Sarah Flannery is just 16 and already she may have revolutionised global communication. Pretty, bright, kind, polite, sporty, she does well at school - though still not top of her class. Her younger brothers get on her nerves. Her dad, David, a maths lecturer, is always on at the family to think. Her mother, Elaine, who lectures in microbiology, reminds them to practise the piano. It could be any nice, middle-class, academic family.

Yet Sarah’s new mathematical method of encoding and decoding electronic information - which started life as her school project - could have consequences as far-reaching as the cracking of the Enigma code used by German U-boats in the Second World War. She has invented a way to speed up secure transmission of electronic information on the Internet. Her discovery has already won her this year’s Esat (Irish Telecom) Young Scientist of the Year competition.

Her school, Scoil Mhuire Gan Sm l in Blarney, county Cork, Ireland, has 510 students aged 12 to 18. It’s a country school, the only one in the village, co-educational - still not the norm in Catholic Ireland - and comprehensive. The bishop gives prizes, local companies sponsor the computers.

Headteacher Donal O’Grady speaks with confidence. “If students respect themselves fully and have confidence in their abilities, the staff will get them there. I’m talking about their whole personal development, of which the academic is part - a huge part, but part,” he says.

Mr O’Grady has a lot to be confident about. This year Scoil Mhuire Gan Sm l’s basketball team got through to the All Ireland finals; its Irish football team are county champions. The top three Masters’ degrees in law at Cork University this year were taken by former pupils. Paula Sheehan, current All Ireland intermediate 3,000 metre champion, also did “astonishingly well” at Junior Cert - even better than her classmate Sarah Flannery.

For the past 15 years, science teacher Sean Foley has been encouraging pupils to enter young scientist competitions. This year, his son, Vincent, won the physics, mathematics and engineering Intel award for science won by Sarah Flannery last year. Vincent has developed a way to make computer images look more natural, less “blocky”. “I regard all these,” says Mr O’Grady, “as equally important to the development of our pupils.”

Pardon? Did he just say that a school sports team reaching the county finals is as important as Sarah’s invention which could revolutionise commerce around the world?

That’s what the man said, and when you meet Sarah Flannery, you can see that this is no paradox. Sarah’s education, at home and at school, has kept her feet on the ground while her intellect and imagination soar to heights that would dizzy most of us. She works at the kitchen table while her brothers squabble over the Playstation; she is kept to her household tasks by her mother, despite wrestling with world-class knots in number theory; she listens to U2 and The Corrs like any Irish girl of her age, as well as tuning in to the maths which 17th-century thinkers termed the “music of the spheres”.

Sarah’s formula, the Cayley Purser algorithm, named after Arthur Cayley, a 19th-century mathematician, and Michael Purser, the founder of the company where Sarah did work experience, is 22 or more times faster than the current algorithm on which most current cyptographies are based. If Sarah is right, international banking will kick into a higher gear. If Sarah is right, and Dr Tony Scott of University College, Dublin, says her work “is up with the best I’ve ever seen” - your credit card will work on the Internet in the blink of an eye. Patents are already pending.

For both Sarah and Vincent it all began during the transition year, an optional year elsewhere but compulsory in their school, in which Irish students pursue a personal project before going on to study for Leaving Cert, the nearest equivalent of A-levels.

Mr Foley likes competitions and their potential for sparking off teenage originality. “I always begin by asking them to come and see me if they are interested in anything at all,” he says. Over the years, he has got pupils to delve into pursuits as varied as recording bird migration, photography and computer games. 1998 was to be a peak year for students’ own interests.

Mr Foley’s son Vincent, who has been hooked on computers since he was six, had become frustrated at the quality of enlarged images on such software as Adobe and Corel Draw. A proficient, self-taught writer of computer programs, Vincent worked on Visual Basic to devise a better way. Vincent, who also likes music and cross-country running - “he’s a very run-of-the-mill boy,” says his father - was, like many another gifted lad, poor on presentation. Much of his year’s work was on refining the presentation.

