highlights of computing: from the ancient indian vedic literature to modern practical cryptography

8/9/2019 Highlights of Computing: From the Ancient Indian Vedic Literature to Modern Practical Cryptography

1/30


2/30

Uwe Wystup - Highlights of Computing 2

Contents1 Numeral Systems 3

1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Vedic Mathematics 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 The 16 Sutras . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja 6

2.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Square Roots (Vargamula) . . . . . . . . . . . . . . . . . . . . 142.6.1 Duplex Process (Dvandvayoga) . . . . . . . . . . . . . 152.6.2 Square Root of a Perfect Square . . . . . . . . . . . . . 152.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Zero Search Methods 173.1 Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Rate of Convergence . . . . . . . . . . . . . . . . . . . 20

3.2 Herons Square Root Finder . . . . . . . . . . . . . . . . . . . 213.3 Exercise: Divisions without Dividing . . . . . . . . . . . . . . 21

4 Credit Cards 214.1 The Luhn Algorithm . . . . . . . . . . . . . . . . . . . . . . . 224.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Cryptography 245.1 The Inventors of the RSA-Algorithm (1978) . . . . . . . . . . 245.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6 Denitions 286.1 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


3/30


7 Contact Information 29

1. Numeral Systems

2. Vedic Mathematics

3. Zero Search Methods

4. Credit Cards

5. Cryptography

6. Denitions

1 Numeral Systems

Fundamentals: Any integer has a unique representation as a weightedsum of powers of a chosen base. Example: 245 = 2 102 + 4 101 + 5 100

The Base 10 is in no way special except that

The Mayas also used the feet (base=20), see http://en.wikipedia.org/wiki/Maya_numerals .

Computers use a dual system (power off or on, base 2) or a hexadecimal system (0 , 1, 2, . . . , 9,A,B,C,D,E,F ) for memory size: 1 byte = F F . More on http://en.wikipedia.org/wiki/Numeral_system

1.1 ExerciseWrite the hexadecimal number CAFE in the dual, decimal and Maya numbersystem.
http://en.wikipedia.org/wiki/Maya_numeralshttp://en.wikipedia.org/wiki/Maya_numeralshttp://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Maya_numeralshttp://en.wikipedia.org/wiki/Maya_numerals


4/30


2 Vedic Mathematics1. Introduction

2. Multiplication

3. Squares

4. Division

5. Divisibility

6. Square Roots

2.1 Introductionveda (Sanskrit) means: knowledge

Veda Upaveda

Rigveda Ayurveda

Samaveda Gandharvaveda

Yajurveda Dhanurveda

Atharvaveda Sthapatyaveda

Table 1: Vedas and Upavedas (supplementary vedas)


5/30


2.1.1 The 16 Sutras

are part of a Parisista (Appendix) of the Atharvaveda

1. By one more than the one before

2. All from 9 and the last from 10

3. Vertically and crosswise

4. Transpose and apply

5. If the Samuccaya is the same it is zero

6. If one is in ratio the other is zero

7. By addition and by subtraction

8. By the completion or non-completion

9. Differential calculus

10. By the deciency

11. Specic and general

12. The remainders by the last digit

13. The ultimate and twice the penultimate

14. By one less than the one before

15. The product of the sum

16. All the multipliers


6/30


2.1.2 Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja

Explained the sutras in his books.

Jagadguru Swami Sri Bharati KrsnaTirthaji Maharaja (March, 1884 -February 2, 1960) was the Jagadguru(literally, teacher of the world; assignedto heads of Hindu mathas) of the Go-vardhana matha of Puri during 1925-1960. He was one of the most signi-

cant spiritual gures in Hinduism dur-ing the 20th century. He is particularlyknown for his work on Vedic mathemat-ics.

