Euler’s totient function is defined as the number of positive integers relatively prime to n (including 1). E.g. φ(12) = 4 ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), and φ(15) = 8. http://www.thescienceforum.com/mathematics/14111-modular-multiplicative-inverse-context-rsa.html https://en.wikibooks.org/wiki/Algorithm_Implementation/Mathematics/Extended_Euclidean_algorithm https://docs.google.com/viewer?url=www.math.utah.edu/~fguevara/ACCESS2013/Euclid.pdf as to why the extended Euclid’s algo can be used to find the modular multiplicative inverse https://docs.google.com/viewer?url=ftp://ftp.rsasecurity.com/pub/pkcs/pkcs-1/pkcs-1v2-1.pdf https://docs.google.com/viewer?url=ftp://ftp.rsasecurity.com/pub/rsalabs/rsa_algorithm/rsa-oaep_spec.pdf In particular, ^ page 9. OAEP is the padding scheme ( http://crypto.stackexchange.com/questions/10145/rsa-pcks1-v2-1-rsaes-oaep-algorithm http://crypto.stackexchange.com/questions/2074/rsa-oaep-input-parameters ) , whereas I2OSP and OS2IP (on page 4); what really helped things come full circle for me is realizing how they represent arbitrary data as an integer (first converting it to an octet string). Without further do, let’s test our generated keys by encrypting and decrypting the number 521 (any number smaller than 527, our modulus, will do. ( http://stackoverflow.com/questions/10061626/message-length-restriction-in-rsa ) )

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$ python
Python 2.7.5+ (default, Feb 27 2014, 19:37:08)
[GCC 4.8.1] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> print 521**41 % 527
346
>>> print 346**281 % 527
521

Note that if we try with a larger number…

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>>> print 1000**41 % 527
411
>>> print 411**281 % 527
473

Nope. What really helped things come full circle once again (or full sphere..) http://en.wikipedia.org/wiki/Pretty_Good_Privacy#Confidentiality http://superuser.com/questions/383732/how-does-ssh-encryption-work It makes more sense to use a symmetric encryption algorithm with high throughput to encrypt the data first, then use PKI to encrypt and transfer the key. And that is how the world works.

What is a proof by contradiction? A proof by contradiction is if $\neg P \Rightarrow F$ is true.

One assumes that a proposition P is False, and uses that to derive until a contradiction is reached, which can’t be True.

A popular example: Let’s prove that $\sqrt2$ is irrational. An irrational number is something that cannot be expanded into a fraction. (A common misconception is that Pi is $\frac{22}{7}$ and therefore rational; no it is not exactly $\frac{22}{7}$.

See http://mathworld.wolfram.com/PiFormulas.html

Assume that $\sqrt2$ is rational; such that we can represent it as $\frac{a}{b}$, where $\frac{a}{b}$ is a fraction in lowest terms.

$\Rightarrow \sqrt2 = \frac{a}{b}$ $\Rightarrow 2 = \frac{a^2}{b^2}$ $\Rightarrow 2b^2 = a^2$ $\Rightarrow 2 \mid a$ $\Rightarrow 4 \mid a^2$ $\Rightarrow 4 \mid 2b^2$ $\Rightarrow 2 \mid b^2$ $\Rightarrow 2 \mid b$ If both $a$ and $b$ are even, they are not in lowest terms, as both can be divided by 2 for further simplification. Hence we have a contradiction. $\square$

http://sbp.so/supercombiner

Helvetica Scenario.

http://www.dailytech.com/Bitcoin+King+Pt+II+Mt+Goxs+Dictator+Karpels+Proves+Tragically+Flawed/article34452.htma What goes on the internet, stays on the internet.

Imagine having Satoshi Nakamoto for a father. http://arstechnica.com/business/2014/03/bitcoin-creator-satoshi-nakamoto-revealed-after-years-of-mystery/ He is your usual cranky old man in his 60s. His supposed daughter had this to say about him:

‘He is very wary of government interference in general,’ she says. ‘When I was little, there was a game we used to play. He would say, ‘Pretend the government agencies are coming after you.’ And I would hide in the closet.’

He was a software engineer for the FAA, has worked on military systems, and has done work for Hughes Aircraft at RCA.

http://arstechnica.com/business/2014/01/protestors-show-up-at-the-doorstep-of-google-self-driving-car-engineer/
http://www.motherjones.com/mojo/2014/01/bay-area-google-protests-uncomfortably-personal

The activists claim high-paid tech employees are squeezing out the middle and working classes from the city.

Perhaps it has never occurred to them that they are living in a meritocracy. Perhaps it has never occurred to them that Google has competitors, too…

http://today.duke.edu/2014/02/reproducibility The proposed solution is integrating the raw data, the computing, and analysis into one package as opposed to using different software tools for different stages of research which may make it cumbersome for students to back-step.

Enter R Markdown, a statistical package that integrates seamlessly with the programming language R. The team chose R Markdown for its ease of use – students wouldn’t have to learn a new computer syntax – and because it combines the raw data, computing and written analysis into one HTML document. The researchers hoped a single HTML file would give students a start-to-finish understanding of assignments, as well as make studying and grading easier.

What the hivemind had to say on this: https://pay.reddit.com/r/science/comments/1z8w5k/scientists_propose_teaching_reproducibility_to/

http://preview.tinyurl.com/kctg2k9 Warning, link contains profanity. Here is an excerpt:

There’s a moral to this story boys and girls. Inform, teach, explore. Don’t gain the elitest attitude that only the few computer guru’s deserve the k-nowledge. That will get society nowhere. If something is not done to show Americans the light, then it will all go to hell. All computers will turn into entertainment systems. Inventiveness will be left to the evil, who’s only goal is to get money from you. If you are in disbelief, just turn on your T.V. Didn’t MTV start out as a new media form? Now-a-days, theres about .1% M in MTV, and about 99% ads and manipulation. Lets give a wakeup call to the world. Lets make it big and loud. NOW, not tomorrow.

http://programmers.stackexchange.com/questions/60544/why-do-game-developers-prefer-windows tl;dr OpenGL had many functions like multitexturing, shaders, hardware T&L support for quite a long time…. with vendor-specific extensions. MS had a lot of time to mess up and learn, and it didn’t help that there were so many complete rewrites of OpenGL by so many different vendors before the ARB came in and provided their half-hearted fixes. Fast forward to the launch of Vista; MS slipped up again by not providing D3D10 support on XP; and Vista was, well, not the best OS of its time. D3D10 hardware runs D3D9 like a walk in the park, and game developers still provide DX9 compatibility (a year later and Crysis still supports DX9.) OpenGL got their stuff together too late with the release of OpenGL 3.0 with GLSL, and Win7+DX11 is taking everyone by storm. New cards like the HD 5850 are nearly twice as fast as the previous ATI RV770 cards, and OpenGL goes back to picking up the mess made after their mad race.