For Sarah, presentation came more naturally. She is, as Esat judge Brendan Supple, a manager at Siemens, says, “totally rounded in her subject”. Like Vincent Foley, Sarah has been driven by interest. At first her father, David, who writes maths problems on a blackboard at home to stimulate his children, suggested she attend a course he was giving at Cork Institute of Technology on group theory, a branch of number theory. Sarah enjoyed the lectures, particularly on matrices and Arthur Cayley’s discoveries. Here was a possible subject for her transition year project. If she could find a prac-tical application, the project could also be entered for the Young Scientist competition.

Modern cryptography, which depends on a system of “double keys”, caught her imagination. Two decades ago three American graduate students at the Massachusetts Institute of Technology published their own formula known as the RSA algorithm. The students, Rivest, Shamir and Adleman, revived number theory from 1640 to devise their system, which relies on exponentiation, non-commutability, and very large numbers. For their system, two very large prime numbers are multiplied together to give a number of more than 200 digits. Holders of the “public key” use this number to encode information. Without the “private key”, however, the information cannot be decoded.

What would happen, Sarah wondered, if she used some of the ideas from her dad’s talks on group theory in double-key coding? And so Sarah browsed the Internet and found that Baltimore Technologies in Dublin sold codes for use on the Internet. She applied for work experience. At Baltimore Technologies, founder and maths lecturer Dr Michael Purser had some key suggestions on refining the current basis of most Internet cryptography. “So I went home,” she says, “and was begging Dad to tell me more about it. I knew we would be doing matrices so I asked him to tell me about ‘nc’ groups.”

So the work of Arthur Cayley was pulled in and Sarah became gripped by her problem. The biggest hurd,le, and one which produced many dead-ends, was to make the system totally secure. Intense bouts of activity in the summer and before Christmas gave Sarah some degree of satisfaction but, she says, “you get so into it that you are doing it all the time”.

Now, with her Cayley Purser algorithm making front-page news in The Times (January 13, 1999) and having received 160 phone calls in a single week, fame and fortune beckon. She is pleased to meet the Taoiseach (Ireland’s prime minister). She is interested to hear the job offers. She is sensibly a little wary of the media. She is amused by approaches from Disney. And she is thrilled at having been rung up by Ron Rivest, one of the originators of the RSA algorithm.

Sarah rejects the “genius” label and is not even sure if she is a mathematician. “I just got into that topic. It’s so applied, so practical, so hot.” She could get just as absorbed in her aunt’s pony, Clydie, and show-jumping, in listening to Aerosmith or Nirvana, in reading Stephen Fry’s autobiography or Tolstoy’s Anna Karenina. She doesn’t watch television - “I like doing.”

Con Hayes, Sarah’s maths teacher, disclaims credit. It’s a great class, he says, a “dream to teach” with two or three setting an extremely high standard, a once-in-every-10-years class. As bright pupils spark off each other, so they also gain from a curriculum more like the old O-level than the current British GCSE, he says. Mixed-ability teaching is the rule up to Inter Cert at age 15, and works well until then, he says. Sarah “does well at exams, but so do others. It’s that creativity, which you don’t measure in exams, where she is right off on her own, in a different league.

“She’s not precious in any way,” says Mr Hayes. “She’s happy to get covered in mud on the football field. She’s not one-dimensional.”

THE TRANSITION YEAR.

Irish pupils are offered a year out of the examination rat race to pursue a personal project at 15, the current leaving age.

Students enter the Irish secondary system at 12. After three years of studying eight to 10 subjects - including Irish, English, maths and science - they sit the Junior Certificate exams (known as Junior Cert).

Pupils who stay on can then take a transition year or go straight into the fifth year to study six to eight, or even 10, subjects for their Leaving Certificate.

