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Blog Post:
Expressing a function f: GF(2⁸) → GF(2⁸) as a polynomial using a Lagrange polynomial
Maurits [MSFT]
I talked about Rijndael in a couple of previous posts: Generating the Rijndael S-box , Efficient multiplication and division in GF(2 8 ) , Sieving irreducible monic polynomials over a finite field , Addition and multiplication table for GF(2 2 ) . I'm going to talk some more about it today. The...
on
4 Apr 2014
Blog Post:
Generating the Rijndael S-box
Maurits [MSFT]
I talked about Rijndael in a couple of previous posts: Efficient multiplication and division in GF(2 8 ) , Sieving irreducible monic polynomials over a finite field , Addition and multiplication table for GF(2 2 ) . I'm going to talk some more about it today. One of the more interesting steps used...
on
3 Apr 2014
Blog Post:
Efficient multiplication and division in GF(2⁸)
Maurits [MSFT]
I talked about Rijndael in a couple of previous posts: Sieving irreducible monic polynomials over a finite field , Addition and multiplication table for GF(2²) I'm going to talk some more about it today. Rijndael does a lot of arithmetic operations (addition and multiplication) on elements...
on
18 Mar 2014
Blog Post:
Sieving irreducible monic polynomials over a finite field
Maurits [MSFT]
Last time we talked about the addition and multiplication tables for GF(2²) ; GF(2 8 ) is relevant for the Rijndael encryption scheme. Joan Daemen and Vincent Rijmen use a representation of GF(2 8 ) where each element is a polynomial of the form b 7 x 7 + b 6 x 6 + b 5 x 5 + b 4 x 4 + b 3 x 3...
on
1 Feb 2014
Blog Post:
Addition and multiplication table for GF(2²)
Maurits [MSFT]
I'm reading Joan Daemen and Vincent Rijmen's book The Design of Rijndael and I'm giving myself a refresher course on group theory. Key to the encryption standard is the Galois field on 256 elements GF(2 8 ). A multiplication table of 256 elements by 256 elements quickly becomes a wall of text, so...
on
30 Jan 2014
Blog Post:
Why is 1 Pascal equal to 94 dB Sound Pressure Level? (1 Pa = 94 dB SPL)
Maurits [MSFT]
Last time we talked about why a full-scale digital sine wave has a power measurement of -3.01 dB FS (Spoiler: because it's not a square wave.) This time we'll discuss why an atmospheric sound which generates a root-mean-square pressure of 1 Pascal has a power measurement 94 dB SPL. As before, dB...
on
14 Jan 2014
Blog Post:
Sample app for RECT functions
Maurits [MSFT]
Riffing on Raymond Chen's post today about SubtractRect I threw together a toy app which demonstrates three rectangle functions: UnionRect, IntersectRect, and SubtractRect. Usage: >rects.exe rects.exe union (left1 top1 right1 bottom1) (left2 top2 right2 bottom2) | intersect (left1 top1 right1...
on
18 Sep 2013
Blog Post:
An attempt to explain the twin prime conjecture to a five-year-old
Maurits [MSFT]
Back in April, Zhang Yitang came up with a result that is a major step toward proving the twin prime conjecture that there are infinitely many primes p for which p + 2 is also prime. In a reddit.com/r/math thread on the subject, I made the following comment as an attempt to explain the twin prime...
on
9 Jun 2013
Blog Post:
Generating sample first names
Maurits [MSFT]
I had a need to write a script that would give me a random first name. I grabbed the top 200 first names for baby boys in the US from 2000-2009, and the same list for baby girls: http://www.ssa.gov/OACT/babynames/decades/names2000s.html Boys Girls Jacob Emily Michael...
on
24 Oct 2012
Blog Post:
Grabbing large amounts of text from STDIN in O(n) time
Maurits [MSFT]
Last time I blogged about an O( n log n ) solution to finding the longest duplicated substring in a given piece of text ; I have since found an O( n ) algorithm, which I linked to in the comments. But my blog post used an O( n 2 ) algorithm to read the text from STDIN! It looked something like this...
on
1 Oct 2012
Blog Post:
Finding the longest substring which occurs twice in a given string
Maurits [MSFT]
I'm reading Jon Bentley's Programming Pearls and one of the interesting exercises was to find the longest substring which occurs twice in a given string of length n . There's a naïve solution where you look at every pair of (distinct) indexes ( i , j ), and calculate the length of the common...
on
28 Sep 2012
Blog Post:
Weighing the Sun and the Moon
Maurits [MSFT]
In an earlier post I mentioned how the Cavendish experiment allowed us to weigh the Earth - to determine the mass of the Earth m E . Newton knew the acceleration due to gravity on the surface of the Earth and was able to use that to find the product G m E ; Cavendish determined G directly, and was thus...
on
20 Apr 2012
Blog Post:
How many numbers are straights?
