reed-solomon, the chinese remainder · reed-solomon,the chinese remainder theorem, andcryptog-raphy...
TRANSCRIPT
![Page 1: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/1.jpg)
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
![Page 2: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/2.jpg)
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
![Page 3: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/3.jpg)
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
![Page 4: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/4.jpg)
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
![Page 5: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/5.jpg)
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
![Page 6: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/6.jpg)
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
![Page 7: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/7.jpg)
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.
![Page 8: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/8.jpg)
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.
![Page 9: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/9.jpg)
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)
![Page 10: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/10.jpg)
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)
![Page 11: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/11.jpg)
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
)
![Page 12: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/12.jpg)
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.
![Page 13: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/13.jpg)
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
![Page 14: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/14.jpg)
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
![Page 15: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/15.jpg)
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
![Page 16: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/16.jpg)
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
![Page 17: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/17.jpg)
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
![Page 18: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/18.jpg)
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
![Page 19: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/19.jpg)
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
![Page 20: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/20.jpg)
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.
![Page 21: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/21.jpg)
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.
![Page 22: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/22.jpg)
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
![Page 23: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/23.jpg)
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)
![Page 24: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/24.jpg)
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
![Page 25: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/25.jpg)
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).
![Page 26: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/26.jpg)
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).
![Page 27: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/27.jpg)
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).
![Page 28: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/28.jpg)
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.
![Page 29: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/29.jpg)
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.
![Page 30: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/30.jpg)
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
![Page 31: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/31.jpg)
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.
![Page 32: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/32.jpg)
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.
![Page 33: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/33.jpg)
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 ]
![Page 34: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/34.jpg)
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 ]
![Page 35: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/35.jpg)
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.
![Page 36: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/36.jpg)
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)
![Page 37: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/37.jpg)
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)
![Page 38: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/38.jpg)
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
![Page 39: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/39.jpg)
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
![Page 40: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/40.jpg)
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)
![Page 41: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/41.jpg)
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)
![Page 42: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/42.jpg)
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)
![Page 43: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/43.jpg)
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)
![Page 44: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/44.jpg)
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)
![Page 45: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/45.jpg)
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)
![Page 46: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/46.jpg)
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)
![Page 47: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/47.jpg)
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)
![Page 48: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/48.jpg)
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)
![Page 49: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/49.jpg)
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
![Page 50: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/50.jpg)
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
![Page 51: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/51.jpg)
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
![Page 52: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/52.jpg)
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
![Page 53: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/53.jpg)
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
![Page 54: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/54.jpg)
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
![Page 55: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/55.jpg)
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
)
![Page 56: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/56.jpg)
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:
![Page 57: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/57.jpg)
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
![Page 58: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/58.jpg)
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
![Page 59: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/59.jpg)
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
![Page 60: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/60.jpg)
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
)
![Page 61: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/61.jpg)
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)
![Page 62: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/62.jpg)
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)
![Page 63: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/63.jpg)
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)
![Page 64: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/64.jpg)
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)
![Page 65: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/65.jpg)
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)
![Page 66: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/66.jpg)
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〉.
![Page 67: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/67.jpg)
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〉.
![Page 68: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/68.jpg)
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〉.
![Page 69: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/69.jpg)
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〉≡
![Page 70: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/70.jpg)
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)
![Page 71: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/71.jpg)
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
≡
![Page 72: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/72.jpg)
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
![Page 73: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/73.jpg)
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〉
![Page 74: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/74.jpg)
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
![Page 75: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/75.jpg)
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
![Page 76: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/76.jpg)
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
![Page 77: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/77.jpg)
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
![Page 78: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/78.jpg)
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
![Page 79: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/79.jpg)
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
![Page 80: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/80.jpg)
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
![Page 81: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/81.jpg)
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
![Page 82: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/82.jpg)
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)))
![Page 83: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/83.jpg)
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
![Page 84: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/84.jpg)
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
![Page 85: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/85.jpg)
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
![Page 86: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/86.jpg)
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
![Page 87: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/87.jpg)
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
![Page 88: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/88.jpg)
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
![Page 89: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/89.jpg)
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
![Page 90: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/90.jpg)
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
![Page 91: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/91.jpg)
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
![Page 92: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/92.jpg)
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
![Page 93: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/93.jpg)
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
![Page 94: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/94.jpg)
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
![Page 95: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/95.jpg)
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
![Page 96: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/96.jpg)
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
![Page 97: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/97.jpg)
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
![Page 98: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/98.jpg)
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
![Page 99: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/99.jpg)
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)
![Page 100: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/100.jpg)
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)
![Page 101: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/101.jpg)
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)
![Page 102: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/102.jpg)
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)
![Page 103: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/103.jpg)
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.
![Page 104: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/104.jpg)
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
![Page 105: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/105.jpg)
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
![Page 106: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/106.jpg)
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
![Page 107: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/107.jpg)
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
![Page 108: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/108.jpg)
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
![Page 109: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/109.jpg)
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
![Page 110: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/110.jpg)
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
![Page 111: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/111.jpg)
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
)
![Page 112: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/112.jpg)
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
)
![Page 113: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/113.jpg)
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
![Page 114: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/114.jpg)
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
![Page 115: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/115.jpg)
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
![Page 116: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/116.jpg)
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
![Page 117: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/117.jpg)
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
![Page 118: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/118.jpg)
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
![Page 119: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/119.jpg)
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
![Page 120: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/120.jpg)
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
![Page 121: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/121.jpg)
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
![Page 122: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/122.jpg)
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
![Page 123: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/123.jpg)
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
![Page 124: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/124.jpg)
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)
![Page 125: Reed-Solomon, the Chinese Remainder · Reed-Solomon,the Chinese Remainder Theorem, andCryptog-raphy Dr. Anna Johnston Overview Code Crypto QRT Why? Da Yen Weave Weave Swap Summary](https://reader030.vdocuments.net/reader030/viewer/2022040110/5f0b7b2e7e708231d430bb1c/html5/thumbnails/125.jpg)
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〉