Codebreaking in Everyday Life John D Barrow

10 x 10 x 10 x 10 seconds  2.75 hours

Trap-door Operations QUICK to encode SLOW to break Big prime number x Big prime number = very large composite number

Are there enough Postcodes to go round? The Pattern is AB3 4CD 45,697,  26  10  10  26  26 = 45,697,600 choices UK population approximately 60,587,000 26,222,000 Approximately 26,222,000 households and 28,500,500 28,500,500 expected by 2020 Enough for houses But not for people!

National Insurance Numbers (are not boring) Eg pattern is NA123456X Eg pattern is NA123456X 26  26  10  10  10  10  10  10  26 = billion 26  26  10  10  10  10  10  10  26 = billion Population of the world is about 6.65 billion and maybe 9 billion by 2050 Population of the world is about 6.65 billion and maybe 9 billion by 2050

Stanley Milgram’s (Other) Experiment (1967) Give many letters addressed to a Boston stockbroker to random people in Omaha, Nebraska and Wichita, Kansas with a profile of stockbroker Give many letters addressed to a Boston stockbroker to random people in Omaha, Nebraska and Wichita, Kansas with a profile of stockbroker They were to send them on to an acquaintance who they felt might know more about the stockbroker, sign a roster and post a card (less junk mail in those days!) They were to send them on to an acquaintance who they felt might know more about the stockbroker, sign a roster and post a card (less junk mail in those days!) About 20% of Milgram’s letters reached the stockbroker About 20% of Milgram’s letters reached the stockbroker On average they took 6 reposting ‘steps’ to get there via the social network of friends (6.6) On average they took 6 reposting ‘steps’ to get there via the social network of friends (6.6)

Small-World Networks If you know 100 people and they know 100 others then you are just one step away from 10,000 people N steps away from 10 2(N+1) people World population is 6.65 billion = (N+1) > 9.8 when N exceeds 4

“Six Degrees of Separation”

Friends of Friends of Friends Are Important Close links (friends and family) are a lot like you and know the sorts of things and people you know. Close links (friends and family) are a lot like you and know the sorts of things and people you know. More distant acquaintances are more likely to know things and people that you do not. More distant acquaintances are more likely to know things and people that you do not. The Prince and the pauper The Prince and the pauper

Average Path Length Erdös number of mathematicians Erdös number of mathematicians It’s a Small World After All It’s a Small World After All Average Dist = (ln N / ln K) = 5.9 Average Dist = (ln N / ln K) = 5.9 where N = total nodes and K = friends per node. And K=30 and N = = 10%  world population

Problems With Names The Soundex Phonetic System (1918) Keep first letter of the name Keep first letter of the name Delete a,e,i,o,u,h,y,w Delete a,e,i,o,u,h,y,w Assign numbers to the rest of the letters Assign numbers to the rest of the letters b,f,p,v = 1 b,f,p,v = 1 c,g,j,k,q,s,x,z = 2 c,g,j,k,q,s,x,z = 2 d,t = 3 and l = 4 d,t = 3 and l = 4 m,n = 5 and r = 6 m,n = 5 and r = 6 If two or more letters with same number are adjacent in original name keep only the first If two or more letters with same number are adjacent in original name keep only the first Keep only first 4 characters, make up to four with 00s if needed Keep only first 4 characters, make up to four with 00s if needed John Barrow  Jn Br  J5 B6  J500 B600 John Barrow  Jn Br  J5 B6  J500 B600 Smith and Smyth  S530 Smith and Smyth  S530 Ericson, Erickson, Eriksen, Erikson  E6225. Robert, Rupert  R163 Ericson, Erickson, Eriksen, Erikson  E6225. Robert, Rupert  R163

Check-Digit Codes To guard against transcription errors To guard against transcription errors Catch naïve fraudsters Catch naïve fraudsters Credit cards, tickets, passports, tax ID nos Credit cards, tickets, passports, tax ID nos 889/899, 1112/112, 43/34,….. 889/899, 1112/112, 43/34,….. Internal self-checking system to validate numbers Internal self-checking system to validate numbers Airline tickets 10 digits + one check digit which is remainder after dividing by 7 Airline tickets 10 digits + one check digit which is remainder after dividing by = x = x So ticket no. is So ticket no. is

Simple Errors Can Cause Problems TypeApprox Frequency a by b79% ab by ba10% abc by cba 1% aa by bb0.5% a0 by 1a (a=2,3,..)0.5% aca by bcb0.3%

Banking – What’s It All About? Banking – What’s It All About?

