They say some are born great, some achieve greatness and some have greatness thrust upon them. In case of Sarah, did she achieve greatness? The answer to this question is both yes and no! How she can achieve greatness, where was the time for it?

She was just sixteen, a tender age not even good enough to fall in love, and yet the word ‘great’ was firmly implanted on her. Was greatness thrust upon her? The answer to this question is firm ‘No!’ The part of the statement that befits her is that she is born great. Her illustrious mathematical journey had arrived at the sterling landmark at that young age.

This work is both knowledge and enjoyment. The brilliance of the students comes to the fore at the Undergraduate and University levels. That is the time when the Professors spot the intelligent, brilliant and the brilliant among the brilliant students. But Sarah’s extraordinary brilliance must have been spotted when she was in the kindergarten. This is so, because, her father was a renowned Professor of Mathematics.

Academic studies do matter for students like Sarah, but such genius children invariably study in the College of Self-Education, where their mind is their Principal. Their initiative, their Professors! Their hard work, their tutors! They are ambitious and industrious. They decide upon a project; start, act and finish.

“The eldest among five children, Flannery went to high school at Scoil Mhuire Gan Smal in Blarney, where she gave a science fair presentation on cryptography. In researching RSA encryption, she created her own encryption algorithm, which earned her an Esat Young Scientist Exhibition and later an Intel Fellows Achievement Award.

She went to college at the University of Cambridge, graduating in 2003 with a degree in computer engineering. Now she works for American video game developer Electronic Arts.”(Planet...)

In brief, what the book is about….

This book is the mathematician’s delight. How the serious subject like mathematics, can be interspersed with humor. Her introduction to the book itself fascinates and kindles curiosity. By reading it, you realize why she attained instant fame which she richly deserves.

In introduction she gives details about public-key cryptography, the RSA algorithm and the alternate algorithm that she created. The lucid style and control on the language required to deal with mathematical explanations is the strong point of the book.

Genius has nothing to do with the age. Therefore, forget for a while that a teenager is the author of the book, yet you can not forget that lovely and inspiring face of the author in the cover page of the book! Unless known in advance, you will find it difficult to believe that she is the owner of such a brilliant brain. You probably think that this book is the creation of an experienced Professor, well researched from a long list of bibliography.

The book begins in an orthodox style. Instead of saying something about the author straightaway, the book gives description with pages of family background. That again is about mathematics, because her father was the Professor of mathematics. A long section, titled, “Early Challenges” follows. This is where the challenges for the readers are thrown. They are recollections of her past.

The mention of about a dozen puzzles is made. She owned them from her father, told for the benefit of Sarah and her brothers. David Flannery, their father was the mentor of mathematics as well. The description for each puzzle is given along with an invitation for the reader to try to solve the same, before trying to locate the solution provided in the book elsewhere. Your chances of success are not very bright. A black-board in the dining room!

That is bound to happen when you have four sons and a daughter, all interested in the serious study of mathematics. On the blackboard, new puzzle was recorded each day for them to work on. Like, "Given a five-liter jar and a three-liter jar and an unlimited supply of water, how do you measure out four liters exactly?"

Some got into more difficult concepts: "How might you determine the average earnings of a group of people in a room (at a class reunion, perhaps) without any individual's divulging his or her salary?" the second puzzle has the latent message, which forms part of the theme all through the book.

That is to try getting information and hiding information. With such intelligent invitations the readers become part of the proceedings in the book. It is no effort to score over the reader but to win the reader in a positive style. It is to encourage him, and no to affront one with the exhibition of intelligence. Those who think that this is a book on the subject of mathematics which is bound to have the serious start will have peasant surprise to be greeted with mind-boggling puzzles.

Fame and Publicity and thereafter….

She received instant fame, became a celebrity in mathematics overnight, but she was humble about her achievements. Here she speaks beyond her age refuses to expand like the balloon with inflated ego. “I have no doubt that I am not a genius,”she declares.

“I am not being falsely modest. Through my father’s classes I have seen examples of true genius and I know I do not possess that ‘insight’ that distinguishes geniuses from those regarded as merely intelligent.”(p. 243). Those who read the book were sure about her impending success in the world of mathematics. She was the worthy mathematics daughter of a worthy mathematics Professor.

Next to puzzles….

She attempts the most original aspect of the subject of mathematics problems that take her to the position and fame sky-rocketing. Overnight she is a world-figure as for mathematics. The subject matter for any national level competition has got to be the unique one. She was to enter the Esat Irish Young Scientist’s Compettion-1998.

The project on cryptography was done by her at the suggestion and instance of her father. Her project will throw light on various cryptographic techniques, providing the account of the famous RSA algorithm. The mention and discussion of all these things is initiated at page 40 of the book, she discuses learning the relevant mathematics and the programming involved.

Then her father takes over the mathematical literature for the next 110 pages, and you catch up with Sarah again in page number 150.These pages contain mathematical exposition authored by David Flannery. They provide the basics of cryptography, and to understand it, is not the easiest of the propositions. He introduces thoroughly RSA algorithms at this stage.

When Sarah takes over at page number 150, she is at the threshold of the fame that is about to engulf her young personality. She gets tremendous response for her project that fetches her several prizes, and she is inspired to prepare for another ambitious and prestigious entry –the Rafe Jones at Brown University.

She is on the second step of the ladder of success. She undergoes a week-long internship at a Dublin cryptography company, and notices several techniques. Thereafter, she devises an alternative algorithm to the RSA and that is the flagship issue of her new project.

The results achieved by her are astounding. Her method is simple matrix multiplication instead of the relatively cumbersome modular exponentiation of the RSA. Her algorithm runs twenty times faster. It is christened by her as the Cayley Purser algorithm, the 19th century British mathematician Arthur Cayley as also Michael Purser, the mathematician whose ideas caught her imagination during her internship.

She runs in to the thick of the issues now-she proves how the new algorithm is secure from certain kinds of attacks and it becomes the mathematical odyssey for her, wherein she is required to explore and master the cobweb of not too familiar mathematics. Such a situation is the testing time for any student of mathematics.