2.2 MultiplicationExample with working base 10:

9 - 1

7 - 3

6 / 3

= 63

7 - 3

6 - 4

3 / 1 2

= 42

13 + 3

12 + 2

15 / 6

= 156

12 + 2

8 - 2

10 / 4

= 96

Reason: (x + a)(x + b) = x(x + a + b) + abOrigin of the

-sign comes from this method


7/30


Example with working base 100:

91 - 9

96 - 4

87 / 36

= 8736

111 + 11

109 + 9

120 / 99

= 12099

108 + 8

97 - 3

105 / 24

= 10476

Other working bases (division case)

100/2=50

49 - 1

49 - 1

2)48 / 01

24 / 01

= 2401

100/2=50

54 + 4

46 - 4

2)50 / 16

25 / 16

= 2484

Other working bases (multiplication case)

102=2019 - 1

19 - 1

2)18 / 136 / 1

= 361

106=6062 + 2

48 - 12

6)50 / 2 4300 / 2 4

= 2976


8/30


2.2.1 Exercises

Multiply the following

a 94 94b 97 89c 8799d 87 98e 87

95

f 9595g 7996h 98 96i 9299 j 8888k 9756l 9763

m 92 196


9/30


Multiply the following mentally

a 667998b 768997c 989998d 885997e 883998f 8

6

g 891989h 88889996i 69999997 j 9090999994k 7898999997l 9876

9989


10/30



a 133103b 107108c 171101d 102104e 132102f 14

12

g 1813h 12221003i 10511007 j 1511110003k 125105l 10607

10008

2.3 SquaresUsing the sutra Ekadhikena Purvena (by one more than the previous one) weget

152 = 1 2/ 25 = 2/ 25 = 225252 = 2 3/ 25 = 6/ 25 = 625352 = 3

4/ 25 = 12/ 25 = 1225

...1152 = 11 12/ 25 = 132/ 25 = 13225


11/30


2.3.1 Exercises


a 652

b 852

c 0.52

d 7.52

e 0.02252

f 10502

g 1752

2.4 Division

Lots of tricks are available. We do only some highlights. Find the exact decimal representation of 119 .

Standard methods are cumbersome. Using the Ekadhika Purva Sutra it is easy: Start with 1 and then work from right to left multiplying by 2.

. 0 5 2 6 3 1 5 7 8

1 1 1 1 1 1

/ 9 4 7 3 6 8 4 2 1

1 1 1


12/30


A further shortcut is the insight that

. 0 5 2 6 3 1 5 7 8

+ 9 4 7 3 6 8 4 2 1

= 9 9 9 9 9 9 9 9 9

The same works for all periodic decimals, e.g. 17

. 1 4 2

+ 8 5 7

= 9 9 9

2.4.1 Exercises

Compute the exact decimal number of

a 129

b 149


13/30


2.5 Divisibility

Use Ekadhika as an osculator. For 9, 19, 29, 39 etc. the Ekadhikas are 1, 2, 3, 4, etc. For 3, 13, 23, 33 etc. multiply them by 3 and you get 1, 4, 7, 10, etc. asthe Ekadhikas. For 7, 17, 27, 37 etc. multiply them by 7 and you get 5, 12, 19, 26, etc.as the Ekadhikas.

For 1, 11, 21, 31 etc. multiply them by 9 and you get 1, 10, 19, 28, etc.as the Ekadhikas.

Now test if 112 is divisible by 7 osculating by 5: 25 + 11 = 21, whichis divisible by 7. Therefore: yes

Is 2774 divisible by 19? Osculate by 2:

2 7 7 4

+ 8

2 8 5

+ 1 0

3 8

+ 1 6

1 9


14/30


One more example: Is 5293240096 divisible by 139?

The Ekadhika (osculator) is 14.

5 2 9 3 2 4 0 0 9 6

139 89 36 131 29 131 19 51 93

Answer: yes

2.5.1 Exercises

Using the osculation method, check if

a 32896 is divisible by 29

b 93148 is divisible by 29

c 4914 is divisible by 39

d 14061 is divisible by 43

2.6 Square Roots (Vargamula)

1, 5, 6 and 0 at the end of a number reproduce themselves as the lastdigits in the square. Squares of complements from 10 have the same last digit; thus 1 2 and 92end in 1, 22 and 82 end in 4 etc. 2, 3, 7 and 8 are out of court altogether.

If the given number has n digits, then the square root will containn

2 orn +12 digits.

Systematic computation of an exact square root requires the Dvandvayoga (Duplex) process.