The transition year was pioneered in the mid-80s and made available nationally from 1994-95, to give students an opportunity to broaden their skills. Its mission, as set out by the Irish Department of Education, is “to promote the personal, social, educational and vocational development of the students and to prepare them to participate as responsible members of society”. It has no fixed curriculum and is marked by no final exams.

Choices can range from the chance to study new foreign languages, go on field trips and take short study “taster” units in new subjects, to such practical tasks as work experience, running mini-businesses or developing arts or sports projects. Involvement with parents and service in the wider community are an integral part of the transition year. Students are encouraged to keep diaries of work shadowing or work placements and share their reflections with their classmates.

Such assignments, as well as projects like those undertaken by Vincent Foley and Sarah Flannery, take the place of traditional homework. At the other end of the academic scale, transition year can also offer the chance for remedial learning.

Denise Kelly, one of six teachers in the Transition Year Curriculum support Service, says: “I’m totally convinced of the value of it.” She cites a cross-curricular project that involved street theatre, development education, economics and lobbying of politicians at her own Dublin school.

The transition year is available in 500 of the Republic’s 753 schools. Extra funding is given to schools running the programme.

CODE OF CONDUCT FOR SECURE MESSAGES

MATHS OF MASSIVE NUMBERS PRESERVES CONFIDENTIALITY

How can you be sure that no one’s going to nick your credit card information off the Internet? The answer lies in cryptography, the science of code-making.

Cryptography has come a long way, from basic letter transposition - where each party has the same key - to complex computations devised by mathematicians but operated by computers.

All words or numbers you see on your screen are held in the computer’s memory as a series of binary numbers, which can be encrypted by using a mathematical formula.

Nowadays, you buy double-key encryption from software companies. Each client has an encoding key, publicly listed in a directory. The decoding key will be secret. It may operate within widely-used software, but the actual key numbers will be hidden from keyboard operators. The most widely used double-key algorithm is the RSA.

An algorithm is a step-by-step procedure that always produces correct results - the times tables, for instance. The RSA cleverly merges procedures from several areas of mathematics: specifically, Euler’s totient function, modular arithmetic and the breaking down - factorising - of very large numbers formed from multiplying together two large prime numbers.

Crucially, encryption makes use of the non-commutative nature of factorising - the fact that it is easy to multiply two numbers together to create a large one, but much harder to take a large number and work out which prime numbers were multiplied together to create it.

With small numbers like 6, prime factors are few (2x3=6 and 3x2=6). With much bigger numbers, the possibilities become vast. Sarah Flannery suggests that to factorise the 200-digit numbers used in the RSA could take a computer 100,000 years.

To make it more difficult, the RSA uses exponents (superscripts which show multiplying “to the power of”) as keys. Reversing exponentiation (finding square roots etc) in a number when the original exponent is hidden is hard. Harder still with the RSA, because it uses two different exponents - one public, one private.

But the RSA formula is made yet more impenetrable by the use of complex aspects of mathematics such as Euler’s totient function and modular arithmetic. Even for an advanced computer, to crunch such numbers can take time - up to 80 seconds every time you send a secure piece of electronic information, for instance, when shopping over the Internet.

Sarah Flannery looked at this algorithm from the context of number theory, which studies the properties and relations of different types of whole numbers - sometimes using algebra. She has substituted matrices for exponents.

Matrix theory deals with numbers in rectangular arrays of elements or numbers related to each other as rows and columns. Matrices are generally used (in top tier of GCSE and A- level) to solve problems like transformation of shapes. Matrices are part of group theory, dealing with the effect on sets of numbers if two numbers within the group operate on each other.

She believes using “nc groups” (non-commutative groups) could offer another way to look at factorising down very large numbers, a crucial part of the encoding and decoding process. Sarah says: “This could make transmission of secure information 22 to 30 times quicker.”

For more on the RSA algorithm, consult web sites: www.rsa.com, the FAQ section; www.cyberlaw. com; and cs.oberlin.educlasses cs115lect24nO.html