It was probing the new waters for the first time with lots of hopes of positive results, but also with the fear of disappointment, should anything go wrong, at the most unsuspected moment. Mathematics is such a subject where there is no scope for errors. You have got to be accurate, as otherwise the whole edifice built by you step by step will collapse.

“All of this was an unusual experience for me,” she writes, “but I had a great feeling of excitement. I think it was because I was working on something that no one had worked on before. I worked constantly for whole days on end, and it was exhilarating” (p. 208).

To get the worthy solution, you need to have a strong problem. That was the situation Sarah, luckily found she in. The thesis problem provided lots of enthusiasm to her to go ahead. With the finalization of the thesis problem, she considered her to be lucky.

The problem was of her creation and she was bent upon to own the responsibility to solve it herself. She began to put extra efforts and worked desperately to find the authentic solution.

She reaped the dividends for her sincere efforts, she was able to prove that Cayley Purser algorithm had strong defenses and it can withstand successfully attack from a large family. She provides the detailed description to judge her new project, both algorithm and proof, in the 1999 Irish Young Scientist competition.

An inventive mind is always excited about any new achievement-you decide on a problem for you and then solve it successfully. Same was the case with Sarah.

When you smell success, when you have positive indications that you are nearing solution, the excitement is all the more. At that stage Sarah burnt the midnight oil to continue her sincere efforts. Success had to kneel before her; she showed the strong defense for the Cayley-Purser algorithm in the face of multi-pronged family attacks. She gives the detailed account, step by step, how her project needs to be judged, and the related explanation for the algorithm, with unassailable proofs.

In the 1999 Irish Young Scientists Competition., she quotes from her journal, “On one occasion,” she writes, “I looked out of our little huddle and it felt really strange—our conversation was so very intense that just to look around was like coming up for air” (p. 222). The finest moment of her judging was, she writes: “Before they left, [the judge] asked me the simplest question of all, and I could see he was wondering whether or not I would be able to answer it.

The answer was the fast exponentiation algorithm, and I must have smiled before I replied, because I knew it was the perfect end to the perfect session. I had been able to defend my project at all levels. The last question was a check to see if I knew the fundamentals. They smiled at each other on my final answer, which I’ll never forget.” (p. 223).

It was the perfect culmination. The excitement was about how gracefully Sarah walked up to the stage to accept the title of Irish Young Scientist of the Year. It was one of the extraordinary moments of her life. The charm of youth was on her side.

And the algorithm she talked held excellent possibility of rich dividends. The inquisitive media stood alerted and the unexpected bonus arrived, when London Times front-paged an article on her mathematical exploits. Overnight she entered the portals of stardom in mathematics. It did not take long time to transform her academic achievements towards the commercial gain.

The would-be cryptograph entrepreneurs were seeking her services. She received many offers to give lectures in Singapore. Mention of her name was made in the official magazine of the Spice Gils. She also received a request from Profile Books in London to write up the experiences and all that prompted to advance on the tough path of mathematics.

The budding young mathematician’s book had the firs print order of 35,000 copies, the marketing budget of $ 65,000 and an eight-city author-tour.

About the contents of the book, Sarah often sweetly apologizes for going deep into number theory; before explaining matrices, she writes, "I promise that from then on there will be no more explicit mathematics, only light explanations of mathematical ideas."

The hard-core mathematics in the book is restricted to two chapters. For those who wish to learn more, there are appendices. Her main project is about how the most famous current encoding system works, and in the meantime, she had invented one of her own. She takes extensive pains to explain both the systems and goes deep into the number theory along the route.

With the winning of the prize, fame and the monetary gains consequential to the fame arrived like an avalanche. Pepsi wanted her to concede that the mathematics brain and that of the family was due to lavish consumption of Pepsi, but the offer of contract was promptly turned down. A good mathematical calculation viewed from the humanitarian angle; they were aware perhaps of the harmful effects of such addictive drinks on the health of the younger generation!

“I have no doubt that I am not a genius,” said the prize winner of the 1999 European Contest for Young Scientists. Bu who would believe her and at the same time remain without deeply appreciating her modesty! She was the media sensation within days after getting the prestigious award. She was about to be hailed as an instant celebrity, for the public key to cryptography, the method used to transmit secure data over the Internet, but destiny played its part.

When everyone in the knowledgeable circles thought that her encryption algorithm is worth the millions, a security hole was discovered. Nevertheless she had done a great job. Now she met her father on equal terms, a mathematician talking to another senor mathematician.

There is an interesting interaction between the father-daughter mathematical duos. To be taught lessons from her father in the drawing room of the house was one thing. To be part of his lecture fraternity, and listen to his mathematical revelations in an organized way sitting as one among the audience was altogether a different experience. The previous day, he had a serious, purposeful conversation with Sarah. Her father said, "Now that you have decided to do transition year, I must do some math with you." He continued, "I'd like to show you how some beautiful but reasonably elementary mathematics is applied, stuff that you wouldn't ordinarily come across in school."

She could not understand the immediate intentions of her father. She thought he was inviting her to the kitchen blackboard, as she was aware of his enthusiastic ways; how he got inspired at the most unsuspected moment and wished to unleash his mathematical knowledge on her, whether she was mentally prepared for it or not.

Perhaps, at that moment she was not ready to receive the tough lessons of mathematics. She replied, "Dad, whatever you do, do something structured!" That set him thinking, as to the proper, most effective and appropriate way to teach and take her to the world of mathematics. He remembered his past. The debt he owed to the one who taught him mathematics. How the torch of mathematical knowledge was passed on to his hand.

It now depended, what he would do, with what his teacher did for him, and from where he left. He strongly felt that he must transfer the knowledge to some one else who richly deserved it. Who else could be that individual except his own daughter, in whom he must have noticed the latent mathematical genius?

He told his daughter, "Of course, only if you are genuinely interested-I wouldn't force it on you." She was genuinely interested. The evening lecture by her father proved to be the foundation stone for the grand mathematical edifice that she was going to build.

