Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy...

Reed-Solomon, the

ChineseRemainderTheorem,

and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto

QRTWhy?Da YenWeaveWeave Swap

Summary

Reed-Solomon, the Chinese RemainderTheorem, and Cryptography

Dr. Anna Johnston

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Outline

1 Overview

2 Code

3 Crypto

4 QRTWhy?Da YenWeaveWeave Swap

5 Summary

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Comparison

Reed-Solomon

I What: Errordetection andcorrection code;

I Purpose:Encodesinformation todetect andcorrect errorsand allow forpartial data loss;

I How:Overdeterminesa polynomial toallow for lost orcorrupted data;

Da-Yen

I What:Isomorphismfrom single largequotient ring tothe directproduct ofsmaller quotientrings;

I Purpose: Usedin other proofsand an enormousnumber ofapplications;

I How: Breakslarge problemsinto smaller,parallelproblems.

Cryptography

I What:Security/Privacycodes;

I Purpose:Protectsinformationagainstdisclosure,verifies sender;

I How: Algorithmsdesigned with amix of math andart.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Comparison

Reed-Solomon




Da-YenI What: Isomorphism from

single large quotient ring tothe direct product of smallerquotient rings;

I Purpose: Used in other proofsand an enormous number ofapplications;

I How: Breaks large problemsinto smaller, parallel problems.

Cryptography




Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen (Chinese Remainder Theorem)

What is it?

Let R be a ring and {Ik | 0 ≤ k < n} be a set of pair-wiseprime ideals;

The quotient ring R/⋂n−1

k=0 Ik is isomorphic to the directproduct

∏n−1k=0 R/Ik .

µ̇0 (I0) µ̇1 (I1) µ̇2 (I2) µ̇3 (I3)

˙〈R〉 (⋂n−1

k=0 Ik)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


What is it?

Let R = Z and {mk | 0 ≤ k < n} be a set of relatively primeintegers;

The quotient ring Z/∏n−1

k=0 mk is isomorphic to the directproduct

∏n−1k=0 Z/mk .

2 mod 3 3 mod 5 1 mod 11 7 mod 13

(683 mod 2145)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


What is it?

Let R = F[x ] and {mk(x) | 0 ≤ k < n} be a set of relativelyprime polynomials over F;

The quotient ring F[x ]/∏n−1

k=0 mk(x) is isomorphic to the directproduct

∏n−1k=0 F[x ]/mk(x).

2 mod (x − 1) 1 mod (x − 2) 0 mod (x − 3) 1 mod (x − 4)

3−1(x3 − 6x2 + 8x + 3

)mod

(x4 − 10x3 + 35x2 − 50x + 24

)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Comparison

Reed-Solomon








Cryptography




Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Great Extension Tree

Da Yen Chinese Remainder Theorem

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Integers (Z) Polynomials

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Integers (Z) OtherNon-Commutative?

Polynomials

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Integers (Z)

Exponential Base

Polynomials

Deg > 1 Deg One

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Integers (Z)

Exponential

Finite Cyclic GroupRoots (square, cube, etc)

Discrete Logs (Pohlig-Hellman)Factoring (Pollard)

Base

Parallel Arithmetic(redundant number systems)Montgomery Reduction

(multiplication)Fast RSA

Integer Secret Sharing

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Polynomials

Deg > 1

Secret Sharing VariantsError correction codes

Polynomial Factorization

Deg One

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Polynomials

Deg OnePolynomialEvaluation

PolynomialInterpolation

Secret Sharing VariantsError correction codes

(Reed-Solomon)Polynomial Factorization

Discrete FFT’sNumber Theoretic Transform

Truncated Taylor Series Derivation

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Comparison

Reed-Solomon








Cryptography




Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Comparison

Reed-SolomonI What: Error detection and

correction code;I Purpose: Encodes information

to detect and correct errorsand allow for partial data loss;

I How: Overdetermines apolynomial to allow for lost orcorrupted data;

Da-Yen




Cryptography




Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Data as Polynomial, Code as Set of Points

