goog_1036010427"></span>Back to ToC</a><span id="goog_1036010428"></span></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-40222948239796425372013-11-26T23:33:00.001-08:002013-12-12T15:32:26.662-08:00Electrical Engineering ToC<h2 style="text-align: center;">Table of Contents</h2><br /><ol><li>Batteries</li><ol><li><a href="http://ctluk.blogspot.com/2013/11/battery-applications-and-parameters.html">Battery Applications and Parameters</a></li><li><a href="http://ctluk.blogspot.com/2013/12/battery-history.html">Battery History</a></li></ol><li>Digital Design</li><ol><li><a href="http://ctluk.blogspot.com/2013/10/electrical-engineering-finfet.html">Finfet</a></li></ol></ol>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-62942631916462236872013-11-21T16:10:00.003-08:002013-11-21T16:20:59.251-08:00Notes - Sustainabiltiy and Energy Storage StystemsThe following are notes on the sustainable energy sector and the potential of energy storage technologies.<br /><ul><li>Sustainability</li><ul><li>Energy storage is useful in order to balance the grid</li><ul><li>grid balancing is making sure that the amount of energy being produced by the grid is equal to the amount of energy being used by it</li></ul><li>Issues it solves</li><ul><li>As the amount of renewable sources continue to increase, the fluctuating nature of the energy production means that we will need to capture any "excess production" in the case that renewable sources produce too much energy</li><li>This becomes profitable if storage beats the costs of turning on and off other generators such as coal, gas, etc.</li></ul><li>Issues facing growth of industry</li><ul><li>We would require variable renewable sources to generate greater than about 20% of our electricity in order for this to become viable</li><ul><li>variable includes wind/solar, production dependent on variable sources</li><li>non variable renewable includes hydropower, biomass, geothermal because their sources are controllable</li></ul></ul></ul></ul><div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-NEidhX0LLj0/Uo6cWrOMLtI/AAAAAAAACMA/flZSYdfVkOY/s1600/outlet-graph-medium.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="293" src="http://2.bp.blogspot.com/-NEidhX0LLj0/Uo6cWrOMLtI/AAAAAAAACMA/flZSYdfVkOY/s320/outlet-graph-medium.jpg" width="320" /></a></div><span id="goog_1092314195"></span><span id="goog_1092314196"></span><br /><br /><ul><ul><ul><li>As is clearly shown in the above solar amounts to .12% of our total energy production, and wind amounts to 3.36% for a total of about 3.48%</li><ul><li>Solar although it is a rapidly growing field is still dependent on subsidies, although this may soon change, something needs to change in the market in order for there to be an increase in buyers</li><li>Wind is completely dependent on government subsidies. There is near 0 production when subsidies are taken away, so wind development relies on smart governmental decisions</li><ul><li>Cue Government shutdown</li></ul><li>Additional Complications</li><ul><li>there is steep competition given by China which has better economies of scale as they are investing heavily into the older silicon based solar which is cheaper to purchase</li><ul><li>Note China as a whole is not yet profiting, but they are gaining a powerful grip on the market</li></ul><li>US is "going big" in solar arena by increasing efficiency of their cells</li><ul><li>this would need to be coupled with trends i.e. requires help from mainstream media in order to become successful</li><ul><li>Exceptions - First Solar, although product has increase in efficiency, a byproduct was that the it "looks better" as electrodes are attached beneath rather than on top of the modules in order to increase efficiency</li><li>This is similar to Apple which used a sleek design in order to sell the product, meeting the social needs of the user</li></ul><li>Similar story to Solyndra this means US companies require large initial investments, reduces amount of entrepreneurs capable of accessing the market</li></ul></ul></ul></ul></ul><li>Energy Storage</li><ul><li>There are two divergent paths a company can take</li><ul><li>Invest into countries outside of the US</li><ul><li>By taking control of third world countries there is a potential to have access to the infrastructure of an entire market over time</li><li>Can sell to a more mature market</li></ul><li>Invest into a longer time span</li><ul><li>assumes that the US will not tolerate other countries exceeding our infrastructure base</li><ul><li>Has some merit i.e. average internet speed competition of Asian countries now being counteracted with Google fiber and other extreme high speed connections</li><li>Fiber slowly replacing copper</li></ul></ul></ul></ul></ul></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-67171056626813301272013-11-03T01:15:00.003-08:002013-11-03T01:15:29.891-08:00Physics - Classical Theory of ConductionThe following is a brief overview of the classical theory of conduction.