Sarah was now part of the class of serious students studying mathematics. The evening classes from seven ten, with the student strength of 8, continued for twenty five nights. The daughter was the youngest student, just 15 years and six months, but a couple of other students too were young.

There were adults who came from various backgrounds. Computer scientists, a secondary school mathematics teacher, a chemistry graduate working for a medical laboratory. They were the ones who loved mathematics, and who regretted their inaction in not pursuing the mathematics study, when young. It was a class that had its own specialties.

The cause of study was great, not the career out of the study. No credits were given. It was not part of any major. No home work or study was demanded. It was David Flannery’s way of “getting back into math” with no holds barred approach. You wee encouraged to come up with your most silly questions. He loved and appreciated those who made fools of themselves, as according to him, only those will learn and had the chance of success.

These assurances, coming from a reputed Professor, were greatly appreciated by the students. They looked forward to the classes with expectancy and with hope that they will be exposed to something interesting about the mathematics in the next session.

Sarah though had peculiar problem that the teacher was her father, she was able to mentally sort out the issue. Her dedication and the serious approach when she meant study, paved the way or her progress.

From the teacher’s perspective, what one teaches is important. But how one teaches, what one teaches is more important. Towards this end David Flannery filled the bill admirably. He was the one who enjoyed his teachings and encouraged his students to learn in style.

No time-bound hard tasks were expected from the students. Each one was encouraged to estimate one’s level of understanding, and progress accordingly. He was able to maintain an atmosphere of affection at home and in his classes.

The brain teasing puzzles enticed the students to know more and more, and the hours spent by the students in the association of David Flannery, proved highly fructifying. His company kindled their curiosity further. They eagerly awaited his next class and firmly believed that something more interesting would be in store for them.

The class was not all fun as was made out initially by David Flannel. That was his style of making the students interested in the subject. Soon, the intensive part of his chartered syllabus for the students began and his forays were in the elementary number theory, with cryptography as the final destination.

He would the take the students entirely to different horizons of mathematics, interesting sights that were rich in content, may not be of use for the immediate application.

When David Flannery was a student, the application of the number theory were so few (industrial and internet revolutions had not taken off), and yet the number theory was of prime importance to the mathematicians, its study was considered pure. By the end of 1970s, the situation had drastically changed.

The various technicalities involved in the message system and their readings gave a shot in the arm for cryptography. It became a much sought after subject and millions of dollars were invested into the development of this subject. The demand for expertise in this area became tremendous.

The book is a treat to read for the simple reason that it is a great human story as well, a success story, a management and pubic relation story etc. The way she prepared for the competitions would set the standard for any youngster who wishes to be an achiever.

What a careful and great teacher her father was! The concepts of teaching itself have undergone metamorphic changes, and mathematics is no exception. The style of teaching has become more student-friendly.

As for Sarah, her mathematician father was her great career-asset. But her mother also continuously encouraged her by telling interesting anecdotes about the subject. ‘Mathematics is the queen of the sciences and number theory is the queen of mathematics.’ Such sweet nothings said about the subject during formative years of a young girl had great positive impact on her.

Her Mom said Mathematics and the number theory was like the Sleeping Beauty Fairly Tale, and Sarah Flannery writes, “I thought about those who had toiled away through the centuries at unraveling the mysteries of this subject, motivated by nothing more than a passionate desire to know. They could have hardly dreamt of the applications that some of their results would one day find.

I wondered what it was they had discovered, and what they would think if they could see how some of these discoveries are now being used. I was eager to learn the subject and surmise for myself whether they would be surprised or not.”

But everything about the number theory is not all that sweet. It is a deceptive theory. When you think that you are on the verge of success, you are suddenly knocked out by a lethal punch. The simple questions that you will ask your Professor may look so simple to you. The same questions were asked by many brilliant students in the past. The answers to such questions have not been found until this day, and the most intelligent ones are raking their brains to find out the solutions.

But the toughest obstacle lay in waiting for Sarah. The sharp attacks on the Cayley-Purser algorithm arrived with Michael Purse alerts, making her mathematical advancement difficult. Sarah was on the defensive; she made efforts to repair the algorithm, but could not succeed. She stated that it is not salvageable as a workable encryption system.

The theoretical interest stood in her favor through this testing time. She included a postscript explanation on the successful attack. It brought her further success when she was conferred the title of European Young Scientist of the Year for 1999. They say, “When the going gets tough, the tough gets going!”

It is very easy to say that Sarah climbed the stage to receive the award. But behind this glorious moment in the life of this sports-loving teenager from Blarney in Country Cork, Ireland, lay the extraordinary talent and matching efforts of relentless research and discoveries in Internet cryptography.

At the age of sixteen to get the international recognition and to be hailed as “brilliant” by the London Times, is no ordinary achievement. Newspapers and periodicals hailed it as "a wonderfully moving story about the thrill of the mathematical chase" (Nature) and "a paean to intellectual adventure" (Times Educational Supplement).

The dinner-time conversation with her father led her to the hall of fame. Her burning curiosity, the inner joy of persistence paid off handsomely. What is the meaning of the wise saying, “Have a will to grow and grow you will!”—ask Sarah! David Flannery lectures on mathematics at Ireland’s Cork Institute of Technology. Sarah Flannery is now a student at Cambridge University.

Some shortcomings of the book….

To say that it is the combination of a set of two books, would not be a far-fetched criticism on the book. Two narrative segments of the book, at the beginning and the end, do not serve the actual cause for which the book stands for. The 150 pages, though there is no doubting the merit of the contents, it is right thing at the wrong place. It makes the tough and prolonged reading, and causes obstruction in understanding the life of Sarah in a systematic, chronological order.

The book goes on an aim-less wandering, and it defeats the structure of the book. But those who do not like mathematics intensely, for those who are not the serious students of the subject, these 150 pages are a good read! However, this can not be considered as the serious lapse of the book, but the professional critics of this literature, would not like to miss this point, as a matter of their duty.