Early, simplified version of Reed-Solomon

−2 −1 0 1 2 3 40

20

40

60

x

f(x)=

2x3+

0x2−

7x+

10

Four coefficientsdefines cubic

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



−2 −1 0 1 2 3 40

20

40

60

x

f(x)=

2x3+

0x2−

7x+

10

Four pointsdefines cubic

x −1 0 2 3f (x) 15 10 12 43

f (x) ≡ 15 mod (x − (−1))≡ 10 mod (x − 0)

≡ −5x + 10 mod (x2 + 1)

≡ 12 mod (x − 2)

≡ 43 mod (x − 3)

≡ 31x − 50 mod (x2 − 5x + 6)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



−2 −1 0 1 2 3 40

20

40

60

x

f(x)=

2x3+

0x2−

7x+

10

Any four relationsdefines cubic

x − 2 −1 0 1 2 3f (x) 8 15 10 5 12 43

f (x) ≡ 8 mod (x − (−2))≡ 15 mod (x − (−1))≡ 10 mod (x − 0)

≡ 5 mod (x − 1)

≡ 12 mod (x − 2)

≡ 43 mod (x − 3)

Over Determined By TwoRelations

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



−2 −1 0 1 2 3 40

20

40

60

x

f(x)=

11 6x

3−

1 2x

2−

22 3x+

10


x − 2 −1 0 1 2 3f (x) 8 15 10 4 12 43

If one point is corrupted and 4relations mapped to polynomial:

1 Two relations will not fitpolynomial or,

2 Corrupted relation is notused (and doesn’t fit).

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



−2 −1 0 1 2 3 40

20

40

60

80

x

f(x)=

5 2x

3−

1 2x

2−

8x+

10


x − 2 −1 0 1 2 3f (x) 8 15 10 4 12 43




Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



−2 −1 0 1 2 3 40

20

40

60

x

f(x)=

2x3+

0x2−

7x+

10


x − 2 −1 0 1 2 3f (x) 8 15 10 4 12 43




Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



−2 −1 0 1 2 3 40

20

40

60

x

f(x)=

2x3+

0x2−

7x+

10


x − 2 −1 0 1 2 3f (x) 8 15 10 5 12 43

Assumption: Less than half thespares are corrupted.

If at least 1/2 the spare relationsare on the curve,

it is correct.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

BCH Reed-Solomon

Most Widely UsedReed-Solomon CodeMost common form ofReed-Solomon is the BCHvariantData is polynomial;Code is polynomial shiftedby the number of sparerelations;And made to be equivalentto 0 for each sparerelation.

Odd DecodingSpare relations must besequential powers ofmultiplicative groupgenerator.Same underlying theory,but goes about it in around-about way.Standard size: Field F28 ;223 data words and 32spare relations.

There are(

25532

)subsets of size 32 – far toomany to check.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Quotient Ring Transform

What is it? (Integers: R = {ak mod mk}, {mk} are all co-prime)

Da Yen/(CRT)

Notation: 〈̂R〉 =∏

mk , ˙〈R〉 ≡ ak mod mk with 0 ≤ ak < mk

R ⇔ ˙〈R〉 mod 〈̂R〉

2mod3

3mod5

1mod11

7mod13

(˙〈R〉 mod 〈̂R〉

)= (683 mod 2145) ;

Converts a set of relationsa mod m

To a larger relationModulo

∏mk

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


What is it? (Integers: R = {ak mod mk}, {mk} are all co-prime)

QRT

Notation: 〈̂R〉 =∏

mk , ˙〈R〉 ≡ ak mod mk with 0 ≤ ak < mk

R ⇔ Q

Reduction gives Q ={(

bi ≡ ˙〈R〉 mod m′k

)}2

mod33

mod51

mod117

mod13

(683 mod 2145) ; (683 mod 52003)

4mod7

3mod17

18mod19

16mod 23

Converts a set of relationsa mod m

To another set of relationsb mod m′

Such that their combinedvalues are equal.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


What is it? (Integers: R = {µ̇k mod µ̂k}, {µ̂k} are all co-prime)