<br /><br />Fluid Analogy<br />First I would like to invoke the fluid analogy to electricity. Basically there are many similarities between different phenomena for fluid mechanics and electronics. The reason why we have this analogy is simply that most people are used to dealing with fluids, especially those of us with access to plumbing and running water. However, most of us don't even think about electricity, we just put a plug in an outlet and stuff works.<br /><br />Now how does a battery work? Its no mystery though Bill O'Reilly might disagree. We can see on a battery's label a + and a - sign. We can think of this as a lake at the top of a mountain and a lake on the bottom of the mountain. They have different potential energy due to gravity. When we plug in a battery its like digging a trench to connect the two lakes and putting a water mill in between it to drive our machine.<br /><br />Relation to Conduction<br />We can then relate two other phenomena. Current in a wire is very similar to trying to get water to move through a pipe. The smaller our pipe, the less current can get through, which is similar to a wire with high resistivity. In electronics we can think of a wire as a lattice of of atoms. Our electrical current is then though of as electrons traveling through this lattice. An electrical field caused by our difference in potential drives electrons through this lattice but the lattice is stationary and stops/causes ricochets when an electron hits it. We can describe this with the following two equations:<br /><br /><div style="text-align: center;">I =ΔQ/Δt = neAv<sub>d</sub> and<br />λ = <v> τ =1/n<sub>a</sub>πr<sup>2</sup> as<br /><br /><div style="text-align: left;">Where for the first equation I is the current described by n the number of electrons, with charge e, passing through an area A with electron drift velocity <span style="text-align: center;">v</span><sub style="text-align: center;">d</sub>. The second equation gives us <span style="text-align: center;">λ is the mean free path, or the average distance a particle travels before a collision with <</span><span style="text-align: center;">v> the average velocity multiplied against </span><span style="text-align: center;">τ the average time between collisions. Through this we can figure out resistivity and conductivity and can derive Ohm's law. </span></div><div style="text-align: left;"><span style="text-align: center;"><br /></span></div><div style="text-align: center;">V = IR</div><div style="text-align: center;"><br /></div><div style="text-align: left;">Defects</div><div style="text-align: left;">This is a great approximation but there are significant errors when scaled to different temperatures and other quantum effects.</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><a href="http://ctluk.blogspot.com/2012/11/physics-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-23171628988175086092013-11-03T01:40:00.001-07:002013-11-03T01:16:55.402-08:00Paper - Skill Acquisition<div style="text-align: center;">Skill Acquisition as a Function of Effort</div><div style="text-align: center;"><br /></div><div style="text-align: left;">This is a discussion of my viewpoints on skill acquisition in relation to effort/time investment. First I'd like to divide skill into two main components, mechanics and execution mainly in relation to a sport or game, though this can applies to pretty much any skill when substituting in different terminology. Mechanics for a sport is the player's trained skills. For an athlete it would be the capability to move his/her body in a given precise way. An example for tennis would be for a player's forehand, the strength of the hit, its precision, timing of the shot after the bounce, etc. Execution is the player's capability to perform those mechanics under pressure, in competitive situations, or when something is "on the line".</div><div style="text-align: left;"><br /></div><div style="text-align: left;">Mechanics<br />Mechanics for a person will usually rise when effort is put in. So classically we can think of skill as a function of effort as something like this.</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-06gOvvMu854/UnYDpMQGQxI/AAAAAAAACK8/OPx3NTSc0bs/s1600/sqrt+x.