The mathematical exposition part of it is flawless in contents and style. The beginning holds the interest of the common reader as well. The elementary examination of the prime numbers is detailed. The idea of primality about Mersenne primes, the Sieve of Eratosthenes and also primalaity testing are the important topics.

Then is the chapter on Modular arithmetic, Fermat’s Little Theorem, and pseudo primes. For complete understanding of the RSA algorithm, the last two of these three mathematical chapters are necessary. But they can be avoided by those who are desirous of the elementary feel for public-key cryptography.

The author has no problem with the English language, the exposition and style of writing is lucid. In fine, Sarah has given a very interesting book. The theme of the book and the author’s attitude towards the theme of mathematics, both evoke and sustain curiosity. The book deserves an outstanding position not only for the awards that it brought to Sarah, but also for the real contribution it made to the cause of mathematics, for all time to come.

There is something uniqueness about the book. Some more remarkable books should be expected from Sarah’s pen. The author has dealt with the topic in an excellent manner and the contents of the book gain stature, without losing its ground and reality. The book is the creation of the child, with father’s blessings. What more is required for the child and what does a father expect from the child? Both see the fulfillment of their respective mathematical missions through this book.

When a child gives the account of her experiences that ultimately led her to great heights of success, the reader’s interest is all the more, for the simple reason that every parent wishes to visualize and aspires for the success of their children.

From the point of view of human psychology and management principles that lead to success, this book is the torchbearer. Her thoughts are playful, yet never missing the mathematical purpose of the book.

The number of high school students who enjoy mathematics is not very high and many do the subject, as they have to do as it is the unavoidable subject up to a certain level. But an introduction and interaction with this book should change their perspective.

Many may not be lucky enough get a mathematics genius Professor-father. In the case of Sarah, her pastime of solving the mathematical puzzles with her father paid her rich dividends. It started as a game and slowly turned into a thrill of the mathematical chase. Nothing succeeds like success and one good thing in her life, led to the other.

A great lesson for the psychologists and sociologists who study the parental impact on the life and future of children! But for the encouragement and influence of her mathematician father, Sarah would have been yet another university student, pursuing the syllabus-oriented degrees for a routine career.

Conclusion:

God made the natural numbers, 1,2,3,4,5,6,7,8,9 and 0 and what all ‘complications’ the human being has created under the subject-banner mathematics! But for becoming the mathematician, Sarah would have been a storyteller.

The art comes naturally to her. She has dealt with a highly intellectual subject, without any pretensions, without the sense of overbearing. At times, the writing takes the serious turn, but that is what the subject matter is. Her childlike descriptions maintain the essential dignity. If she tries her hand at other subjects of writing, we may be in for surprises.

That’s what her imagination, wit and charm throughout the book reveal. “In Code”, has the makings of a very good novel and her pen holds out great promises for even better works. It is a twice-blessed book. Primarily it is a book on mathematics, and more importantly it is an interesting book of human endeavor, the human spirit, the book on positive qualities like grit and determination. Few children are lucky to have such great upbringing.

The black board in the kitchen truly speaks about the studious family. She and her four brothers made it the perfect class at home, and mostly issues related to mathematics were discussed at the dining table. At lunch, the distraction would be not be like the television set in the modern drawing room, but the attractions were puzzles appearing on the blackboard days after day, without intermission. David Flannery had a clear purpose about the puzzles.

They encouraged the children into enjoying abstract reasoning. But do no imagine, Sarah liked all these intensely. She was not a book worm. She liked to ride horses, played hurling and basketball, did boating and liked other adventurous team sports.

She carried that adventurous spirit to her mathematics research as well.

So to say, Sarah was a philanthropist mathematician and did not possess any motivated desires about her accomplishments. Mathematical community offered the talented Sarah all co-operations. Experts in the field of cryptology were eager to help her.

Initial reactions to her code system, was that it was patentable and she had the possibilities of becoming the millionaire. The exchange of vital information with other mathematicians could have damaged her interest for financial gains. Yet, she shared the information bearing in mind the overall interest of mathematics.

She was only 17, when she was a guest speaker at an IBM leadership conference for women. Sarah was also given an invitation to attend the Nobel Prize ceremonies in Stockholm. To write an interesting readable book on mathematics, normally considered a dry and brain-racking subject is no mean achievement. She has set the trend for the budding young scientists, and great scholars in any subject need not be men and women with silver hair.

The book provides to all concerned, the parents, the teachers and above all the combustible younger generation, who wish to achieve something in life, but do not know the correct procedures and steps to achieve very valuable information. Sarah provides good solutions to one’s ambitions and the way to achieve them.

This makes the book even greater, than her contribution to cryptography. Bertrand Russell once talked about ‘the silent beauty’ of mathematics. Sarah has demonstrated how the skills of the mathematician and the skills of the fiction & fantasy writer can be clubbed together to create an outstanding contribution to the world of literature. Here is the combination of great human experience mixed with intellectual stuff. It is very easy to record and offer her congratulations for all that she achieved at the young age of 16.

But think of the hard work she did, the relentless pressure she was able to endure at such a young age, and all this she did at the same time enjoying and pursuing her hobbies. So, if you have young children, and do not have many ideas as to how to inspire them, give this book to them—it would be more appropriate if the parents read and discuss the contents of this book for their benefit. The results are bound to be far fetching as for their future. The contents of the book have the lessons in ‘moral counseling’ as well.

The application of cryptography has caught up fast with the internet revolution. Many of the big companies are willing to sponsor researches and Sarah is eminently suited to take advantage of this situation. Her achievements have changed her perspective of life. The career opportunities that arrive at her doors are perhaps too much for her to handle. She has traveled to important destinations all over the world, met cryptography figureheads like Ronald Rivest and Whitfield Diffey. But the best is yet to come, and Sarah Flannery knows it well.

