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Prime Numbers and Cryptography the sunday tImes of malta maRCh 11, 2018 | 51 LIFE &WELLBEING SCIENCE Prime numbers and cryptography ALEXANDER FARRUGIA factored into its two primes by Benjamin Moody – that is a number having 155 digits! (By comparison, the number You are given a number n, and you 6,436,609 has only seven digits.) The first 50 prime numbers. are asked to find the two whole It took his desktop computer 73 numbers a and b, both greater days to achieve this feat. Later, than 1, such that a times b is equal during the same year, a 768-bit MYTH to n. If I give you the number 21, number, having 232 digits, was say, then you would tell me “that factored into its constituent DEBUNKED is 7 times 3”. Or, if I give you 55, primes by a team of 13 academ - you would say “that is 11 times 5”. ics – it took them two years to do Now suppose I ask you for the so! This is the largest number Is 1 a prime two whole numbers, both greater that has been factored to date. than 1, whose product (multipli - Nowadays, most websites number? cation) equals the number employing the RSA algorithm If this question was asked 6,436,609. This is a much harder use at least a 2,048-bit (617- before the start of the 20th problem, right? I invite you to try digit) number to encrypt data. century, one would have to work it out, before you read on. Some websites such as Facebook invariably received a ‘yes’. A prime number is a number and GMail use a different Most mathematicians before that is only divisible by 1 and by encryption algorithm called ECC the 20th century listed 1 as itself. The number 21 is not (elliptic curve cryptography), one of the prime numbers. prime, because it is divisible by which is not based on the prime However, at the start of the 3 and by 7. However, the num - factorisation problem. 20th century, 1 was debarred bers 3 and 7 themselves are both By the way, what are the prime from being a prime number. prime. On the other hand, the The 2,160-bit public key, in hexadecimal, of a local website that employs factors of 6,436,609? I will not Instead, it was placed in its numbers 21, 55 and 6,436,609 the RSA algorithm to encrypt its data. As its name suggests, this number spoil you with the answer, but I own category, that of a unit. have this thing in common: each is public information and is downloaded by your web browser and stored will give you a hint: one of its Why did this change occur? of them is equal to the product of on your device as soon as you visit the website. prime factors has three digits. To answer this, we need to two prime numbers. first understand what a unit The problem of splitting up a Alexander Farrugia is a lecturer at is. A unit is a whole number number into its constituent amount of time. This fact is the Even though the prime fac - the University of Malta Junior Col - that has a ‘partner’, also a primes is called the prime factor - basis of internet security schemes torisation problem for the num - lege with a PhD in mathematics and whole number, such that ization problem. It is known that such as the RSA algorithm used to ber 6,436,609 looks difficult, it a top writer on the website when we multiply the unit by if these prime numbers are large encrypt online bank transactions is a relatively straightforward www.quora.com, where he writes its partner, the answer is 1. enough, then there is no efficient and social network websites such problem for a computer. Indeed, primarily about various aspects of From this, we can confirm way of finding them in a feasible as Twitter. in 2009, a 512-bit number was mathematics. that 1 is a unit, having the part - ner 1; indeed, 1x1=1. Moreover, no other positive whole num - ber is a unit, because we can - not multiply any whole num - PHOTO OF THE WEEK DID YOU KNOW? ber greater than one by any whole number ‘partner’ and The front page of the very first printed • G. H. Hardy, in his 1940 book A Mathe - get 1. version of the mathematical treatise matician’s Apology , wrote: “No one has The whole numbers satisfy Elements . The Elements is a 13-volume yet discovered any purpose to be a very important property book entirely written by Euclid of served by the theory of numbers and called the Fundamental Alexandria (circa 300BC). This treatise is it seems unlikely that anyone will do so Theorem of Arithmetic. This considered to contain the very first study for many years.” Sixty years later, the property states that any posi - of prime numbers, which Euclid defines in theory of numbers is at the heart of tive whole number that is not Book 7, Definition 11 as ‘those which are internet security! Read the main arti - a unit is either a prime num - measured by a unit alone’. He also proves important results on prime numbers, such cle for more details. ber or made up of the multi - as his famous theorem in Book 9, • The sieve of Eratosthenes is an old plication of some unique list Pro position 20, asserting that ‘prime method used to generate all the prime of prime numbers. For exam - numbers are more than any assigned numbers up to some number N. Essen - ple, the number 59 is a prime multitude of prime numbers’ – in other tially, a list of all the numbers from 2 number, but the number 60 is words, there are infinitely many prime to N is produced, then the first number made up of the primes 2, 2, 3 numbers. The version of the Elements in in this list is placed among the prime and 5 such that 2x2x3x5=60. the photo is in both Greek and Latin and numbers and all the multiples of this In fact, not only is 60 equal to was published in 1573 in Paris. It is number are crossed out from the list. 2x2x3x5, but no other list of currently preserved in the Archives and This process is repeated until the list prime numbers, when multi - Special Collections of Dickinson College, Carlisle, Pennsylvania. An online version is exhausted. plied by each other, can result of an English translation of the entire • The number 11 is the smallest so-called in 60. Elements is available for free at repunit prime, whose decimal repre - Why do mathematicians https://mathcs.clarku.edu/~djoyce/java/ sentation 11 is a string of ones. The exclude 1 from being a prime elements/elements.html and is next decimal repunit prime is number? If 1 were prime, maintained by Prof. David Joyce of Clark 1,111,111,111,111,111,111, the number rep - then the number 60 would University, Worcester. PHOTO: resented by a string of 19 ones. This also be equal to the multipli - https://www.maa.org/sites/default/files/i means, for example, that the number cation 1x2x2x3x5 of the mages/upload_library/46/Swetz_2012_Ma 11,111 is not prime. What are the two primes 1, 2, 2, 3 and 5. This th_Treasures/Dickinson/EuclidisElementor um_title.png factors of 11,111? means that our list of prime numbers whose multiplica - For more trivia, see: www.um.edu.mt/think. tion is equal to 60 would not be unique, and this would contradict the Fundamental tively. Mr Pace’s Mersenne prime is found the natural logarithm of n. The merit of a Theorem of Arithmetic. SOUND BITES by multiplying 77,232,917 twos, and then prime gap is thus defined as the prime gap subtracting 1! This number, if written divided by this average prime gap. For • The day after Christmas of 2017, a 51-year down, would have a whopping 23,249,425 example, the merit of the prime gap old electrical engineer from Germantown, digits! That is almost a million digits more between 317 and 331 is 2.43, which means Tennessee discovered what, up to this day, than the next largest discovered prime, that this prime gap is more than twice the is the largest known prime number. which was discovered in January 2016, average prime gap among those between Jonathan Pace has been a volunteer of the also by GIMPS. 1 and 317. In December 2017, the GapCoin GIMPS project (Great Internet Mersenne • A prime gap is the difference between two network discovered a prime gap of length Prime Search) for 14 years, using free soft - successive prime numbers. For example, 8350 following an 87-digit prime. This ware available by GIMPS to search for so- the numbers 317 and 331 are both prime, prime gap has merit 41.94, which means called Mersenne primes. A Mersenne but no number in between is prime, so we that it is almost 42 times as large as the prime is a prime number that is equal to have a prime gap of 14. By the Prime Num - average prime gap. This is the largest one less than a power of two. For example, ber Theorem, the average prime gap prime gap merit discovered to date. 3 and 7 are Mersenne primes, because they between any two consecutive prime num - For more science news, listen to Radio Mocha are equal to 2x2-1 and 2x2x2-1, respec - bers among the first n whole numbers is on Radju Malta 2 every Saturday at 11.05am..
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