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="157" src="http://3.bp.blogspot.com/-06gOvvMu854/UnYDpMQGQxI/AAAAAAAACK8/OPx3NTSc0bs/s320/sqrt+x.png" width="320" /></a></div><br /></div><div style="text-align: left;">However, this as with everything has major pitfalls. It is very possible for this to go negative like so after some initial improvement. </div><div style="text-align: left;"><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-yeR6U7AZsu8/UnYDw_emypI/AAAAAAAACLE/9nXt-d4Z-7g/s1600/negative+sqrt+x.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="156" src="http://3.bp.blogspot.com/-yeR6U7AZsu8/UnYDw_emypI/AAAAAAAACLE/9nXt-d4Z-7g/s320/negative+sqrt+x.png" width="320" /></a></div><br /></div><div style="text-align: left;">How does this happen?<br />Unfortunately it has everything to do with luck and our environment. When acquiring a skill everyone usually learns from somewhere. It usually builds upon something that is taught to us directly, or builds on past experiences that we are able to connect with the skill we want to improve. With the internet we can also be "self-taught" and do our own research but that information was still written by someone else in the first place. In some rare occasions people are pioneers and must figure out things the first time through, but we won't go into that too much here.<br /><br />The issue with this is that our information source can be wrong. Whether it be a coach or a teacher, or some random article on the internet we might just be going about something the wrong way and be ingraining into ourselves incorrect muscle memory. We could also be guided by correct information but interpret it the wrong way which will have the same effect, worsening our skill.<br /><br />Benefits?<br />The only thing we can really get out of this situation is experience. Usually by doing something for a long period of time, we can then make a realization on how to correct our errors, or observe or be observed by others and make the correction by fixing that information returning us to our original rate of improvement. However, there is a possibility of getting worse. This is a very important concept to understand, there is no fairness in the world, spending effort does not necessarily mean you will improve, but it does give you the chance to improve/change where you are.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-06gOvvMu854/UnYDpMQGQxI/AAAAAAAACK8/OPx3NTSc0bs/s1600/sqrt+x.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="157" src="http://3.bp.blogspot.com/-06gOvvMu854/UnYDpMQGQxI/AAAAAAAACK8/OPx3NTSc0bs/s320/sqrt+x.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Also not spending effort will always result in a decrease in skill over the course of a long period of time. Again things are not fair here, breaks can sometimes result in a breakthrough as you gain a new perspective. This phenomena is often talked about by academics, but it also applies to sports and other motor based skills as your muscles have an opportunity to loosen up on their habits/memory which can lead to greater control with additional information/accidental improvements. This means that a person's skill level is often riddled by positive and negative interjections like this, where mechanics raise and fall temporarily. Sometimes a person is able to grasp their sudden improvement and retain parts of it/all of it and increase their skill over a very rapid period of time, but it can also sink and again result in a decrease in mechanics if analyzed incorrectly.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-E7eQ8lOqLU0/UnYIiQZZ5gI/AAAAAAAACLU/icra_Rnkh5s/s1600/DeltaPlot.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="245" src="http://1.bp.blogspot.com/-E7eQ8lOqLU0/UnYIiQZZ5gI/AAAAAAAACLU/icra_Rnkh5s/s320/DeltaPlot.gif" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">Execution</div><div class="separator" style="clear: both; text-align: left;">Execution is essentially a person's mental game. Very few people have managed to obtain a stable mental game and it is affected by a wide range of events. A person's relationships, their confidence, their job, anything that causes mental stress or happiness outside of the pursuit of that skill can affect their ability to perform that skill. A graph that represents this would be something like the following.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-uvBNXqrFzpQ/UnYK38m4I9I/AAAAAAAACLg/hs2jXK4sSMs/s1600/1++sin.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="151" src="http://4.bp.blogspot.com/-uvBNXqrFzpQ/UnYK38m4I9I/AAAAAAAACLg/hs2jXK4sSMs/s320/1++sin.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This is the roller coaster ride of ups and downs that everyone experiences which is linked to our maturity. Speaking of which as we mature, we are able to clamp down on these variations, and improve based which can be represented with the following.