Maurits [MSFT]
As many people, I have to deal with a lot of numbers all the time (bug numbers, changelist numbers, tracking numbers, ...) and as a mathematician I sometimes notice when numbers have peculiar properties. For example, I recently ran into 152463, which is a straight (made up of consecutive digits, not...
on
8 Mar 2012
Blog Post:
Teaching someone to fish and the AKS primality test
Maurits [MSFT]
This morning my wife (whom I love and adore) woke me up at 3:00 AM with an urgent question. "Hey!" she said, shaking me awake. "Is 19 prime?" ... Like a fool, I answered her. "Yes. Yes it is." Off she went. This is a true response, but not a correct response. I realized shortly afterwards...
on
8 Feb 2012
Blog Post:
What is a perfect score in Yahtzee?
Maurits [MSFT]
The dice game Yahtzee takes thirteen turns. Each turn involves rolling a set of five dice (with two possible rerolls, the player having the option to reroll only a subset of the dice), then marking one of thirteen boxes and scoring points according to whether the numbers on the dice meet the rules of...
on
30 Nov 2011
Blog Post:
Generating primes using the Sieve of Eratosthenes plus a few optimizations
Maurits [MSFT]
When solving Project Euler problems I frequently need to iterate over prime numbers less than a given n . A Sieve of Eratosthenes method quickly and easily finds the small prime numbers; there are more complicated methods that find larger prime numbers, but with a couple of tweaks the Sieve of Eratosthenes...
on
11 Nov 2011
Blog Post:
Linearity of Windows volume APIs - render session and stream volumes
Maurits [MSFT]
We have talked about some of the volume APIs Windows exposes . We have also talked about what it means for a volume control to be linear in magnitude, linear in power, or linear in dB . We have also talked about how to read IAudioMeterInformation and how the limiter can attenuate full-scale signals...
on
1 Jun 2011
Blog Post:
Car Talk puzzler analysis - palindromic odometers
Maurits [MSFT]
Last week's Car Talk question was again of mathematical interest - it dealt with palindromic numbers on a car's odometer. (This is not the first palindromic odometer question they've done.) Read Car Talk's "Tommy's Drive to Work" puzzler question Short version: Tommy goes for a short drive ...
on
28 May 2011
Blog Post:
Translating Ada Lovelace - mathematical science is an instrument
Maurits [MSFT]
Lady Augusta Ada, Countess of Lovelace is credited with being the first computer programmer. The short version: her associate Charles Babbage gave a lecture on his Analytical Engine in Italy; an Italian engineer wrote up a report on his lecture, in French; Lady Ada then translated the report...
on
26 May 2011
Blog Post:
Car Talk puzzler analysis - Limerick equation
Maurits [MSFT]
Car Talk's puzzler of the week has some mathematical interest. See Car Talk's transcript of the question. In brief, we are given the equation... (12 + 144 + 20 + 3√4) / 7 + 5(11) = 9 2 + 0 (which holds true, by the way) ... and asked to transcribe it as a limerick. ...
on
14 May 2011
Blog Post:
Linearity of Windows volume APIs - IAudioMeterInformation and full-scale signals
Maurits [MSFT]
We have talked about some of the volume APIs Windows exposes . We have also talked about what it means for a volume control to be linear in magnitude , linear in power , or linear in dB . The attachment to this blog post contains: An app I wrote to exercise the IAudioStreamVolume, ISimpleAudioVolume...
on
11 May 2011
Blog Post:
Basic audio volume theory
Maurits [MSFT]
Last time we talked about the different Windows Audio Session APIs for setting volume. Let's talk a little about what volume means. For purposes of illustration, let's take our signal to be a full-scale square wave: Incidentally, the answer to the exercise completely characterize the set...
on
4 May 2011
Blog Post:
Generating formulae for polygonal numbers (triangular, square, pentagonal...)
Maurits [MSFT]
The formula for square numbers is easy to remember: s n = n 2 There are also triangular numbers, pentagonal numbers, hexagonal numbers, and in general k -gonal numbers. These have formulae too... for example, triangular numbers: t n = n ( n + 1) / 2 ... but they can be hard to remember. If you see...
on
29 Apr 2011
Blog Post:
How much is a perfect Jeopardy! score?
Maurits [MSFT]
Some sports and games have a notion of a "perfect score." For example, a perfect score in bowling is 300 (twelve consecutive strikes;) a perfect score in golf would be 18 (18 holes-in-one.) After watching Watson destroy Ken Jennings and Brad Rutter I began to wonder what a perfect score in Jeopardy...
on
2 Mar 2011
Blog Post:
Phase and stereo-to-mono downmix
Maurits [MSFT]
Warning: trigonometry ahead. Last time we looked at how to downmix a stereo signal to mono ( M = L /2 + R /2). The resulting M signal can only be as powerful as the L and R signals if they are perfectly correlated ( L = R ); if the L and R signals are uncorrelated (no relationship between L and R...
on
27 Dec 2010
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