QRT

Notation: 〈̂R〉 =∏µ̂k , ˙〈R〉 ≡ µ̇k mod µ̂k with 0 ≤ µ̇k < µ̂k

R ⇔ Q

Q = {νk}

µ̇0modµ̂0

µ̇1modµ̂1

µ̇2modµ̂2

µ̇3modµ̂3

ν̇0modν̂0

ν̇1modν̂1

ν̇2modν̂2

ν̇3mod ν̂3

Without computingthe combined value.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


What is it? (E is Euc. Dom.: R = {µ̇k mod µ̂k}, {µ̂k} are all co-prime)

QRT


R ⇔ Q

Q = {νk} = R(Q)

µ̇0modµ̂0

µ̇1modµ̂1

µ̇2modµ̂2

µ̇3modµ̂3

ν̇0modν̂0

ν̇1modν̂1

ν̇2modν̂2

ν̇3mod ν̂3

Without computingthe combined value.

Better E : F[x ]Best E : F2n [x ]

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


What is it? (E is Euc. Dom.: R = {µ̇k mod µ̂k}, {µ̂k} are all co-prime)

QRT


R ⇔ Q

Q = {νk} = R(Q)

2mod3

3mod5

1mod11

7mod13

4mod7

3mod17

18mod19

〈̂R〉 = 2145〈̂Q3〉 = 2261

Recoverable withlost relation

Better E : F[x ]Best E : F2n [x ]

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Comparison

Reed-Solomon




Da-Yen




Cryptography

I What: Security/Privacycodes;

I Purpose: Protectsinformation against disclosure,verifies sender;

I How: Algorithms designedwith a mix of math and art.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Encryption/Encipherment Basics

(Secret Variable)

Cryptovariable/Key

(Usable data)PlainText Encrypt CipherText

(Gobbldygook)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


(Secret Variable)

Cryptovariable/Key

(Usable data)PlainText Decrypt CipherText

(Gobbldygook)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


(Secret Variable)

Cryptovariable/Key

(Usable data)PlainText CipherText

(Gobbldygook)

Key Stream

Key Generator

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


(Secret Variable)

Cryptovariable

Key Stream

Key Generator

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Overdetermined Polynomial: determining relationsr = 2, s = 1

−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

Four relations determinecubic

0 1 2 3 4 5ck −2 −1 1 3µ̇k −32 −15 −5 13

f (x) ≡ −32 mod (x − (−2))≡ −15 mod (x − (−1))≡ −5 mod (x − 1)

≡ 13 mod (x − 3)

≡ −120 mod (x − (−4))≡ − 8 mod (x)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)=

x3−

2x2+

4x−

8

Four relations determinecubic

0 1 2 3 4 5ck −2 −1 1 3µ̇k −32 −15 −5 13

f (x) ≡ −32 mod (x − (−2))≡ −15 mod (x − (−1))≡ −5 mod (x − 1)

≡ 13 mod (x − 3)

≡ −120 mod (x − (−4))≡ − 8 mod (x)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)=

x3−

2x2+

4x−

8

Any four relationson f

0 1 2 3 4 5ck −2 −1 1 3µ̇k −32 −15 −5 13dk −4 − 3 0 2 4 5ν̇k −120 − 65 −8 0 40 87

f (x) ≡ −32 mod (x − (−2))≡ −15 mod (x − (−1))≡ −5 mod (x − 1)

≡ 13 mod (x − 3)

≡ −120 mod (x − (−4))≡ − 65 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

Any four relationsdetermine f

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 65 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 65 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

Less than fourdoes not

0 1 2 3 4 5dk −4 0 4ν̇k −120 −8 40

f (x) ≡ −120 mod (x − (−4))≡ − 8 mod (x)

≡ 40 mod (x − 4)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

200

x

f(x)

If a bad relationoccurs

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 60 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

200

x

f(x)

More than half ofextra relations fail

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 60 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 61.333 6≡ 40 mod (x − 4)

≡ 132 6≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

More than half ofextra relations fail

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 60 mod (x − (−3))≡ − 0.857 6≡ − 8 mod (x)

≡ 3.214 6≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

At least half ofextra relations pass

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 65 6≡ − 60 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Application Examples

