### Christmas 2019

December 23, 2019

Tikalon is on a year-end holiday. Our next article will be posted on Monday, January 6, 2020. Here are a few things to think about on that long drive to grandma's house. Christmas is about nativity, so birthdays are the theme of this article. It's interesting to note that physicist and mathematician, Isaac Newton, was born on Christmas Day, December 25, 1642.

Humans have always been oppressed by conventions adopted from earlier times. An example of this is the number of days in a week. One of my childhood speculations involved how different our lives would be if there were six, instead of seven, days in a week. Would our weeks be structured as four school and working days, followed by two leisure days, thereby giving us weekends expanded by 4.7%? There's the alternative dark scenario in which we have five working days and just one leisure day in that six day week. That might be the topic of a dystopian novel.

Leap years cause a problem for the 0.07% of the population born on leap year day, February 29. In principle, they have a birthday every four years, which would be a tragedy for a child seeing everyone except him/her get an annual party and associated gifts. This problem is solved by using a different enumeration of date. It's either the day after February 28, or the day before March 1.

Saint Nicholas of Myra (March 15, 270 - December 6, 343) anonymously donating dowry money for a penniless man's three daughters by pushing gold coins through a window.

Santa Claus is associated with Saint Nicholas in the Dutch tradition of Sinterklaas; so, Santa's birthday would be March 15, which is also the Ides of March.

(Detail from a 1437 tempera on wood painting by Fra Angelico in the Vatican collection, via Wikimedia Commons image. Click for larger image.)

Birthdays are quite uniformly distributed, although there are some factors that lead to slightly more birthdays one gestation period (nine months) after significant events such as New Year's Day, or the northeast blackout of 1965. Assuming uniformity and ignoring leap years, two people will share the same birthday with the probability, 1/365, or about 0.27%. That's because the second person has just one chance in 365 of having the same birthday as the first.

When we consider a large number of people, such as students in a classroom, we can explore another aspect of birthdays called the birthday problem. I was introduced to this problem in high school during a series of mathematics seminars hosted by Colgate University (Hamilton, New York). In this case, we don't ask whether someone has a birthday on a specific day, but whether any two of those assembled share a birthday. If N is the number of people in addition to the first, the probability P(N) of any two having the same birthday is given as
Of course, having 365 people will give a probability of one by the pigeonhole principle. The surprising thing is that you need just 23 people to give a 50% probability. A graph of probabilities as a function of the number of people in a group is given below.

Graph of birthday problem probabilities. In this graph, N is the total number of people, not those in addition to the first person, as expressed in the equation.

Only 23 people are needed to give a 50.7% probability, or better than a 50:50 chance.

(Graphed using Gnumeric. Click for larger image.)

The birthday problem also has application to cryptography. Cryptographic hash functions are used for storing passwords in coded form and confirming that a document has not been altered (e.g. using a hash in a digital signature). It's difficult to generate a document having the same hash value as another, but it's much easier to generate two documents having the same hash value, an attribute called a collision. A so-called birthday attack produces a good chance of a collision after only 2N/2 codes, while the complete hash space has 2N codes. A 64-bit hash has 1.8 x 1019 possible values, but only 5 billion attempts are needed to generate a collision in a 64-bit hash with 50% probability.

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