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Codebreaking in Everyday Life

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In "Codebreaking in Everyday Life," John D. Barrow explores the critical role of codes and patterns in modern communication. His analysis includes the mathematics behind postal systems, national insurance numbers, and social networks, highlighting how these systems impact our lives. Barrow examines the intricate relationships between average path lengths in social networks and the concept of "six degrees of separation." Additionally, practical applications of code validation methods in banking and product identification are discussed, demonstrating the importance of error checking and data integrity.

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Codebreaking in Everyday Life

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  1. Codebreaking in Everyday Life John D Barrow

  2. 10 x 10 x 10 x 10 seconds 2.75 hours

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

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

  5. National Insurance Numbers(are not boring) • Eg pattern is NA123456X • 262610101010101026 = 17.576 billion • Population of the world is about 6.65 billion and maybe 9 billion by 2050

  6. 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 • 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 • On average they took 6 reposting ‘steps’ to get there via the social network of friends (6.6)

  7. 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 102(N+1) people World population is 6.65 billion = 109.8 2(N+1) > 9.8 when N exceeds 4

  8. “Six Degrees of Separation”

  9. 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. • More distant acquaintances are more likely to know things and people that you do not. • The Prince and the pauper

  10. Average Path Length • Erdös number of mathematicians • It’s a Small World After All • Average Dist = (ln N / ln K) = 5.9 where N = total nodes and K = friends per node. And K=30 and N = 108.8 = 10%  world population

  11. Problems With NamesThe Soundex Phonetic System (1918) • Keep first letter of the name • Delete a,e,i,o,u,h,y,w • Assign numbers to the rest of the letters • b,f,p,v = 1 • c,g,j,k,q,s,x,z = 2 • d,t = 3 and l = 4 • m,n = 5 and r = 6 • 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 • John Barrow  Jn Br  J5 B6  J500 B600 • Smith and Smyth  S530 • Ericson, Erickson, Eriksen, Erikson  E6225. Robert, Rupert  R163

  12. Check-Digit Codes • To guard against transcription errors • Catch naïve fraudsters • Credit cards, tickets, passports, tax ID nos • 889/899, 1112/112, 43/34,….. • Internal self-checking system to validate numbers • Airline tickets 10 digits + one check digit which is remainder after dividing by 7 • 4851913640 = 693130521 x 7 + 3 • So ticket no. is 4851913640 3

  13. Simple Errors Can Cause Problems Type Approx Frequency a by b 79% ab by ba 10% abc by cba 1% aa by bb 0.5% a0 by 1a (a=2,3,..) 0.5% aca by bcb 0.3%

  14. Banking – What’s It All About?

  15. IBANs International Bank Account Numbers GB82 WEST 1234 5698 7654 32 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 97 5. Valid IBANS give a remainder = 1. eg WEST 1234 5698 7654 32 GB82  3214282912345698765432161182 = 1 mod 97 This is a valid IBAN Moved from front

  16. Credit Cards and the Luhn Test • 12 or 16 digits, for example: 4000 1234 5678 9314 • L to R, double the digits in odd slots, add the two if bigger than 9 (14  5 etc) 8 0 2 6 10 14 18 2  8 0 2 6 1 5 9 2 Sum = 33 + original digits in even slots 8+2+6+1+5+9+2+0+0+2+4+6+8+3+4 = 60 The total must be divisible by 10 for a valid card Card number 4000 1234 5678 9010 fails (sum = 57) Catches all single digit errors, most adjacent swops (not 09/90 though) Invented by Peter Luhn at IBM in 1954

  17. Check validity Fails ! 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!

  18. 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 productmanufacturersize/colour/modelcheck digit 6 44209 42095 7 0,1,6,7,9 = any 4 = sale items 2 = food by weight 5 = special offers/coupons 3 = drugs, health

  19. 3 474370 01631 7 Check digit ? Add digits in odd positions: 3 + 7 + 3 + 0 + 1 + 3 = 17 Multiply by 3: 3  17 = 51 Add digits in even positions: 51 + 4 + 4 +7 + 0 + 6 +1 = 73 Divide by 10: check digit is 10 minus the remainder = 7

  20. 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 0-19-280569-X the sum is 199 + 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.

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

  22. 49015420323751? Calculate check digit

  23. Check Digit Added IMEI is 490154203237518 Sum must be divisible by 10

  24. The General Picture • Check digits are computed from a product codea1a2a3a4…..anbymultiplying by weights w1w2w3w4…..wnand evaluating the dot product and remainder on dividing by r C =r -(a1 ,a2 ,a3 ,…an)(w1,w2,w3 ,…wn) = r - aw (mod r) • UPC : n = 12, 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 • 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

  25. Sometimes You Don’t Need Numbers

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