15/30


2.6.1 Duplex Process (Dvandvayoga)

D(4) = 4 2 = 16 (1)D(43) = 2 4 3 = 24 (2)

D(137) = 2 1 7 + 32 = 23 (3)

D(1034) = 2 1 4 + 2 0 3 = 8 (4)D(10345) = 2 1 5 + 2 0 4 + 3

2 = 19 (5)

Got it?

2.6.2 Square Root of a Perfect Square

Find 1849.Group in pairs, taking a single extra digit on the left as extra digit.

1 8 4 9

8) 2

4

4 is the largest integer whose square does not exceed 18.18/ 4 is 4 with remainder 2.The divisor 8 is two times 4.

Next we divide 24 by the divisor 8. This gives 3 remainder 0, placed as

1 8 4 9

8) 2 0

4 3

Now we see 09 and we deduct from this the duplex of the last answer gure 3,i.e. 09 D(3) = 09 32 = 09 9 = 0. This means that the answer is exactly43.


16/30


1 3 6 9

6) 4

3

3 is the largest integer whose square does not exceed 13.13/ 3 is 3 with remainder 4.The divisor 6 is two times 3.

Next we divide 46 by the divisor 6. This gives 7 remainder 4, placed as

1 3 6 9

6) 4 4

3 7

49 D(7) = 0, so 37 is the exact square root of 1369.


17/30


2.6.3 Exercises

Find the square root of the following.

a 3136

b 3969

c 5184

d 3721

e 6889

f 1296

Find out how the method extends to 6-digit squares and nd the square roof of the numbers

a 119025

b 524176

c 519841

d 375769

3 Zero Search Methods1. Newtons Method

2. Herons Square Root Finder

3. Exercise: Divisions without Dividing


18/30


Sir Isaac Newton1643 - 1727

3.1 Newtons Method

Given a differentiable function f (x) we want to determine x such thatf (x) = 0 . (6)

Starting with x0 we take the tangent to the curve through the point ( x0 , f (x0 ))and use its intersection with the x-axis x1 as a new starting point. We repeatthis method until no further changes occur. The recursive relation is

xn +1 = xn f (xn )f (xn )

(7)

and the result islim

n xn = x. (8)


19/30


Newtons Method - graphically

3.1.1 Problems

Problems can occur due to

1. multiple solutions

2. non convex f , reection points

3. solutions at extreme values4. |f | = 5. pathological cases


20/30


3.1.2 Rate of Convergence

Considering all these problems, why is Newtons method still so popular? Thereason lies in the rate of convergence. Dene the error by

en= xn x (9)

From the denition of the Newton iteration, we have

en +1 = xn +1 x= xn

f (xn )f (xn ) x

= en f (xn )f (xn )

=en f (xn ) f (xn )

f (xn )(10)

By Taylors Theorem, we have

0 = f (x) = f (xn en ) (11)= f (xn )

en f (xn ) +

1

2e2n f (n ) (12)

where n is a number between xn and x. A rearrangement of this equationyields

en f (xn ) f (xn ) =12

f (n )e2n (13)

Putting this in ( 10) leads to

en +1 =12

f (n )f (xn )

e2n 12

f (x)f (x)

e2n = Ce2n (14)

This equation tells us that en +1 is roughly a constant times e2n . This desirablestate of affairs is called quadratic convergence .


21/30


3.2 Herons Square Root Finder

Goldman Sachs standard interview question: Find the square root of 17. The Greek Engineer Heron who lived sometime between 100 B.C. and 100A.D. had used the recursion

xn +1 =12

xn +Rxn

(15)

to nd the square root of R.

This is based on Newtons method.

Taking R = 17 and starting with x0 = 4 we nd x4 =4.123105625617660549821409856 which is correct to 28 gures.

3.3 Exercise: Divisions without Dividing

Use Newtons method to determine an algorithm that computes the re-ciprocal of a given number x, without ever performing any division. apply your method to compute 119 exact to 8 decimal places.

4 Credit CardsWe follow An Introduction to the Mathematics of Money by Lovelock, Mendeland Wright [ 3].

9876 BIN

5432 1987 654

cardholder ID3

checksum(16)

BIN: Bank Identication Number More on http://en.wikipedia.org/wiki/Credit_card_numbers checksum: designed to protect against accidental errors, not maliciousattacks. How? Luhn algorithm
http://www.gs.com/http://en.wikipedia.org/wiki/Credit_card_numbershttp://en.wikipedia.org/wiki/Credit_card_numbershttp://www.gs.com/


22/30


4.1 The Luhn Algorithm

Hans Peter Luhn

The Luhn algorithm or Luhn formula , also known as the modulus 10 or mod 10 algorithm , is a simple checksum formula used to validate a variety of identication numbers, such as credit card numbers and Canadian SocialInsurance Numbers.