References Cited:

In Code: A Mathematical Journey (Paperback)

by Sarah Flannery (Author), David Flannery (Author}

Paperback: 352 pages

Publisher: Algonquin Books (December 30, 2002)

Language: English

ISBN-10: 1565123778

ISBN-13: 978-1565123779

PlanetMath: Sarah Flannery

planetmath.org/encyclopedia/SarahFlannery.html - 19k – Cached, Retrieved on January29,2008

She was just sixteen, a tender age not even good enough to fall in love, and yet the word ‘great’ was firmly implanted on her. Was greatness thrust upon her? The answer to this question is firm ‘No!’ The part of the statement that befits her is that she is born great. Her illustrious mathematical journey had arrived at the sterling landmark at that young age.

This work is both knowledge and enjoyment. The brilliance of the students comes to the fore at the Undergraduate and University levels. That is the time when the Professors spot the intelligent, brilliant and the brilliant among the brilliant students. But Sarah’s extraordinary brilliance must have been spotted when she was in the kindergarten. This is so, because, her father was a renowned Professor of Mathematics.

Academic studies do matter for students like Sarah, but such genius children invariably study in the College of Self-Education, where their mind is their Principal. Their initiative, their Professors! Their hard work, their tutors! They are ambitious and industrious. They decide upon a project; start, act and finish.

“The eldest among five children, Flannery went to high school at Scoil Mhuire Gan Smal in Blarney, where she gave a science fair presentation on cryptography. In researching RSA encryption, she created her own encryption algorithm, which earned her an Esat Young Scientist Exhibition and later an Intel Fellows Achievement Award.

She went to college at the University of Cambridge, graduating in 2003 with a degree in computer engineering. Now she works for American video game developer Electronic Arts.”(Planet...)

In brief, what the book is about….

This book is the mathematician’s delight. How the serious subject like mathematics, can be interspersed with humor. Her introduction to the book itself fascinates and kindles curiosity. By reading it, you realize why she attained instant fame which she richly deserves.

In introduction she gives details about public-key cryptography, the RSA algorithm and the alternate algorithm that she created. The lucid style and control on the language required to deal with mathematical explanations is the strong point of the book.

Genius has nothing to do with the age. Therefore, forget for a while that a teenager is the author of the book, yet you can not forget that lovely and inspiring face of the author in the cover page of the book! Unless known in advance, you will find it difficult to believe that she is the owner of such a brilliant brain. You probably think that this book is the creation of an experienced Professor, well researched from a long list of bibliography.

The book begins in an orthodox style. Instead of saying something about the author straightaway, the book gives description with pages of family background. That again is about mathematics, because her father was the Professor of mathematics. A long section, titled, “Early Challenges” follows. This is where the challenges for the readers are thrown. They are recollections of her past.

The mention of about a dozen puzzles is made. She owned them from her father, told for the benefit of Sarah and her brothers. David Flannery, their father was the mentor of mathematics as well. The description for each puzzle is given along with an invitation for the reader to try to solve the same, before trying to locate the solution provided in the book elsewhere. Your chances of success are not very bright. A black-board in the dining room!

That is bound to happen when you have four sons and a daughter, all interested in the serious study of mathematics. On the blackboard, new puzzle was recorded each day for them to work on. Like, "Given a five-liter jar and a three-liter jar and an unlimited supply of water, how do you measure out four liters exactly?"

Some got into more difficult concepts: "How might you determine the average earnings of a group of people in a room (at a class reunion, perhaps) without any individual's divulging his or her salary?" the second puzzle has the latent message, which forms part of the theme all through the book.

That is to try getting information and hiding information. With such intelligent invitations the readers become part of the proceedings in the book. It is no effort to score over the reader but to win the reader in a positive style. It is to encourage him, and no to affront one with the exhibition of intelligence. Those who think that this is a book on the subject of mathematics which is bound to have the serious start will have peasant surprise to be greeted with mind-boggling puzzles.

Fame and Publicity and thereafter….

She received instant fame, became a celebrity in mathematics overnight, but she was humble about her achievements. Here she speaks beyond her age refuses to expand like the balloon with inflated ego. “I have no doubt that I am not a genius,”she declares.

“I am not being falsely modest. Through my father’s classes I have seen examples of true genius and I know I do not possess that ‘insight’ that distinguishes geniuses from those regarded as merely intelligent.”(p. 243). Those who read the book were sure about her impending success in the world of mathematics. She was the worthy mathematics daughter of a worthy mathematics Professor.

Next to puzzles….

She attempts the most original aspect of the subject of mathematics problems that take her to the position and fame sky-rocketing. Overnight she is a world-figure as for mathematics. The subject matter for any national level competition has got to be the unique one. She was to enter the Esat Irish Young Scientist’s Compettion-1998.

The project on cryptography was done by her at the suggestion and instance of her father. Her project will throw light on various cryptographic techniques, providing the account of the famous RSA algorithm. The mention and discussion of all these things is initiated at page 40 of the book, she discuses learning the relevant mathematics and the programming involved.

Then her father takes over the mathematical literature for the next 110 pages, and you catch up with Sarah again in page number 150.These pages contain mathematical exposition authored by David Flannery. They provide the basics of cryptography, and to understand it, is not the easiest of the propositions. He introduces thoroughly RSA algorithms at this stage.

When Sarah takes over at page number 150, she is at the threshold of the fame that is about to engulf her young personality. She gets tremendous response for her project that fetches her several prizes, and she is inspired to prepare for another ambitious and prestigious entry –the Rafe Jones at Brown University.

She is on the second step of the ladder of success. She undergoes a week-long internship at a Dublin cryptography company, and notices several techniques. Thereafter, she devises an alternative algorithm to the RSA and that is the flagship issue of her new project.

The results achieved by her are astounding. Her method is simple matrix multiplication instead of the relatively cumbersome modular exponentiation of the RSA. Her algorithm runs twenty times faster. It is christened by her as the Cayley Purser algorithm, the 19th century British mathematician Arthur Cayley as also Michael Purser, the mathematician whose ideas caught her imagination during her internship.

She runs in to the thick of the issues now-she proves how the new algorithm is secure from certain kinds of attacks and it becomes the mathematical odyssey for her, wherein she is required to explore and master the cobweb of not too familiar mathematics. Such a situation is the testing time for any student of mathematics.

