Sarah Flannery became famous at a young age of 16 when her algorithm for cryptography was speculated to be a far better alternative to the widely adopted RSA algorithm. She presented an algorithm at the Young Scientist Competition with no fan fare and she won the competition. This book would not have been written but for one reason , the competition result got picked up by “London Times “ and an article appeared on the front page with a nerdy picture and a catchy title , ”Sarah Flannery, 16 , who baffled the judges with her grasp of cryptography”. Subsequently “Reuters” published the story , which put Sarah, her family, her life till then, under public eyes. She was inundated with calls from journalists, TV shows, firms offering her employment , VCs wanting to give seed capital , universities asking her to give a talk to their students etc.
This book is an attempt by Sarah to shed light in to her mathematical journey and the various factors that helped her to create the algorithm, to a far wider audience. The book has been coauthored by her father David Flannery.
Sarah Flannery hails from a village of Blarney,County Cork, Ireland where she attended a local girls school, a very ordinary school by the usual standards. Her father was a math lecturer at Cork Institute of technology. What brought her to the world of math ? Sarah credits her father for introducing her to the world of math in a unique way, “using puzzles”. David Flannery believed that math needs to enjoyed for one to be creative. One way to enjoy math is to solve puzzles which bring out mathematical concepts. So, he cultivated this puzzle culture at home.
At a blackboard in the kitchen, the father & daughter would discuss math puzzles and like a good coach, Sarah’s father merely provided directions and subtle cues and almost never gave the entire solution at once. This culture of solving puzzles and engaging one’s mind over puzzles was the main reason for Sarah to develop a liking towards math. It made math more tangible and interesting . Also , Sarah’s father never allowed to let go an opportunity to extend a puzzle once it was solved. He used to extend the solved puzzle to create another puzzle , a little difficult one , and scribble it on the blackboard. What he was doing through this culture at home was basically trying to inculcate the axiomatic method of mathematics where you build up theories over a nice structure of axioms, theorems, lemmas. For Sarah, learning through Puzzle was one of the fundamental ways to understand math.
In the book she says
Puzzles, like humour , have an universal appeal and know no boundaries – cultural , educational or otherwise. People of all ages and levels of education are attracted to the puzzle as they are to the joke. In a sense, there is an affinity between the two in that a vital ingredient of both is the element of surprise. No problem is worthy of the name “puzzle” if its solution is obvious, just as the joke whose punch line is easily anticipated is soon forgotten. The true puzzle should be accessible to all, its solution should require no specialist knowledge other than, at times, the rudiments of arithmetic and algebra. It is perhaps, the unconscious feeling that we all start out equal that gives puzzles their charm
Transition Year
This book talks about a specific aspect of Irish educational system that I found very interesting . After the end of high school, the students in Ireland can opt for a one year Transition Year Programme before moving on to 2 year college education. The purpose of transition year is described as follows at an Irish school site.
The overall mission of the Transition Year Programme is to promote the personal, social, educational and vocational development of students and to prepare them for working life. The course is designed to cater for the needs and capacities of students within a framework of broad general education which would have a substantial academic base. The course also includes a wide range of practical applications, activitybased learning, team and group work and work experience.
The Transition Year Programme acts as a "bridge" from the highly structured environment of the Junior Cycle and to the much more independent, responsible and mature attitude towards work, study, school and interpersonal relationships of the Leaving Certificate Programme.
Almost every parent in India is anxious at the time his/her kid moves from school to college. From a highly structured and regulated environment, the kid moves to college environment where the learning typically is selfregulated/ semistructured. Most of the kids cannot handle this sudden burst of freedom and mostly fail to use it constructively. This transition year concept sounds very appealing to me and it is definitely an idea that can be explored in the context of Indian education system. Imagine a scenario where the kid after 10th class has one year time to explore his/her interests / take time out to read up stuff from varied fields / take up mini community projects / explore some programming skills / experiment with stuff etc with no pressure of performance..It would be immensely beneficial to parents too as they can get an idea of what their kids truly enjoy.. They can then guide the kids in that direction.A one year break at the right time in a person’s life might do wonders and at least increases the chance of finding interest areas. However I see resistance to this idea in Indian educational system where everybody is in a hurry to graduate out of some place or the other:).,I hope at least a few parents encourage their kids to take a year off and explore stuff.
It comes as no surprise that Sarah found her calling during the Transition year where she developed a liking towards number theory and Mathematica package. Sarah spent the entire transition year learning about prime numbers, their distribution, their properties etc. By the end of the transition year , she had pretty much decided that she would explore cryptography field besides going through college education.
Sarah then mentions a few essential math stuff needed to understand her improved algo. The details mentioned does not need any prerequisite knowledge for understanding. She goes over the following stuff :
1. How were enciphering and deciphering systems created in the olden times ?

The basic math behind enciphering and deciphering text depends on the modulus function. It starts off from the Caesor system which can be summarized as
“ C = ( P + s ) mod 26, P = ( C  s ) mod 26 “. C represents enciphering and P represents deciphering system for the 26 alphabets.The above system is essentially a single key system,meaning the only thing that is essential to crack the system is key s.

The above system can be refined as follows :
“ C = ( m1*P ) mod 26 , P = ( m2*C ) mod 26 “.This is a two key system where instead of additive shift, there is a multiplicative shift in m. Even for such a simple system one needs to know the multiplicative inverse of m1*mod 26. Appendix provides the Euclid’s algorithm for generating the multiplicative inverse of m1*mod 26

The above system again can be refined as follows :
“ C = ( m1*P + s ) mod 26 , P = ( m2*C + s ) mod 26”. This is a three key system where there is additive shift, and a multiplicative shift in m.

The above system again can be refined as follows:
” C = ( m1*P + s ) mod 676, P = ( m2*C + s ) mod 676”. Here the key space is improved by considering digrams

The above system again can be refined as follows:
” C = ( m1*P + s ) mod 17576 , P = ( m2*C + s ) mod 17576 ”.Here the key space is improved by considering trigrams At a broader level, there is a onetoone function mapping C and P and hence the system components can be summarized as keys,key space and an invertible function.
2. What sort of Arithmetic is used in Cryptography ?
3. Why are oneway functions & trapdoor functions important ?
4. What is Public Key Cryptography and What is the concept behind RSA algorithm ?
With this background provided in the book, Sarah resumes her story.Sarah working as a apprentice in Baltimore technologies meets Dr.William White who then shows a paper by his colleague Dr. Michael Purser. Sarah picks up a lot of stuff from the paper and then tries to come up with an alternative algo to RSA , an algo which is 30 times faster than RSA by exploiting the noncommutative property of matrices. Basically RSA uses exponential calculations whereas Sarah’s also uses multiplication operations.
The last part of the book shows the various phases of preparation that Sarah goes through before winning the Young Scientist award. She decides not to patent the algo and opens it for Peer review. Very soon she finds that there are cracks in the algo and all the publicity she got was unwarranted. She realizes that media will jump on to anything saucy and actually don’t care about the veracity of the news item. Meanwhile, cracks in her algorithm make her return to sanity and get back to life where she can do further research in math. Her decision of not patenting algo actually was pivotal as it helped her see chinks in her math and thus provided the much needed breathing space to continue research in cryptography.
This book is a great story of a youngster who uses her time to learn and implement an algorithm , to challenge the prevalent RSA algo , which was originally designed by 3 MIT guys. A school kid having the guts to challenge the RSA algo is definitely an inspiring read. It goes on to show that , if you have to have the right attitude towards math , you can churn out kickass stuff, irrespective of your age.