It was created by IBM scientist Hans Peter Luhn and described in US Patent2,950,048, led on January 6, 1954, and granted on August 23, 1960.


23/30


9876 BIN 5432 1987 654 cardholder ID 3 checksum9 8 7 6 5 4 3 2 1

2 2 2 2 2 18 8 14 6 10 4 6 2 2

9 8 7 6 5 4 add up

2 2 2 digits9 16 7 12 5 8 82

Table 2: Example of the Luhn Algorithm

Note: 18 counts as 1 + 8, i.e. count only the digits

9876

BIN5432 1987 654

cardholder ID3

checksum Add the checksum 3 to the 82 and obtain 85. If the new total is divisible by 10, then the credit card number has passedthe validation test. our number fails, whereas

9876

BIN5432 1987 654

cardholder ID8

checksumpasses.

Note: Amex has one digit less and starts with the second digit.


24/30


4.2 Exercises1. Implement the Luhn-Algorithm in Excel.

2. Apply the Luhn-Algorithm to your own credit card number.

3. In your group think of a credit card number and ask another group toverify it.

4. Does the Luhn Algorithm pick up any incorrect entry of a single digit?

5. Will the Luhn Algorithm pick up any incorrect transposition of adjacentdigits?

5 Cryptography1. The Inventors of the RSA-Algorithm

2. Method

3. Examples

4. Exercise

5.1 The Inventors of the RSA-Algorithm (1978)Ron Rivest, Adi Shamir, and Leonard Adleman [ 5]


25/30


Ronald L. RivestAndrew and Erna Viterbi Professor of Electrical Engineering and ComputerScience in MITs Department of Electrical Engineering and Computer Science

http://theory.lcs.mit.edu/ ~rivest/

Adi ShamirPaul and Marlene Borman Professor of Applied Mathematics at WeizmannInstitute

http://www.weizmann.ac.il/math/profile04/scientists/

shamir-prof04.html
http://theory.lcs.mit.edu/~rivest/http://theory.lcs.mit.edu/~rivest/http://theory.lcs.mit.edu/~rivest/http://www.weizmann.ac.il/math/profile04/scientists/shamir-prof04.htmlhttp://www.weizmann.ac.il/math/profile04/scientists/shamir-prof04.htmlhttp://www.weizmann.ac.il/math/profile04/scientists/shamir-prof04.htmlhttp://www.weizmann.ac.il/math/profile04/scientists/shamir-prof04.htmlhttp://theory.lcs.mit.edu/~rivest/


26/30


Professor Leonard AdlemanDepartment of Computer Science University of Southern California

http://www.usc.edu/dept/molecular-science/fm-adleman.htm

5.2 MethodWe follow Francis Litterio instructions on http://world.std.com/ ~franl/crypto/rsa-guts.html .

1. Find P and Q, two large (e.g., 1024-bit) prime numbers.

2. Choose E ]1, P Q [, no prime factors in common with (P 1)(Q 1).3. Compute D (the multiplicative inverse) such that DE = 1 mod (P 1)(Q1). (Find an integer X which causes D = ( X (P 1)(Q1)+1) /E to be an integer, then use that value of D.)4. The encryption function is C = T E mod P Q, where C is the ciphertext

(a positive integer), T is the plaintext (a positive integer). The messagebeing encrypted, T , must be less than the modulus, P Q.

5. The decryption function is T = C D mod P Q, where C is the ciphertext

(a positive integer), T is the plaintext (a positive integer).
http://www.usc.edu/dept/molecular-science/fm-adleman.htmhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://world.std.com/~franl/crypto/rsa-guts.htmlhttp://www.usc.edu/dept/molecular-science/fm-adleman.htm


27/30


Your public key is the pair (PQ,E ). Your private key is the number D(reveal it to no one). The product P Q is the modulus (often called N in theliterature). E is the public exponent. D is the secret exponent.