Purpose

Flexible error correction/detection: R ←⇒ Q

Paired with stream cipher, becomes new ’mode’:

Adds integrity, disperses data and protects againstDDoS/Ransomware attacksPotential quantum resistant PKC

Cipher chooses moduliand salt relations

|R| = 2r + t|Q| = 2(r + s)

t salt relations in R

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose


Paired with stream cipher, becomes new ’mode’:Adds integrity, disperses data and protects againstDDoS/Ransomware attacks

Potential quantum resistant PKC

0 1 2 3

4 5 6 7

8 9 10 11

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 1 2 3

4 5 6 7

8 9 10 11

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2Corrects 2 Errors

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 2 3

5 6 7

8 9

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2Up to 4 relations lost

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 2 3

5 7

8 9

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2> 4 lost

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Problems and Solutions

Problem: Converting relations to polynomialsis computationally expensive

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Doing it repeatedly to find errorsis combinatorially worse

(2(r + s)

2r

)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




(2(r + s)

2r

)

Solution:

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




(2(r + s)

2r

)

Solution: Bypass polynomials using QRT

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




(2(r + s)

2r

)


All work is modulo mj

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




(2(r + s)

2r

)



Intermediate weave valuesreduced computation per check

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




(2(r + s)

2r

)



Intermediate weave valuesreduced computation per check

reduced number of checks

(r + sr

)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen, Weaving, and the QRT

Conversion from {modµ̂i} to {modν̂j}Weave enables efficient QRT

Based on the Da Yen (Chinese remainder theorem)

Single Sum

Iterative

For monic, degreeone polynomial

modulithis technique is

Newtoninterpolation

Enables weave andefficient transform

(and Montgomery multiplication)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




Single SumMost common formula

˙〈R〉 =∑j

〈̂R〉µ̂j

( 〈̂R〉µ̂j

)−1µ̇j mod µ̂j

mod 〈̂R〉

Iterative



Newtoninterpolation



Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary




Single Sum/Lagrange

For monic, degree one polynomial moduli,this equation is Lagrange interpolation

˙〈R〉 =∑j

〈̂R〉µ̂j

( 〈̂R〉µ̂j

)−1µ̇j mod µ̂j

mod 〈̂R〉

Iterative



Newtoninterpolation



Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Based on Iterative (Newton) Da Yen Formula

Single Sum/Lagrange

For monic, degree onepolynomial moduli,

this equation is Lagrangeinterpolation

Iterative/Newton

For monic, degree one polynomialmoduli

this technique is Newton interpolation



Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary



Based on Iterative (Newton) Da Yen Formula

Single Sum/Lagrange

For monic, degree onepolynomial moduli,

this equation is Lagrangeinterpolation

Iterative/Newton

For monic, degree one polynomialmoduli

this technique is Newton interpolationEnables weave andefficient transform


Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula

Iterative: Let〈̂Rk〉 =

∏k−1j=0 µ̂j

˙〈Rk〉 ≡ µ̇j mod µ̂j for j = 0, 1, . . . , k − 1; 0 ≤ ˙〈Rk〉 < 〈̂Rk〉.

1 Start with ˙〈R1〉 = µ̇0;2 Let ˙〈Rk〉 =

˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)3 Final result is ˙〈Rn〉 < 〈̂Rn〉.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j


1 Start with ˙〈R1〉 = µ̇0;

2 Let ˙〈Rk〉 =˙〈Rk−1〉+ 〈̂Rk−1〉

(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1


Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)

3 Final result is ˙〈Rn〉 < 〈̂Rn〉.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)


Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 0 mod 〈̂Rk−1〉≡

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)


Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 0 mod 〈̂Rk−1〉≡ µ̇j mod µ̂j ∀ 0 ≤ j < (k − 1)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)


Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 〈̂Rk−1〉〈̂Rk−1〉−1 (

˙µk−1 − ˙〈Rk−1〉)mod µ̂k−1

≡

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)


Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 〈̂Rk−1〉〈̂Rk−1〉−1 (

˙µk−1 − ˙〈Rk−1〉)mod µ̂k−1

≡ ˙µk−1 mod µ̂k−1

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1

)