It was probing the new waters for the first time with lots of hopes of positive results, but also with the fear of disappointment, should anything go wrong, at the most unsuspected moment. Mathematics is such a subject where there is no scope for errors. You have got to be accurate, as otherwise the whole edifice built by you step by step will collapse.

“All of this was an unusual experience for me,” she writes, “but I had a great feeling of excitement. I think it was because I was working on something that no one had worked on before. I worked constantly for whole days on end, and it was exhilarating” (p. 208).

To get the worthy solution, you need to have a strong problem. That was the situation Sarah, luckily found she in. The thesis problem provided lots of enthusiasm to her to go ahead. With the finalization of the thesis problem, she considered her to be lucky.

The problem was of her creation and she was bent upon to own the responsibility to solve it herself. She began to put extra efforts and worked desperately to find the authentic solution.

She reaped the dividends for her sincere efforts, she was able to prove that Cayley Purser algorithm had strong defenses and it can withstand successfully attack from a large family. She provides the detailed description to judge her new project, both algorithm and proof, in the 1999 Irish Young Scientist competition.

An inventive mind is always excited about any new achievement-you decide on a problem for you and then solve it successfully. Same was the case with Sarah.

When you smell success, when you have positive indications that you are nearing solution, the excitement is all the more. At that stage Sarah burnt the midnight oil to continue her sincere efforts. Success had to kneel before her; she showed the strong defense for the Cayley-Purser algorithm in the face of multi-pronged family attacks. She gives the detailed account, step by step, how her project needs to be judged, and the related explanation for the algorithm, with unassailable proofs.

In the 1999 Irish Young Scientists Competition., she quotes from her journal, “On one occasion,” she writes, “I looked out of our little huddle and it felt really strange—our conversation was so very intense that just to look around was like coming up for air” (p. 222). The finest moment of her judging was, she writes: “Before they left, [the judge] asked me the simplest question of all, and I could see he was wondering whether or not I would be able to answer it.

The answer was the fast exponentiation algorithm, and I must have smiled before I replied, because I knew it was the perfect end to the perfect session. I had been able to defend my project at all levels. The last question was a check to see if I knew the fundamentals. They smiled at each other on my final answer, which I’ll never forget.” (p. 223).

It was the perfect culmination. The excitement was about how gracefully Sarah walked up to the stage to accept the title of Irish Young Scientist of the Year. It was one of the extraordinary moments of her life. The charm of youth was on her side.

And the algorithm she talked held excellent possibility of rich dividends. The inquisitive media stood alerted and the unexpected bonus arrived, when London Times front-paged an article on her mathematical exploits. Overnight she entered the portals of stardom in mathematics. It did not take long time to transform her academic achievements towards the commercial gain.

The would-be cryptograph entrepreneurs were seeking her services. She received many offers to give lectures in Singapore. Mention of her name was made in the official magazine of the Spice Gils. She also received a request from Profile Books in London to write up the experiences and all that prompted to advance on the tough path of mathematics.

The budding young mathematician’s book had the firs print order of 35,000 copies, the marketing budget of $ 65,000 and an eight-city author-tour.

About the contents of the book, Sarah often sweetly apologizes for going deep into number theory; before explaining matrices, she writes, "I promise that from then on there will be no more explicit mathematics, only light explanations of mathematical ideas."

The hard-core mathematics in the book is restricted to two chapters. For those who wish to learn more, there are appendices. Her main project is about how the most famous current encoding system works, and in the meantime, she had invented one of her own. She takes extensive pains to explain both the systems and goes deep into the number theory along the route.

With the winning of the prize, fame and the monetary gains consequential to the fame arrived like an avalanche. Pepsi wanted her to concede that the mathematics brain and that of the family was due to lavish consumption of Pepsi, but the offer of contract was promptly turned down. A good mathematical calculation viewed from the humanitarian angle; they were aware perhaps of the harmful effects of such addictive drinks on the health of the younger generation!

“I have no doubt that I am not a genius,” said the prize winner of the 1999 European Contest for Young Scientists. Bu who would believe her and at the same time remain without deeply appreciating her modesty! She was the media sensation within days after getting the prestigious award. She was about to be hailed as an instant celebrity, for the public key to cryptography, the method used to transmit secure data over the Internet, but destiny played its part.

When everyone in the knowledgeable circles thought that her encryption algorithm is worth the millions, a security hole was discovered. Nevertheless she had done a great job. Now she met her father on equal terms, a mathematician talking to another senor mathematician.

There is an interesting interaction between the father-daughter mathematical duos. To be taught lessons from her father in the drawing room of the house was one thing. To be part of his lecture fraternity, and listen to his mathematical revelations in an organized way sitting as one among the audience was altogether a different experience. The previous day, he had a serious, purposeful conversation with Sarah. Her father said, "Now that you have decided to do transition year, I must do some math with you." He continued, "I'd like to show you how some beautiful but reasonably elementary mathematics is applied, stuff that you wouldn't ordinarily come across in school."

She could not understand the immediate intentions of her father. She thought he was inviting her to the kitchen blackboard, as she was aware of his enthusiastic ways; how he got inspired at the most unsuspected moment and wished to unleash his mathematical knowledge on her, whether she was mentally prepared for it or not.

Perhaps, at that moment she was not ready to receive the tough lessons of mathematics. She replied, "Dad, whatever you do, do something structured!" That set him thinking, as to the proper, most effective and appropriate way to teach and take her to the world of mathematics. He remembered his past. The debt he owed to the one who taught him mathematics. How the torch of mathematical knowledge was passed on to his hand.

It now depended, what he would do, with what his teacher did for him, and from where he left. He strongly felt that he must transfer the knowledge to some one else who richly deserved it. Who else could be that individual except his own daughter, in whom he must have noticed the latent mathematical genius?

He told his daughter, "Of course, only if you are genuinely interested-I wouldn't force it on you." She was genuinely interested. The evening lecture by her father proved to be the foundation stone for the grand mathematical edifice that she was going to build.