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-MqJZ8sf-8Fw/UnYLNNYC8kI/AAAAAAAACLo/kqcntUQYr0A/s1600/e+-x+sin+x.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="125" src="http://2.bp.blogspot.com/-MqJZ8sf-8Fw/UnYLNNYC8kI/AAAAAAAACLo/kqcntUQYr0A/s320/e+-x+sin+x.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Basically as we mature we become more predictable and are able to reduce the impact of our emotions into our skills, thereby making our performance more reliant on our mechanics rather than state of our emotions. However this is one of the most difficult things to do, and honestly achieving this does not necessarily mean you will be the most successful, but I will discuss that at another time as it relates more to successful personalities, and less to the process of acquiring a skill.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://ctluk.blogspot.com/2012/11/papers-and-research-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-86045262861321809592013-11-01T22:20:00.001-07:002013-11-02T00:40:44.547-07:00Physics - Quantum TunnelingThe following is a brief overview of Quantum tunneling.<br /><br />What is it?<br />Quantum Tunneling is the effect where a particle tunnels through a barrier it could not penetrate in classical physics.<br /><br />Why does it happen?<br />Quantum physics tells us that matter exhibits both particle and wave-like behavior. Therefore if we apply this idea to an electron going up against an potential barrier, we can relate its behavior similarly to a more well known event, light traveling through a material such as glass. In this case, the brightness of the light passing through the surface is reduced and some is reflected back away from the observer behind the glass. With our electron, some of its probability function travels through the potential well, and some of it reflected, depending on the width of the energy barrier, which we can relate the opacity of an object.<br /><br />Brief Overview of Formulation<br />The time independent Schrodinger equation is as follows<br /><br /><div style="text-align: center;">(-ħ<sup>2</sup>/2m)(d<sup>2</sup>Ψ(x)/dx<sup>2</sup>) + U(x)Ψ(x) = EΨ(x)</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The full derivation will be done elsewhere but we can use these to get the solutions for traveling waves, which represents the probability function of a particle in motion. This represents our electron moving. Using the WKB approximation and a few other mathematical tricks such as series expansion, we arrive at the following equation for T the transmission coefficient or percentage chance at quantum tunneling.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-V0Gl-m1eync/UnSFmYVFgaI/AAAAAAAACKs/h8-YQAM5B3w/s1600/transmission+coefficient.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-V0Gl-m1eync/UnSFmYVFgaI/AAAAAAAACKs/h8-YQAM5B3w/s1600/transmission+coefficient.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">The important part of this equations are, x1 and x2 which are the classical turning points for the energy barrier. We can therefore see that this equation depends primarily on two characteristics, the difference in energy between the particle/wave represented as E. V(x) which is the potential of the energy barrier given x, and the width of the barrier x where a higher energy difference results in a lower transmission coefficient, and a greater "width" again results in a lower transmission coefficient.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Relation to Electrical Engineering</div><div class="separator" style="clear: both; text-align: left;">This is a material science concept and when used in reference to tunneling in electrical components, we don't have to go into the full Feynman path integral method to just get an idea of how it works. We can get a good idea of the effects with the explanation above. Tunneling occurs with barriers the size of around 1-3 nanometers which will be extremely significant when dealing with finfets which promise to move the total size of the transistor to around the 10 nm range. Efforts must be made to maintain detection of the on and off state due to thermal noise and summation of inductance effects of nearby inductance effects. This means design rules must be made with greater spacing margins than might be otherwise done for normal mosfet design. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://ctluk.blogspot.com/2012/11/physics-toc.html">Back to ToC</a></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-91839030025896036092013-10-29T13:58:00.000-07:002013-11-02T00:24:03.085-07:00Electrical Engineering - FinfetThe following will be notes taken on the subject of FinFets<br /><ul><li>Summary</li><ul><li>Classical Mosfet</li><ul><li>Transistor that operates by using an Electrical field to invert the channel allowing conduction from source to drain</li></ul></ul></ul><ul><ul><ul></ul></ul></ul><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-lo0q_sqgzb4/Um9Q6XlAkiI/AAAAAAAACJg/6jar-aTrx9I/s1600/Mosfet.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="174" src="http://2.bp.blogspot.com/-lo0q_sqgzb4/Um9Q6XlAkiI/AAAAAAAACJg/6jar-aTrx9I/s320/Mosfet.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div><ul><ul><ul><li>The Classical Mosfet Dimensions</li><ul><li>Include length the short distance in this picture, Width is the longer dimension</li></ul><li>Functionality</li><ul><li>Length reduction results in loss of control over the channel</li><ul><li>performance reduction</li></ul></ul></ul></ul></ul><ul><ul><li>Finfet</li><ul><li>Goal</li><ul><li>solve performance reduction caused by smaller sizes </li></ul><li>Wrap the gate electrode around the channel</li></ul></ul></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-dJ4KF0DLbho/UnAWu327-rI/AAAAAAAACJw/3nhv63I4adY/s1600/Finfet.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="173" src="http://4.bp.blogspot.com/-dJ4KF0DLbho/UnAWu327-rI/AAAAAAAACJw/3nhv63I4adY/s320/Finfet.png" width="320" /></a></div><div><ul><ul><ul><li>Thin fin of sillicon acts as the channel and it is encased by the gate electrode</li><li>Silicon fin surrounded by extension implant and poly oxide</li><li>Premise</li><ul><li>di-electric and metal gate allows for a stronger e-field to be formed as it increases the dimensions of the gate effective using the height/thickness of the fin as the channel length allowing for a decrease in overall size when compared to a typical Mosfet</li></ul></ul></ul></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-4H3xqYQ765M/UnAX3382V2I/AAAAAAAACJ4/uR3iWTC8ZZA/s1600/Dimension+Finfet.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="170" src="http://2.bp.blogspot.com/-4H3xqYQ765M/UnAX3382V2I/AAAAAAAACJ4/uR3iWTC8ZZA/s320/Dimension+Finfet.png" width="320" /></a></div><div><ul><ul><ul><li>Source and Drain can be wrapped in silicon germanium or silicon carbon stressors just like classical transistors</li><li>Shown below is a representation of a finfet design used for production</li></ul></ul></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-xv_4g6D1wyE/UnAZ_VjkgFI/AAAAAAAACKE/Tc5oSgxc4jI/s1600/Finfet+Construction.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="http://4.bp.blogspot.com/-xv_4g6D1wyE/UnAZ_VjkgFI/AAAAAAAACKE/Tc5oSgxc4jI/s320/Finfet+Construction.png" width="320" /></a></div><div><ul><ul><ul><li>The gate electrode is uniform for ease of construction</li></ul></ul></ul><ul><li>Clarification of terminology</li><ul><li>After seeing the design used for production, we can see that it is nearly identical to the construction of the tri-gate</li></ul></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-o7UilTH25IY/UnAbrMbCvOI/AAAAAAAACKQ/8Jxf_FL8iWE/s1600/Trigate.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="126" src="http://1.bp.blogspot.com/-o7UilTH25IY/UnAbrMbCvOI/AAAAAAAACKQ/8Jxf_FL8iWE/s320/Trigate.jpg" width="320" /></a></div><div><br /></div><div><ul><ul><li>The only difference is that a tri-gate includes multiple sources and drains whereas the finfet description shows only a single source and drain but the process of construction seems identical, and terminology could be used interchangeably assuming you're fine with annoying Intel</li><li>The original papers indicate that the original process of the finfet did not wrap around on top of the fin, lowering the overall z or height of the overall transistor but this is too complicated of a process for mass production so the standard finfet is equivalent to the tri-gate which does wrap around </li></ul></ul><ul><li>Benefits</li><ul><li>Maintain performance which includes</li><ul><li>Conductivity when turned on</li><li>Insulation when turned off</li><ul><li>Finfet in relation to a standard mosfet reduces electron tunneling effect when insulation is small</li><ul><li>Main issues are - weaker dielectric, small size</li><li>Finfet significantly increases volume of dielectric reducing leakage currents</li></ul></ul><li>increase switching speed</li><ul><li>due to lower size due to gate capacitance being smaller</li><li>note - non-issue in modern constructions, interface delay between metals is more significant as we get to smaller scale</li></ul><li>Lower's Voltage requirements</li><ul><li>increases lifetime/efficiency of product per charge</li></ul></ul><li>Company Promises</li><ul><li>estimation, 2-5% higher price in exchange