Why?: ˙〈Rk〉 < 〈̂Rk〉

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+ 〈̂Rk−1〉(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1


Why?: ˙〈Rk〉 ≡ µ̇j mod µ̂j

0 ≤ j ≤ k

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

8 mod 15 1 mod 11 7 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

8 mod 15 1 mod 11 7 mod 13˙〈R3〉 = 8+ 15

((1− 8)15−1 mod 11

)= 8+ 15(1) = 23 mod 3 · 5 · 11

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

8 mod 15 1 mod 11 7 mod 13˙〈R3〉 = 8+ 15

((1− 8)15−1 mod 11

)= 8+ 15(1) = 23 mod 3 · 5 · 11

23 mod 165 7 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

8 mod 15 1 mod 11 7 mod 13˙〈R3〉 = 8+ 15

((1− 8)15−1 mod 11

)= 8+ 15(1) = 23 mod 3 · 5 · 11

23 mod 165 7 mod 13

˙〈R4〉=23+ 165((7− 23)165−1 mod 13

)= 23+ 165 (3(10) mod 13)

=23+ 165 (4) = 683 mod 3 · 5 · 11 · 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

8 mod 15 1 mod 11 7 mod 13˙〈R3〉 = 8+ 15

((1− 8)15−1 mod 11

)= 8+ 15(1) = 23 mod 3 · 5 · 11

23 mod 165 7 mod 13

˙〈R4〉=23+ 165((7− 23)165−1 mod 13

)= 23+ 165 (3(10) mod 13)

=23+ 165 (4) = 683 mod 3 · 5 · 11 · 13

683 mod 2145

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Da Yen Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13˙〈R2〉=2+ 3

((3− 2)3−1 mod 5

)=2+ 3 (2) = 8 mod 3 · 5

8 mod 15 1 mod 11 7 mod 13˙〈R3〉 = 8+ 15

((1− 8)15−1 mod 11

)= 8+ 15(1) = 23 mod 3 · 5 · 11

23 mod 165 7 mod 13

˙〈R4〉=23+ 165((7− 23)165−1 mod 13

)= 23+ 165 (3(10) mod 13)

=23+ 165 (4) = 683 mod 3 · 5 · 11 · 13

683 mod 2145˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4)))

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4))))

WeaveComputes intermediate relationsR = {2 mod 3, 2 mod 5, 1 mod 11, 4 mod 13} ;

Uses only modulo µ̂k operations;Operations: ;Follows iterative algorithm for ωk

But only computes ωk values

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave

˙〈R〉 = ω0 + µ̂0 (ω1 + µ̂1 (ω2 + µ̂2 (ω3))))

WeaveComputes intermediate relationsR = {ωk mod µ̂k | 0 ≤ k < 2r };Uses only modulo µ̂k operations;

Operations: ;Follows iterative algorithm for ωk


Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave

˙〈R〉 = ω0 + µ̂0 (ω1 + µ̂1 (ω2 + µ̂2 (ω3))))

WeaveComputes intermediate relationsR = {ωk mod µ̂k | 0 ≤ k < 2r };Uses only modulo µ̂k operations;Operations: (2r − 1)r in serial;

Follows iterative algorithm for ωk


Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave

˙〈R〉 = ω0 + µ̂0 (ω1 + µ̂1 (ω2 + µ̂2 (ω3))))

WeaveComputes intermediate relationsR = {ωk mod µ̂k | 0 ≤ k < 2r };Uses only modulo µ̂k operations;Operations: (2r − 1) in parallel;Follows iterative algorithm for ωk


Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Iterative Formula


∏k−1j=0 µ̂j



˙〈Rk−1〉+〈̂Rk−1〉

ωk−1︷︸︸︷(〈̂Rk−1〉

−1 (˙µk−1 − ˙〈Rk−1〉

)mod µ̂k−1


Why?: ˙〈Rk〉 ≡ µ̇j mod µ̂j

0 ≤ j ≤ k

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave

˙〈R〉 = ω0 + µ̂0 (ω1 + µ̂1 (ω2 + µ̂2 (ω3))))