IBANs International Bank Account Numbers GB82 WEST GB82 WEST Country - cheque no - bank - sort – account no. 1. Move first 4 characters to the end 2. Replace letters by A=10, B=11,…Z=35 3. Interpret digit string as a decimal 4. Divide by Valid IBANS give a remainder = 1. eg WEST GB82  = 1 mod 97 This is a valid IBAN Moved from front

Credit Cards and the Luhn Test 12 or 16 digits, for example: 12 or 16 digits, for example: L to R, double the digits in odd slots, add the two if bigger than 9 (14  5 etc) L to R, double the digits in odd slots, add the two if bigger than 9 (14  5 etc)  Sum = 33 + original digits in even slots Sum = 33 + original digits in even slots = 60 The total must be divisible by 10 for a valid card Card number fails (sum = 57) Catches all single digit errors, most adjacent swops (not 09/90 though) (not 09/90 though) Invented by Peter Luhn at IBM in 1954

Need to change the final check digit from 3 to 8 to validate the number This is a necessary, but not a sufficient, condition for the card to be valid! Fails ! Check validity

Universal Product Code Began in 1973 for grocery products now used for most retailed goods 12 digit number represented by bars for laser scanning Two strings of five between two single digits Type of product manufacturer size/colour/model check digit ,1,6,7,9 = any 4 = sale items 2 = food by weight5 = special offers/coupons 3 = drugs, health

Check digit ? Add digits in odd positions: = 17 Multiply by 3: 3  17 = 51 Add digits in even positions: = 73 = 7 Divide by 10: check digit is 10 minus the remainder = 7

ISBN-10 Multiply each digit by its position from the right. Add and check digit must make the sum divisible by 11. If the remainder is 10 use X eg for X the sum is X. If X is 10 the total is 209 = 19  11 ISBN-13 Is like UPC but multiplies even digits by 3 instead of odd ones. Check digit must make sum divisible by 10.

International Mobile Equipment Identity IMEI This is what you cancel when your phone is stolen This is what you cancel when your phone is stolen Changing it is a criminal offence Changing it is a criminal offence 14 digits + check digit 14 digits + check digit The software version IMEISV has 16 digits The software version IMEISV has 16 digits

? IMEI ? DoubleEveryOther418 = =552 Sum ? Calculatecheckdigit

IMEI is IMEI ? DoubleEveryOther418  552 Sum Sum must be divisible by 10 CheckDigitAdded

The General Picture Check digits are computed from a product code a 1 a 2 a 3 a 4 …..a n by multiplying by weights w 1 w 2 w 3 w 4 …..w n and evaluating the dot product and remainder on dividing by r Check digits are computed from a product code a 1 a 2 a 3 a 4 …..a n by multiplying by weights w 1 w 2 w 3 w 4 …..w n and evaluating the dot product and remainder on dividing by r C = r - (a 1,a 2,a 3,…a n )  (w 1,w 2,w 3,…w n ) = r - a  w (mod r) C = r - (a 1,a 2,a 3,…a n )  (w 1,w 2,w 3,…w n ) = r - a  w (mod r) UPC : n = 12, w = (3,1,3,1…), r= 10 UPC : n = 12, w = (3,1,3,1…), r= 10 EAN-8: n = 7, w = (3,1,3,1…), r= 10 EAN-8: n = 7, w = (3,1,3,1…), r= 10 Airline: n = 10, w = (1,1,1,1…), r= 7 Airline: n = 10, w = (1,1,1,1…), r= 7 ISBN-10: n = 9, w = (10,9,8,7…,2,1), r= 11 and X =10 ISBN-10: n = 9, w = (10,9,8,7…,2,1), r= 11 and X =10 ISBN-13 and EAN-13: n = 12, w = (1,3,1,3…), r = 10 ISBN-13 and EAN-13: n = 12, w = (1,3,1,3…), r = 10

Sometimes You Don’t Need Numbers