for 37% speed increase and 90% reduction in leakage current</li></ul></ul></ul><ul><li>Complications</li><ul><li> Adds complexity to construction process</li><ul><li>possible reduction in yield</li><ul><li>can increase price significantly above 2-5% estimate for the company overall</li></ul></ul><li>Depending on process</li><ul><li>high k-dielectrics are more expensive, but already required as transistor size decreases</li></ul><li>Increased modeling difficulty</li><ul><li>geomotries now are first order effects</li><ul><li>random fluctuations in manufacturing can now cause deviations in result</li><li>impacts possible econcomic forecasts for the division</li></ul><li>Verification</li><ul><li>requires new software to ensure design rules for fin to fin spacing is followed</li></ul></ul></ul></ul><ul><li>History</li><ul><li>construction process has roots in 1990s</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/new-developments-toc.html">Back to ToC</a></div></div></div></div></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-38090484615712503292013-10-22T16:34:00.002-07:002013-10-22T16:34:45.231-07:00Notes - Square Roots Mod nThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>Suppose we are told that x<sup>2</sup> ≡ 71 (mod 77) has a solution</li><ul><li>How do we find one solution/all solutions?</li></ul><li>Proposition</li><ul><li>if y has a square root mod p, then teh square roots of y mod p are +-x.</li><li>If y has no square root mod p, then -y has a square root mod p, and the square roots of -y are +-x</li></ul><li>Example</li><ul><li>Lets find the square roots of 5 mod 11. Since (P+1)/4 = 3, we compute x ≡ 5<sup>3</sup> ≡ 4 (mod 11). Since 4<sup>2</sup> ≡ 5 (mod 11), the square roots of 5 mod 11 are <complete id="goog_1270574515">+-4 or 5,7 mod 11</complete></li><li>Let's try to find the square root of 2 mod 11. Since (p+1)/4 = 3, we compute 2<sup>3</sup> ≡ 8 (mod 11). But 8<sup>2</sup> ≡ 9 ≡ -2 (mod 11), so we have found a square root of -2 rather than 2</li></ul><li>Example</li><ul><li>Composite Modulus square roots</li><li>x<sup>2</sup> ≡ 71 (mod 77)</li><li>x<sup>2</sup> ≡ 71 ≡ 1 (mod 7) and x<sup>2</sup> ≡ 71 ≡ 5 (mod 11)</li><li>x ≡ +-1 (mod 7) and x ≡ +-4 (mod 11)</li><ul><li>The chinese remainder theorem tells us that a congruence mod 7 and a congruence mod 11 can be recombined into a congruence mod 77</li><ul><li>m<sub>1</sub> = 7, m<sub>2</sub> = 11, m = m<sub>1</sub>m<sub>2 </sub>= 77</li><li>M<sub>1</sub> = m/m<sub>1</sub>= 11, M<sub>2</sub> = m/m<sub>2</sub>= 7</li><li>M<sub>1</sub>N<sub>1</sub> ≡ 1 (mod m<sub>1)</sub> => 11 N<sub>1 </sub>≡ 1 (mod 7) => 4 N<sub>1 </sub>≡ 1 (mod 7) => N<sub>1 </sub>≡ 2 (mod 7)</li></ul><li>this will get us results of x ≡ +-15,+-29 (mod 77) </li></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-64286487615773697482013-10-08T15:47:00.002-07:002013-10-08T15:47:38.715-07:00Notes - Primitive RootsThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>Consider the powers of 3 (mod 7)</li><ul><li>3<sup>1</sup>≡ 3, 3<sup>2</sup>≡ 2, 3<sup>3</sup>≡ 6, 3<sup>4</sup>≡ 4, 3<sup>5</sup>≡ 5, 3<sup>6</sup>≡ 1</li><li>Note that we obtain all nonzero congruence classes mod 7 as powers of 3</li></ul><li>Generally when p is a prime, a primitive root mod p is a number whose powers yield every nonzero class mod p</li><li>Primitive Roots</li><ul><li>Let g be a primitive root for the prime p</li><li>Let n be an integer. Then g<sup>n</sup>≡ 1 (mod p) if and only if n ≡ 0 (mod p - 1)</li><li>If j and k are integers, then g<sup>j </sup>≡ g<sup>j </sup>(mod p) if and only if j ≡ k (mod p - 1)</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-11776384355475849322013-10-08T11:00:00.000-07:002013-10-08T11:00:05.436-07:00Notes - Fermat's Little Theorem and Euler's TheoremThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>Fermat's Little Theorem</li><ul><li>if p is a prime and p does not divide a, then</li><li>a <sup>p-1</sup>≡ (1 mod p)</li><li>Example</li><ul><li>2<sup>10 </sup>= 1024 ≡ 1 (mod 11)</li><li>can then evaluate 2<sup>53</sup> (mod 11)</li><li>2<sup>53 </sup>= (2<sup>10</sup>)<sup>5</sup>2<sup>3</sup> ≡ 1<sup>5</sup>2<sup>3</sup> ≡ 8 (mod 11)</li></ul></ul><li>We now require an analog for composite modulus n</li><ul><li>Define φ(n) as the number of integers a between 1 and n such that gcd(a,n) = 1</li><li>Example</li><ul><li>φ(10) = 4 [1,3,7,9]</li></ul><li>Also can be deduced from chinese remainder theorem</li><ul><li>φ(n) = n Π<sub>p|n</sub> (1-1/p)</li></ul><li>φ(p) = p - 1 where p is a prime number</li><ul><li>when n = pq where p and q are primes φ(pq) = (p - 1)(q - 1)</li></ul></ul><li>Euler's Theorem</li><ul><li>If gcd(a,n) = 1, then a<sup><span style="font-size: small;">φ(n)</span> </sup>≡ 1 (mod n)</li><li>Example</li><ul><li>What are the last 3 digits of 7<sup>803</sup></li><ul><li>same as working mod 1000</li><li>1000 = 2<sup>3</sup>5<sup>3</sup></li><li>φ(1000) = 1000(1-1/2)(1-1/5) = 400</li></ul></ul><li>Example</li><ul><li>Compute 2<sup>43210 </sup> (mod 101)</li><ul><li>Note that 101 is prime, therefore 2<sup>100 </sup> ≡ 1 (mod 101)</li><li>Therefore (2<sup>100</sup>)<sup>432</sup>2<sup>10</sup> = 1<sup>432</sup>2<sup>10</sup> ≡ 1024 (mod 101) ≡ 14</li></ul></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-72917393515466362782013-10-08T00:54:00.002-07:002013-10-08T00:54:56.506-07:00Notes - The Chinese Remainder TheoremThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>Chinese remainder theorem is used to break a congruence mod n into a system of congruences mod factors of n</li><ul><li>Example</li><ul><li>Number satisfies x≡ 25(mod 42)</li><li>means we can write x = 25 + 42k (k is some integer)</li><li>we can rewrite 42 as 7*6</li><li>x = 25 + 7(6k)</li><li>x = 25 + 6(7k)</li></ul></ul><li>Chinese Remainder Theorem</li><ul><li>suppose gcd(m,n) = 1. If an integer c is a multiple of both m and n, then c is a multiple of mn. given integers a and b there exists exactly one solution x (mod mn)to the simultaneous congruences</li><ul><li>x ≡ a (mod m)</li></ul><li>x ≡ a (mod m)</li><ul><li>x ≡ b (mod n)</li></ul></ul><li>Lemma</li><ul><li>Let m,n be integers with gcd(m,n) = 1</li><li>if an integer c is a multiple of both m and n, then c is a multiple of mn</li></ul><li>Example</li><ul><li>Solve x ≡ 3 (mod 7), x ≡ 5 (mod 15)</li><li>7*15 = 105</li><li>list congruences</li><ul><li>5,20,35,50,65,80</li><li>3,10,17,24,31,38,45,52,59,66,73,80</li><li>x ≡ 80 mod 105</li></ul><li>solve</li><ul><li>nk ≡ a - b (mod m)</li><li>k ≡ (a - b)i (mod m) [i is the multiplicative inverse of n]</li></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-40290416654154340592013-10-02T00:03:00.001-07:002013-10-02T00:03:16.958-07:00Notes - Pseudo-random Bit GenerationThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>Ways to generate random bits</li><ul><li>sampling thermal noise from semiconductor resistor</li><li>flipping coins</li></ul><li>Sampling natural processes however is very slow and its difficult to ensure that the enemy does not view the process</li><li>Pseudo-Random generation</li><ul><li>example rand()</li><ul><li>takes a seed as an input produces output bitstream</li><li>based on linear congruential generators</li><li>x<sub>n</sub> = a x <sub>n-1</sub> + b (mod m)</li><li>number x<sub>0</sub> is the initial seed</li></ul><li>this is suitable for experimental purposes but not for cryptographic purposes because it is too predictable</li><li>one way functions</li><ul><li>functions f(x) that are easy to compute but is computationally infeasible to determine y = f(x)</li><li>therefore we can define a x<sub>j</sub> = f(s+j) for j = 1,2,3</li><li>DES, SHA</li></ul><li>Blum-Blum Shub pseudo-random bit generator</li><ul><li>quadratic residue generator</li><li>generate two large primes p, q that are congruent to 3 mod 4</li><li>then set n = pq and chose random integer x that is relatively prie to n</li><li>BBS produces a sequence by</li><ul><li>x<sub>j </sub>≡ x<sup>2</sup><sub>j-1</sub> (mod n)</li><li>b<sub>j</sub> is the least significant bit of x<sub>j</sub></li></ul><li>However BBS is slow to calculate due to size of primes</li></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-28968194351081285232013-10-01T16:47:00.001-07:002013-10-01T23:44:38.555-07:00Notes - Block CipherThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>also known as the hill cipher</li><li>choose an integer n, the key is an n x n matrix</li><ul><li>M =</li><li>(1 2 3)</li><li>(4 5 6)</li><li>(11 9 8)</li></ul><li>Message is written as a series of row vectors</li><ul><li>abc becomes (0 1 2) which gets </li><li>(012)(1 2 3)</li><li> (4 5 6)</li><li> (11 9 8)</li><li>mod 26</li><li>(0,23,22)</li></ul><li>In order to decrypt we need the determinant of matrix M to satisfy</li><ul><li>gcd(det(M),26)=1</li><li>Find the inverse of matrix M</li><li> (-14 11 -3)</li><li>-1/3(34 -25 6)</li><li> (-19 13 -3)</li><ul><li>Since 17 is the inverse of -3 mod 26</li></ul><li> (22 5 1 )</li><li> (6 17 24)</li><li> (15 13 1)</li><ul><li>can return plaintext by mulitplying inverse matrix by (0,23,22) mod 26</li></ul></ul><li>In order to perform the block cipher we just divide the plaintext into blocks of n characters and if the ciphertext does not divide evenly, the blanks are left as 0 but the matrix multiplication is still done</li><li>changing one letter changes n letters of plaintext therefore frequency decryption is very difficult</li><li>Claude Shannon</li><ul><li>the fundamental foundations of cryptography include</li><ul><li>Diffusion</li><ul><li>means that changing one character of the plaintext, then several characters</li></ul><li>Confusion</li><ul><li>means that key does not relate in a simple way to the ciphertext</li><li>each ciphertext should depend on several parts of the key</li></ul></ul></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-92072561964531825632013-09-28T01:18:00.001-07:002013-09-28T01:18:10.605-07:00Notes - The Vigenere CipherThe following are notes from Introduction to Cryptography with Coding Theory.