Sarah was now part of the class of serious students studying mathematics. The evening classes from seven ten, with the student strength of 8, continued for twenty five nights. The daughter was the youngest student, just 15 years and six months, but a couple of other students too were young.

There were adults who came from various backgrounds. Computer scientists, a secondary school mathematics teacher, a chemistry graduate working for a medical laboratory. They were the ones who loved mathematics, and who regretted their inaction in not pursuing the mathematics study, when young. It was a class that had its own specialties.

The cause of study was great, not the career out of the study. No credits were given. It was not part of any major. No home work or study was demanded. It was David Flannery’s way of “getting back into math” with no holds barred approach. You wee encouraged to come up with your most silly questions. He loved and appreciated those who made fools of themselves, as according to him, only those will learn and had the chance of success.

These assurances, coming from a reputed Professor, were greatly appreciated by the students. They looked forward to the classes with expectancy and with hope that they will be exposed to something interesting about the mathematics in the next session.

Sarah though had peculiar problem that the teacher was her father, she was able to mentally sort out the issue. Her dedication and the serious approach when she meant study, paved the way or her progress.

From the teacher’s perspective, what one teaches is important. But how one teaches, what one teaches is more important. Towards this end David Flannery filled the bill admirably. He was the one who enjoyed his teachings and encouraged his students to learn in style.

No time-bound hard tasks were expected from the students. Each one was encouraged to estimate one’s level of understanding, and progress accordingly. He was able to maintain an atmosphere of affection at home and in his classes.

The brain teasing puzzles enticed the students to know more and more, and the hours spent by the students in the association of David Flannery, proved highly fructifying. His company kindled their curiosity further. They eagerly awaited his next class and firmly believed that something more interesting would be in store for them.

The class was not all fun as was made out initially by David Flannel. That was his style of making the students interested in the subject. Soon, the intensive part of his chartered syllabus for the students began and his forays were in the elementary number theory, with cryptography as the final destination.

He would the take the students entirely to different horizons of mathematics, interesting sights that were rich in content, may not be of use for the immediate application.

When David Flannery was a student, the application of the number theory were so few (industrial and internet revolutions had not taken off), and yet the number theory was of prime importance to the mathematicians, its study was considered pure. By the end of 1970s, the situation had drastically changed.

The various technicalities involved in the message system and their readings gave a shot in the arm for cryptography. It became a much sought after subject and millions of dollars were invested into the development of this subject. The demand for expertise in this area became tremendous.

The book is a treat to read for the simple reason that it is a great human story as well, a success story, a management and pubic relation story etc. The way she prepared for the competitions would set the standard for any youngster who wishes to be an achiever.

What a careful and great teacher her father was! The concepts of teaching itself have undergone metamorphic changes, and mathematics is no exception. The style of teaching has become more student-friendly.

As for Sarah, her mathematician father was her great career-asset. But her mother also continuously encouraged her by telling interesting anecdotes about the subject. ‘Mathematics is the queen of the sciences and number theory is the queen of mathematics.’ Such sweet nothings said about the subject during formative years of a young girl had great positive impact on her.

Her Mom said Mathematics and the number theory was like the Sleeping Beauty Fairly Tale, and Sarah Flannery writes, “I thought about those who had toiled away through the centuries at unraveling the mysteries of this subject, motivated by nothing more than a passionate desire to know. They could have hardly dreamt of the applications that some of their results would one day find.

I wondered what it was they had discovered, and what they would think if they could see how some of these discoveries are now being used. I was eager to learn the subject and surmise for myself whether they would be surprised or not.”

But everything about the number theory is not all that sweet. It is a deceptive theory. When you think that you are on the verge of success, you are suddenly knocked out by a lethal punch. The simple questions that you will ask your Professor may look so simple to you. The same questions were asked by many brilliant students in the past. The answers to such questions have not been found until this day, and the most intelligent ones are raking their brains to find out the solutions.

But the toughest obstacle lay in waiting for Sarah. The sharp attacks on the Cayley-Purser algorithm arrived with Michael Purse alerts, making her mathematical advancement difficult. Sarah was on the defensive; she made efforts to repair the algorithm, but could not succeed. She stated that it is not salvageable as a workable encryption system.

The theoretical interest stood in her favor through this testing time. She included a postscript explanation on the successful attack. It brought her further success when she was conferred the title of European Young Scientist of the Year for 1999. They say, “When the going gets tough, the tough gets going!”

It is very easy to say that Sarah climbed the stage to receive the award. But behind this glorious moment in the life of this sports-loving teenager from Blarney in Country Cork, Ireland, lay the extraordinary talent and matching efforts of relentless research and discoveries in Internet cryptography.

At the age of sixteen to get the international recognition and to be hailed as “brilliant” by the London Times, is no ordinary achievement. Newspapers and periodicals hailed it as "a wonderfully moving story about the thrill of the mathematical chase" (Nature) and "a paean to intellectual adventure" (Times Educational Supplement).

The dinner-time conversation with her father led her to the hall of fame. Her burning curiosity, the inner joy of persistence paid off handsomely. What is the meaning of the wise saying, “Have a will to grow and grow you will!”—ask Sarah! David Flannery lectures on mathematics at Ireland’s Cork Institute of Technology. Sarah Flannery is now a student at Cambridge University.

Some shortcomings of the book….

To say that it is the combination of a set of two books, would not be a far-fetched criticism on the book. Two narrative segments of the book, at the beginning and the end, do not serve the actual cause for which the book stands for. The 150 pages, though there is no doubting the merit of the contents, it is right thing at the wrong place. It makes the tough and prolonged reading, and causes obstruction in understanding the life of Sarah in a systematic, chronological order.

The book goes on an aim-less wandering, and it defeats the structure of the book. But those who do not like mathematics intensely, for those who are not the serious students of the subject, these 150 pages are a good read! However, this can not be considered as the serious lapse of the book, but the professional critics of this literature, would not like to miss this point, as a matter of their duty.