You can publish your public key freely, because there are no known easy meth-ods of calculating D, P , or Q given only (PQ,E ) (your public key). If P and Q are each 1024 bits long, the sun will burn out before the most powerfulcomputers presently in existence can factor your modulus into P and Q.

5.3 Examples

prime 1 P 5 11 37 61

prime 2 Q 7 7 41 53

public modulus N = P Q 35 77 1517 3233

P 1 4 10 36 60Q 1 6 6 40 52(P 1)(Q 1) 24 60 1440 3120

public exponent E 5 7 7 17

inverse of E D 5 43 823 2753

check DE 25 301 5761 46801

check DE 1 24 300 5760 46800check [DE 1]/ [(P 1)(Q 1)] 0 0 0 0plaintext T 5 6 100 123

encryption function C = T E mod P Q 10 41 1062 855

decryption function T = C D mod P Q 5 6 100 123


28/30


5.4 Exercise1. Set up an RSA tool in Excel/VBA and verify the examples.

2. Take two prime numbers P and Q between 100 and 200, e.g. from http://primes.utm.edu and compute E and D.

3. Based on a plain text of your choice compute its ciphertext.

4. Give your public key and the ciphertext to the next group and determinethe plaintext of another group.

6 Denitions6.1 Prime Numbersa number that can only be divided evenly by 1 and the number itself.

6.2 ModulusFor integers K , R and N the equation

K = R mod N (17)means that R is the remainder of the division of K by N or alternatively thereexists an integer L such that

N L + R = K. (18)
http://primes.utm.edu/http://primes.utm.edu/http://primes.utm.edu/http://primes.utm.edu/


29/30


7 Contact InformationProfessor Dr. Uwe WystupAnsua Dutta-WystupManaging Directors

MathFinance AGMainluststrae 460329 Frankfurt am MainGermanyPhone +49-700-MATHFINANCE

More papers are available athttp://www.mathfinance.com/wystup/papers.phpThese slides and handouts are available athttp://www.mathfinance.com/seminars/vedic.php

References[1] Burden , R. L. and Faires , J. D. (1993). Numerical analysis . PWS

Publishing Company.

[2] Datta , B. and Singh , A.N. (1962). History of Hindu Mathematics .Asia Publishing House, Calcutta.

[3] Lovelock , D., Mendel , M. and Wright , A.L. (2007). An Introduc-tion to the Mathematics of Money , Springer , New York.

[4] Maharaja , Bharati Krsna Tirthaji (1992). Vedic Mathematics , MotilalBanarsidass Publishers Private Ltd, Delhi.

[5] Rivest , R.L., Shamir , A. and Adleman , L.M. (1978). A Method for

Obtaining Digital Signatures and Public-Key Cryptosystems. Commu-nications of the ACM 21,2 , 120126.

[6] Schonard , A. and Kokot , C. (2006). Der Mathekn uller . http://www. matheknueller.de .

[7] Williams , K.R. (2002). Vedic Mathematics - Teachers Manual . Ad-vanced Level. Motilal Banarsidass Publishers Private Limited, Delhi.http://www.mlbd.com
http://www.mathfinance.com/wystup/papers.phphttp://www.mathfinance.com/seminars/vedic.phphttp://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-17%%@%203660912-0http://www.matheknueller.de/http://www.matheknueller.de/http://www.mlbd.com/http://www.mlbd.com/http://www.matheknueller.de/http://www.matheknueller.de/http://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-17%%@%203660912-0http://www.mathfinance.com/seminars/vedic.phphttp://www.mathfinance.com/wystup/papers.php


30/30

Indexbank identication number (BIN), 21BIN, 21

division, 11, 13dual system, 3duplex, 15dvandvayoga, 15

ekadhika, 13

Herons Square Root Finder, 21hexadecimal system, 3

Luhn algorithm, 22

Maya numerals, 3modulus, 28multiplication, 6

Newtons Method, 18

numeral systems, 3osculator, 13

prime number, 28

quadratic convergence, 20

RSA-Algorithm, 24

square root, 14squares, 10

sutras, 5vedas, 4

30

highlights of computing: from the ancient indian vedic literature to modern practical cryptography

Documents