WeaveComputes intermediate relationsR = {ωk mod µ̂k | 0 ≤ k < 2r };Uses only modulo µ̂k operations;Operations: (2r − 1) in parallel;Follows iterative algorithm for ωk


Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

2 mod 3 (3− 2)3−1 mod 5 (1− 2)3−1 mod 11 (7− 2)3−1 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

2 mod 3 (3− 2)3−1 mod 5 (1− 2)3−1 mod 11 (7− 2)3−1 mod 13

2 mod 5 7 mod 11 6 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

2 mod 3 (3− 2)3−1 mod 5 (1− 2)3−1 mod 11 (7− 2)3−1 mod 13

2 mod 5 7 mod 11 6 mod 13

2 mod 5 (7− 2)5−1 mod 11 (6− 2)5−1 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

2 mod 3 (3− 2)3−1 mod 5 (1− 2)3−1 mod 11 (7− 2)3−1 mod 13

2 mod 5 7 mod 11 6 mod 13

2 mod 5 (7− 2)5−1 mod 11 (6− 2)5−1 mod 13

1 mod 11 6 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

2 mod 3 (3− 2)3−1 mod 5 (1− 2)3−1 mod 11 (7− 2)3−1 mod 13

2 mod 5 7 mod 11 6 mod 13

2 mod 5 (7− 2)5−1 mod 11 (6− 2)5−1 mod 13

1 mod 11 6 mod 13

1 mod 11 (6− 1)11−1 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Example

2 mod 3 3 mod 5 1 mod 11 7 mod 13

2 mod 3 (3− 2)3−1 mod 5 (1− 2)3−1 mod 11 (7− 2)3−1 mod 13

2 mod 5 7 mod 11 6 mod 13

2 mod 5 (7− 2)5−1 mod 11 (6− 2)5−1 mod 13

1 mod 11 6 mod 13

1 mod 11 (6− 1)11−1 mod 13

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4))) 4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Example

2 mod 3 2 mod 5 1 mod 11 4 mod 13

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4) ) )

(1+ 4 · 4) mod 7 3 mod 7

(1+ 11 · 4) mod 17 11 mod 17

(1+ 11 · 4) mod 19 7 mod 19

(1+ 11 · 4) mod 23 22 mod 23

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Example

2 mod 3 2 mod 5 1 mod 11 4 mod 13

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4) ) )

(2+ 5 · 3) mod 7 3 mod 7

(2+ 5 · 11) mod 17 6 mod 17

(2+ 5 · 7) mod 19 18 mod 19

(2+ 5 · 22) mod 23 20 mod 23

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Example

2 mod 3 2 mod 5 1 mod 11 4 mod 13

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4) ) )

(2+ 3 · 3) mod 7 4 mod 7

(2+ 3 · 6) mod 17 3 mod 17

(2+ 3 · 18) mod 19 18 mod 19

(2+ 3 · 20) mod 23 16 mod 23

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

200

x

f(x)

Only four relationsdetermine all

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 60 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

200

x

f(x)

Using only first fourwoven relations

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 60 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 61.333 6≡ 40 mod (x − 4)

≡ 132 6≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

Swap woven orderand try alternate sets

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 60 mod (x − (−3))≡ − 0.857 6≡ − 8 mod (x)

≡ 3.214 6≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


−6 −4 −2 0 2 4 6

−200

−100

0

100

x

f(x)

Until at least halfrelations pass

0 1 2 3 4 5dk −4 − 3 0 2 4 5ν̇k −120 − 60 −8 0 40 87

f (x) ≡ −120 mod (x − (−4))≡ − 65 6≡ − 60 mod (x − (−3))≡ − 8 mod (x)

≡ 0 mod (x − 2)

≡ 40 mod (x − 4)

≡ 87 mod (x − 5)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap

Goal: Change woven orderwith minimal work.