The mathematical exposition part of it is flawless in contents and style. The beginning holds the interest of the common reader as well. The elementary examination of the prime numbers is detailed. The idea of primality about Mersenne primes, the Sieve of Eratosthenes and also primalaity testing are the important topics.

Then is the chapter on Modular arithmetic, Fermat’s Little Theorem, and pseudo primes. For complete understanding of the RSA algorithm, the last two of these three mathematical chapters are necessary. But they can be avoided by those who are desirous of the elementary feel for public-key cryptography.

The author has no problem with the English language, the exposition and style of writing is lucid. In fine, Sarah has given a very interesting book. The theme of the book and the author’s attitude towards the theme of mathematics, both evoke and sustain curiosity. The book deserves an outstanding position not only for the awards that it brought to Sarah, but also for the real contribution it made to the cause of mathematics, for all time to come.

There is something uniqueness about the book. Some more remarkable books should be expected from Sarah’s pen. The author has dealt with the topic in an excellent manner and the contents of the book gain stature, without losing its ground and reality. The book is the creation of the child, with father’s blessings. What more is required for the child and what does a father expect from the child? Both see the fulfillment of their respective mathematical missions through this book.

When a child gives the account of her experiences that ultimately led her to great heights of success, the reader’s interest is all the more, for the simple reason that every parent wishes to visualize and aspires for the success of their children.

From the point of view of human psychology and management principles that lead to success, this book is the torchbearer. Her thoughts are playful, yet never missing the mathematical purpose of the book.

The number of high school students who enjoy mathematics is not very high and many do the subject, as they have to do as it is the unavoidable subject up to a certain level. But an introduction and interaction with this book should change their perspective.

Many may not be lucky enough get a mathematics genius Professor-father. In the case of Sarah, her pastime of solving the mathematical puzzles with her father paid her rich dividends. It started as a game and slowly turned into a thrill of the mathematical chase. Nothing succeeds like success and one good thing in her life, led to the other.

A great lesson for the psychologists and sociologists who study the parental impact on the life and future of children! But for the encouragement and influence of her mathematician father, Sarah would have been yet another university student, pursuing the syllabus-oriented degrees for a routine career.

Conclusion:

God made the natural numbers, 1,2,3,4,5,6,7,8,9 and 0 and what all ‘complications’ the human being has created under the subject-banner mathematics! But for becoming the mathematician, Sarah would have been a storyteller.

The art comes naturally to her. She has dealt with a highly intellectual subject, without any pretensions, without the sense of overbearing. At times, the writing takes the serious turn, but that is what the subject matter is. Her childlike descriptions maintain the essential dignity. If she tries her hand at other subjects of writing, we may be in for surprises.

That’s what her imagination, wit and charm throughout the book reveal. “In Code”, has the makings of a very good novel and her pen holds out great promises for even better works. It is a twice-blessed book. Primarily it is a book on mathematics, and more importantly it is an interesting book of human endeavor, the human spirit, the book on positive qualities like grit and determination. Few children are lucky to have such great upbringing.

The black board in the kitchen truly speaks about the studious family. She and her four brothers made it the perfect class at home, and mostly issues related to mathematics were discussed at the dining table. At lunch, the distraction would be not be like the television set in the modern drawing room, but the attractions were puzzles appearing on the blackboard days after day, without intermission. David Flannery had a clear purpose about the puzzles.

They encouraged the children into enjoying abstract reasoning. But do no imagine, Sarah liked all these intensely. She was not a book worm. She liked to ride horses, played hurling and basketball, did boating and liked other adventurous team sports.

She carried that adventurous spirit to her mathematics research as well.

So to say, Sarah was a philanthropist mathematician and did not possess any motivated desires about her accomplishments. Mathematical community offered the talented Sarah all co-operations. Experts in the field of cryptology were eager to help her.

Initial reactions to her code system, was that it was patentable and she had the possibilities of becoming the millionaire. The exchange of vital information with other mathematicians could have damaged her interest for financial gains. Yet, she shared the information bearing in mind the overall interest of mathematics.

She was only 17, when she was a guest speaker at an IBM leadership conference for women. Sarah was also given an invitation to attend the Nobel Prize ceremonies in Stockholm. To write an interesting readable book on mathematics, normally considered a dry and brain-racking subject is no mean achievement. She has set the trend for the budding young scientists, and great scholars in any subject need not be men and women with silver hair.

The book provides to all concerned, the parents, the teachers and above all the combustible younger generation, who wish to achieve something in life, but do not know the correct procedures and steps to achieve very valuable information. Sarah provides good solutions to one’s ambitions and the way to achieve them.

This makes the book even greater, than her contribution to cryptography. Bertrand Russell once talked about ‘the silent beauty’ of mathematics. Sarah has demonstrated how the skills of the mathematician and the skills of the fiction & fantasy writer can be clubbed together to create an outstanding contribution to the world of literature. Here is the combination of great human experience mixed with intellectual stuff. It is very easy to record and offer her congratulations for all that she achieved at the young age of 16.

But think of the hard work she did, the relentless pressure she was able to endure at such a young age, and all this she did at the same time enjoying and pursuing her hobbies. So, if you have young children, and do not have many ideas as to how to inspire them, give this book to them—it would be more appropriate if the parents read and discuss the contents of this book for their benefit. The results are bound to be far fetching as for their future. The contents of the book have the lessons in ‘moral counseling’ as well.

The application of cryptography has caught up fast with the internet revolution. Many of the big companies are willing to sponsor researches and Sarah is eminently suited to take advantage of this situation. Her achievements have changed her perspective of life. The career opportunities that arrive at her doors are perhaps too much for her to handle. She has traveled to important destinations all over the world, met cryptography figureheads like Ronald Rivest and Whitfield Diffey. But the best is yet to come, and Sarah Flannery knows it well.

References Cited:

In Code: A Mathematical Journey (Paperback)

by Sarah Flannery (Author), David Flannery (Author}

Paperback: 352 pages

Publisher: Algonquin Books (December 30, 2002)

Language: English

ISBN-10: 1565123778

ISBN-13: 978-1565123779

PlanetMath: Sarah Flannery

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