<br /><div><ul><li>A variation of the shift cipher</li><ul><li>key for the encryption is a vector of a chosen length</li><li>Example</li><ul><li>key length 6</li><li>(21,4,2,19,14,17)</li><li>corresponds to word vector</li><li>To encrypt we shift by the word vector</li></ul></ul><li>A known plain text attack will succeed if enough characters are known</li><li>Cryptanalysis uses the fact that english letter frequency is not equal</li><ul><li>a - 0.082</li><li>b - 0.015</li><li>c - 0.028</li><li>d - 0.043</li><li>e - 0.127</li><li>f - 0.022</li><li>g - 0.020</li><li>h - 0.061</li><li>i - 0.070</li><li>j - 0.002</li><li>k - 0.008</li><li>l - 0.040</li><li>m - 0.024</li><li>n - 0.067</li><li>o - 0.075</li><li>p - 0.019</li><li>q - 0.001</li><li>r - 0.060</li><li>s - 0.063</li><li>t - 0.091</li><li>u - 0.028</li><li>v - 0.010</li><li>w - 0.023</li><li>x - 0.001</li><li>y - 0.020</li><li>z - 0.001</li></ul><li>Variations can occur but usually takes effort to produce, example the book Gadsby does not contain the letter e</li><li>However, when using a Vigenere Cipher the frequencies are distributed so that the above is not nearly as useful in decryption</li><li>Finding Key Length</li><ul><li>We find the key length by matching the number of coincidences</li><li>write the ciphertext, then rewrite it with a displacement</li><li>Example</li><li> VVHQ</li><li>VVHQW</li><li>Then we mark each time there is a match between the two rows, and after trying multiple displacements the largest number of coincidences will be closest to the key length</li></ul><li>Finding the Key: First Method</li><ul><li>take a look at the 1+5n letters and find the frequency of letters then repeat for 2+5n, 3+5n, 4+5n, 5+5n we then guess at the code by finding the difference between the most frequent letter and e to find the overall shift that is used as the key</li><li>After finding the key we then test it by decrypting to confirm if the key was correct</li></ul><li>Soft Proof</li><ul><li>place the frequencies of the english letters into a vector A<sub>0</sub> and let A<sub>i</sub> represent the frequencies shifted by i</li><li>When taking the dot product of these two vectors we arrive at 0.066 with a full match but significantly less than that for any other shift</li><li>We can also use this knowledge to approximate the number of coincidences we should find for a given match, which would be 0.066*the number of comparisons</li></ul></ul><div><a href="http://ctluk.blogspot.com/2013/09/math-cyrptography-toc.html">Back to ToC</a></div></div>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0tag:blogger.com,1999:blog-6959467840546371805.post-35615207869716922162013-09-28T01:12:00.001-07:002013-09-28T01:12:43.226-07:00Physics - DielectricsThis will be an introduction to dielectrics. Dielectrics are insulators that are mainly used to fill the gaps within capacitors in order to increase the capacitance it can hold. The Dielectric actually weakens the electric field within the capacitor as the electrodes induce a charge on the surfaces of the insulator, but this effect will never completely reverse the electric field.<br /><br />A Dielectric has a property called electric susceptibility which is designated by Χ<sub>e</sub>. This is a multiplier of the electrical permittivity of free space ε<sub>0</sub>. This susceptibility is pretty much always an increase meaning its value is greater than one and can range up to 300 times larger depending on what you use as a dielectric. Therefore its effect on a capacitor can be thought of as simply changing the permittivity of free space to the permittivity of the material of the dielectric while keeping the rest of the equations that govern a capacitor the same.<br /><br />There are other phenomena to be examined but in general they correspond to the insertion or removal of a dielectric, which does not have many applications as they are mainly used to enhance capacitors.<br /><br /><a href="http://ctluk.blogspot.com/2012/11/physics-toc.html">Back to ToC</a>Chris Lukhttps://plus.google.com/117560829084147159365noreply@blogger.com0