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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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Outline
1 Overview
2 Code
3 Crypto
4 QRTWhy?Da YenWeaveWeave Swap
5 Summary
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Outline
1 Overview
2 Code
3 Crypto
4 QRTWhy?Da YenWeaveWeave Swap
5 Summary
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Outline
1 Overview
2 Code
3 Crypto
4 QRTWhy?Da YenWeaveWeave Swap
5 Summary
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Outline
1 Overview
2 Code
3 Crypto
4 QRTWhy?Da YenWeaveWeave Swap
5 Summary
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Outline
1 Overview
2 Code
3 Crypto
4 QRTWhy?Da YenWeaveWeave Swap
5 Summary
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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-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
I What:Security/Privacycodes;
I Purpose:Protectsinformationagainstdisclosure,verifies sender;
I How: Algorithmsdesigned with amix of math andart.
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Da Yen (Chinese Remainder Theorem)
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Da Yen (Chinese Remainder Theorem)
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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-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
I What:Security/Privacycodes;
I Purpose:Protectsinformationagainstdisclosure,verifies sender;
I How: Algorithmsdesigned with amix of math andart.
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
Integers (Z) Polynomials
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
Integers (Z) OtherNon-Commutative?
Polynomials
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
Integers (Z)
Exponential Base
Polynomials
Deg > 1 Deg One
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
Polynomials
Deg > 1
Secret Sharing VariantsError correction codes
Polynomial Factorization
Deg One
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Great Extension Tree
Da Yen Chinese Remainder Theorem
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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-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
I What:Security/Privacycodes;
I Purpose:Protectsinformationagainstdisclosure,verifies sender;
I How: Algorithmsdesigned with amix of math andart.
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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)=
11 6x
3−
1 2x
2−
22 3x+
10
Any four relationsdefines cubic
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
80
x
f(x)=
5 2x
3−
1 2x
2−
8x+
10
Any four relationsdefines cubic
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
Any four relationsdefines cubic
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
Any four relationsdefines cubic
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Quotient Ring Transform
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Quotient Ring Transform
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Quotient Ring Transform
What is it? (E is Euc. Dom.: 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} = 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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Quotient Ring Transform
What is it? (E is Euc. Dom.: 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} = 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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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: Protectsinformation against disclosure,verifies sender;
I How: Algorithms designedwith a mix of math and art.
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Encryption/Encipherment Basics
(Secret Variable)
Cryptovariable/Key
(Usable data)PlainText Encrypt CipherText
(Gobbldygook)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Encryption/Encipherment Basics
(Secret Variable)
Cryptovariable/Key
(Usable data)PlainText Decrypt CipherText
(Gobbldygook)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Encryption/Encipherment Basics
(Secret Variable)
Cryptovariable/Key
(Usable data)PlainText CipherText
(Gobbldygook)
Key Stream
Key Generator
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Encryption/Encipherment Basics
(Secret Variable)
Cryptovariable
Key Stream
Key Generator
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 1 2 3
4 5 6 7
8 9 10 11
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 1 2 3
4 5 6 7
8 9 10 11
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2Corrects 2 Errors
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 2 3
5 6 7
8 9
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2Up to 4 relations lost
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 2 3
5 7
8 9
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2> 4 lost
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Doing it repeatedly to find errorsis combinatorially worse
(2(r + s)
2r
)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Doing it repeatedly to find errorsis combinatorially worse
(2(r + s)
2r
)
Solution:
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Doing it repeatedly to find errorsis combinatorially worse
(2(r + s)
2r
)
Solution: Bypass polynomials using QRT
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Doing it repeatedly to find errorsis combinatorially worse
(2(r + s)
2r
)
Solution: Bypass polynomials using QRT
All work is modulo mj
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Doing it repeatedly to find errorsis combinatorially worse
(2(r + s)
2r
)
Solution: Bypass polynomials using QRT
All work is modulo mj
Intermediate weave valuesreduced computation per check
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Problems and Solutions
Problem: Converting relations to polynomialsis computationally expensive
Doing it repeatedly to find errorsis combinatorially worse
(2(r + s)
2r
)
Solution: Bypass polynomials using QRT
All work is modulo mj
Intermediate weave valuesreduced computation per check
reduced number of checks
(r + sr
)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 SumMost common formula
˙〈R〉 =∑j
〈̂R〉µ̂j
( 〈̂R〉µ̂j
)−1µ̇j mod µ̂j
mod 〈̂R〉
Iterative
For monic, degreeone polynomial
modulithis technique is
Newtoninterpolation
Enables weave andefficient transform
(and Montgomery multiplication)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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/Lagrange
For monic, degree one polynomial moduli,this equation is Lagrange interpolation
˙〈R〉 =∑j
〈̂R〉µ̂j
( 〈̂R〉µ̂j
)−1µ̇j mod µ̂j
mod 〈̂R〉
Iterative
For monic, degreeone polynomial
modulithis technique is
Newtoninterpolation
Enables weave andefficient transform
(and Montgomery multiplication)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Da Yen, Weaving, and the QRT
Conversion from {modµ̂i} to {modν̂j}Weave enables efficient QRT
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
Enables weave andefficient transform
(and Montgomery multiplication)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Da Yen, Weaving, and the QRT
Conversion from {modµ̂i} to {modν̂j}Weave enables efficient QRT
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
(and Montgomery multiplication)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉.
Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 0 mod 〈̂Rk−1〉≡
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉.
Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 0 mod 〈̂Rk−1〉≡ µ̇j mod µ̂j ∀ 0 ≤ j < (k − 1)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉.
Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 〈̂Rk−1〉〈̂Rk−1〉−1 (
˙µk−1 − ˙〈Rk−1〉)mod µ̂k−1
≡
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉.
Why?: ˙〈Rk〉 ≡ ˙〈Rk−1〉+ 〈̂Rk−1〉〈̂Rk−1〉−1 (
˙µk−1 − ˙〈Rk−1〉)mod µ̂k−1
≡ ˙µk−1 mod µ̂k−1
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉.
Why?: ˙〈Rk〉 < 〈̂Rk〉
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉.
Why?: ˙〈Rk〉 ≡ µ̇j mod µ̂j
0 ≤ j ≤ k
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Da Yen Example
2 mod 3 3 mod 5 1 mod 11 7 mod 13
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
But only computes ωk values
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
But only computes ωk values
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
But only computes ωk values
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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〉
ωk−1︷ ︸︸ ︷(〈̂Rk−1〉
−1 (˙µk−1 − ˙〈Rk−1〉
)mod µ̂k−1
)3 Final result is ˙〈Rn〉 < 〈̂Rn〉.
Why?: ˙〈Rk〉 ≡ µ̇j mod µ̂j
0 ≤ j ≤ k
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
But only computes ωk values
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Example
2 mod 3 3 mod 5 1 mod 11 7 mod 13
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Overdetermined Polynomial: determining relationsr = 2, s = 1
−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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap
Goal: Change woven orderwith minimal work.
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap
Goal: Change woven orderwith minimal work.
Woven Relations
Order of laterdoes not effect earlier
Order of earlierdoes not effect later
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap Example
2 mod 3
7 mod 11 2 mod 5
4 mod 13
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap
Goal: Change woven orderwith minimal work.
Woven Relations
Order of laterdoes not effect earlier
Order of earlierdoes not effect later
µk =(µ̇k µ̂k−1 + ˙µk−1 mod µ̂k
)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap
Goal: Change woven orderwith minimal work.
Woven Relations
Order of laterdoes not effect earlier
Order of earlierdoes not effect later
µk =(µ̇k µ̂k−1 + ˙µk−1 mod µ̂k
)µk−1 =
(( ˙µk−1 − µ̇k) µ̂k−1 mod µ̂k−1
)
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap
Goal: Change woven orderwith minimal work.
Woven Relations
Order of laterdoes not effect earlier
Order of earlierdoes not effect later
µk =(µ̇k µ̂k−1 + ˙µk−1 mod µ̂k
)µk−1 =
(( ˙µk−1 − µ̇k) µ̂k−1 mod µ̂k−1
)Swap relations k , k − 1
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Weave Swap: Monic, Degree one moduli
Goal: Change woven orderwith minimal work.
Woven Relations
Order of laterdoes not effect earlier
Order of earlierdoes not effect later
t = µ̇k
µk =(µ̇k µ̂k−1 + ˙µk−1 mod µ̂k
)µk−1 =
(t mod µ̂k−1
)Swap relations k , k − 1
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 1 2 3
4 5 6 7
8 9 10 11
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 1 2 3
4 5 6 7
8 9 10 11
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2Corrects 2 Errors
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 2 3
5 6 7
8 9
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2Up to 4 relations lost
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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 attacks
Potential quantum resistant PKC
0 2 3
5 7
8 9
|R| = 2r + t|Q| = 2(r + s)
t salt relations in R
Ex: r = 4, s = 2> 4 lost
Reed-Solomon, the
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
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
ChineseRemainderTheorem,
and Cryptog-raphy
Dr. AnnaJohnston
Overview
Code
Crypto
QRTWhy?Da YenWeaveWeave Swap
Summary
Potential Hard Problem
Given {µ̇k | 0 ≤ k < 2(r + s)}Determined by any 2r relations
Find µ̂k or ˙〈R〉
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