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap Example

2 mod 3

2 mod 5

2 mod 5 (7− 2)5−1 mod 11

1 mod 11

4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap


Woven Relations

Order of laterdoes not effect earlier

Order of earlierdoes not effect later

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap Example

2 mod 3

2 mod 5

2 mod 5 (7− 2)5−1 mod 11

1 mod 11

4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap Example

2 mod 3

2 mod 5 7 mod 11

2 mod 5 1 · 5 + 2 mod 11

1 mod 11

4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap Example

2 mod 3

7 mod 11 2 mod 5

4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap Example

2 mod 3

7 mod 11 2 mod 5

7 mod 11 (2− 7)11−1 mod 5

4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap Example

2 mod 3

7 mod 11 2 mod 5

7 mod 11 (2− 7)11−1 mod 5

0 mod 5

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4)))= 2+ 3 (7+ 11 (0+ 5 (4)))

4 mod 13

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap


Woven Relations



µk =(µ̇k µ̂k−1 + ˙µk−1 mod µ̂k

)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap


Woven Relations




)µk−1 =

(( ˙µk−1 − µ̇k) µ̂k−1 mod µ̂k−1

)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap


Woven Relations




)µk−1 =

(( ˙µk−1 − µ̇k) µ̂k−1 mod µ̂k−1

)Swap relations k , k − 1

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Weave Swap: Monic, Degree one moduli


Woven Relations



t = µ̇k


)µk−1 =

(t mod µ̂k−1

)Swap relations k , k − 1

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Summary

Points To RememberQuotient Ring Transform;

Encodes with even dispersal of information;Positive and negative thresholding;May be used as a cryptographic scheme; adds randomaccess to data, added integrity, increased security;Using E = F[x ] the system is homomorphic under addition.

Converts data from one set of relationsto an entirely new set of relationsWithout large data computations

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Summary

Points To RememberQuotient Ring Transform;Encodes with even dispersal of information;

Positive and negative thresholding;May be used as a cryptographic scheme; adds randomaccess to data, added integrity, increased security;Using E = F[x ] the system is homomorphic under addition.

Each encode relation is a snapshotof original relation set

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Example

2 mod 3 2 mod 5 1 mod 11 4 mod 13

˙〈R〉 = 2+ 3 (2+ 5 (1+ 11 (4) ) )

(2+ 3 · 3) mod 7 4 mod 7

(2+ 3 · 6) mod 17 3 mod 17

(2+ 3 · 18) mod 19 18 mod 19

(2+ 3 · 20) mod 23 16 mod 23

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Summary

Points To RememberQuotient Ring Transform;Encodes with even dispersal of information;Positive and negative thresholding;

May be used as a cryptographic scheme; adds randomaccess to data, added integrity, increased security;Using E = F[x ] the system is homomorphic under addition.

Less than threshold hides dataMore than threshold corrects errors

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Summary

Points To RememberQuotient Ring Transform;Encodes with even dispersal of information;Positive and negative thresholding;May be used as a cryptographic scheme; adds randomaccess to data, added integrity, increased security;

Using E = F[x ] the system is homomorphic under addition.

Different subsets of data every decryption.Errors detectable, correctable

Not enough points, no decryption

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 1 2 3

4 5 6 7

8 9 10 11

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 1 2 3

4 5 6 7

8 9 10 11

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2Corrects 2 Errors

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 2 3

5 6 7

8 9

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2Up to 4 relations lost

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary


Purpose




0 2 3

5 7

8 9

|R| = 2r + t|Q| = 2(r + s)


Ex: r = 4, s = 2> 4 lost

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

QRT Summary

Points To RememberQuotient Ring Transform;Encodes with even dispersal of information;Positive and negative thresholding;May be used as a cryptographic scheme; adds randomaccess to data, added integrity, increased security;Using E = F[x ] the system is homomorphic under addition.

E (a) + E (b) = E (a+ b)

Reed-Solomon, the


and Cryptog-raphy

Dr. AnnaJohnston

Overview

Code

Crypto


Summary

Potential Hard Problem

Given {µ̇k | 0 ≤ k < 2(r + s)}Determined by any 2r relations

Find µ̂k or ˙〈R〉

Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy...

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Transcript of Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy...