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Outline
The complexity class coNP
Piotr Wojciechowski1
1Lane Department of Computer Science and Electrical EngineeringWest Virginia University
Wojciechowski coNP
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Outline
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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Outline
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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Outline
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
What is coNPExamples of problems in coNP
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
What is coNPExamples of problems in coNP
coNP as related to NP
Definition (coNP)
coNP is the complexity class which contains the complements of problems found in NP.
Another way of looking at coNP
Just as NP can be considered to be the set of problems with succinct ”yes” certificates,coNP can be considered to be the set of problems with succinct ”no” certificates. Thismeans that a ”no” instance of a problem in coNP has a short proof of it being a ”no”instance.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
What is coNPExamples of problems in coNP
coNP as related to NP
Definition (coNP)
coNP is the complexity class which contains the complements of problems found in NP.
Another way of looking at coNP
Just as NP can be considered to be the set of problems with succinct ”yes” certificates,coNP can be considered to be the set of problems with succinct ”no” certificates. Thismeans that a ”no” instance of a problem in coNP has a short proof of it being a ”no”instance.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
What is coNPExamples of problems in coNP
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
What is coNPExamples of problems in coNP
Examples
1 coSAT = {〈b〉 : b is a boolean expression with no satisfying assignments}2 PRIMES = {〈p〉 : p is a prime number}
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
What is coNPExamples of problems in coNP
Examples
1 coSAT = {〈b〉 : b is a boolean expression with no satisfying assignments}2 PRIMES = {〈p〉 : p is a prime number}
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Properties
Problems have both succinct ”yes” and succinct ”no” certificates.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Examples
1 PRIMES2 All problems in P
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Examples
1 PRIMES2 All problems in P
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
PRIMES is in NP ∩ coNP
Goal
We first want to develop a different way of determining primality.Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p
such that rp−1 = 1 mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Definition (Relative Primality)
Two numbers a and b are relatively prime iff their greatest common divisor, (a, b), is 1.
Examples
5 and 234 are relatively prime,57 and 95 are not, 19 is a common factor.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
PRIMES is in NP ∩ coNP
Goal
We first want to develop a different way of determining primality.Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p
such that rp−1 = 1 mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Definition (Relative Primality)
Two numbers a and b are relatively prime iff their greatest common divisor, (a, b), is 1.
Examples
5 and 234 are relatively prime,57 and 95 are not, 19 is a common factor.
Wojciechowski coNP
![Page 18: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/18.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
PRIMES is in NP ∩ coNP
Goal
We first want to develop a different way of determining primality.Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p
such that rp−1 = 1 mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Definition (Relative Primality)
Two numbers a and b are relatively prime iff their greatest common divisor, (a, b), is 1.
Examples
5 and 234 are relatively prime,57 and 95 are not, 19 is a common factor.
Wojciechowski coNP
![Page 19: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/19.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
PRIMES is in NP ∩ coNP
Goal
We first want to develop a different way of determining primality.Want to show that a number p > 1 is prime if and only if there is a number 1 < r < p
such that rp−1 = 1 mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Definition (Relative Primality)
Two numbers a and b are relatively prime iff their greatest common divisor, (a, b), is 1.
Examples
5 and 234 are relatively prime,57 and 95 are not, 19 is a common factor.
Wojciechowski coNP
![Page 20: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/20.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Definition (Φ(n))
Φ(n) = {m : 1 ≤ m < n, (m, n) = 1}.
Definition (Euler φ function)
φ(n) = |Φ(n)| and φ(1) = 1.In other words, φ(n) is the number of numbers between 1 and n − 1 which arerelatively prime to n
Lemma (1)
φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Proof.
Assume that p1, p2 . . . , pk are the prime divisors of n. Observe that each pi knocks offone in every pi candidates for φ(n), leaving n · (1− 1
pi) candidates for φ(n). It therefore
follows that φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Definition (Φ(n))
Φ(n) = {m : 1 ≤ m < n, (m, n) = 1}.
Definition (Euler φ function)
φ(n) = |Φ(n)| and φ(1) = 1.In other words, φ(n) is the number of numbers between 1 and n − 1 which arerelatively prime to n
Lemma (1)
φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Proof.
Assume that p1, p2 . . . , pk are the prime divisors of n. Observe that each pi knocks offone in every pi candidates for φ(n), leaving n · (1− 1
pi) candidates for φ(n). It therefore
follows that φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Definition (Φ(n))
Φ(n) = {m : 1 ≤ m < n, (m, n) = 1}.
Definition (Euler φ function)
φ(n) = |Φ(n)| and φ(1) = 1.In other words, φ(n) is the number of numbers between 1 and n − 1 which arerelatively prime to n
Lemma (1)
φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Proof.
Assume that p1, p2 . . . , pk are the prime divisors of n. Observe that each pi knocks offone in every pi candidates for φ(n), leaving n · (1− 1
pi) candidates for φ(n). It therefore
follows that φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Definition (Φ(n))
Φ(n) = {m : 1 ≤ m < n, (m, n) = 1}.
Definition (Euler φ function)
φ(n) = |Φ(n)| and φ(1) = 1.In other words, φ(n) is the number of numbers between 1 and n − 1 which arerelatively prime to n
Lemma (1)
φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Proof.
Assume that p1, p2 . . . , pk are the prime divisors of n. Observe that each pi knocks offone in every pi candidates for φ(n), leaving n · (1− 1
pi) candidates for φ(n). It therefore
follows that φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Wojciechowski coNP
![Page 24: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/24.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Definition (Φ(n))
Φ(n) = {m : 1 ≤ m < n, (m, n) = 1}.
Definition (Euler φ function)
φ(n) = |Φ(n)| and φ(1) = 1.In other words, φ(n) is the number of numbers between 1 and n − 1 which arerelatively prime to n
Lemma (1)
φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Proof.
Assume that p1, p2 . . . , pk are the prime divisors of n. Observe that each pi knocks offone in every pi candidates for φ(n), leaving n · (1− 1
pi) candidates for φ(n). It therefore
follows that φ(n) = nQ
p|n(1− 1p ) where p is a prime.
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Examples
Φ(8) = {1, 3, 5, 7}φ(8) = 8 · (1− 1
2 ) = 4
Theorem
If (m, n) = 1, then φ(m · n) = φ(m) · φ(n).
Proof.
Follows from the previous lemma as m and n share no common prime factors. Thusthe terms in the product m · n
Qp|m·n(1− 1
p ) are distributed without overlap to
nQ
p|n(1− 1p ) and m
Qp|m(1− 1
p ).
Example
φ(95) = 95 · (1− 15 ) · (1− 1
19 ) = 72 = 4 · 18 = φ(5) · φ(19)
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Examples
Φ(8) = {1, 3, 5, 7}φ(8) = 8 · (1− 1
2 ) = 4
Theorem
If (m, n) = 1, then φ(m · n) = φ(m) · φ(n).
Proof.
Follows from the previous lemma as m and n share no common prime factors. Thusthe terms in the product m · n
Qp|m·n(1− 1
p ) are distributed without overlap to
nQ
p|n(1− 1p ) and m
Qp|m(1− 1
p ).
Example
φ(95) = 95 · (1− 15 ) · (1− 1
19 ) = 72 = 4 · 18 = φ(5) · φ(19)
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Examples
Φ(8) = {1, 3, 5, 7}φ(8) = 8 · (1− 1
2 ) = 4
Theorem
If (m, n) = 1, then φ(m · n) = φ(m) · φ(n).
Proof.
Follows from the previous lemma as m and n share no common prime factors. Thusthe terms in the product m · n
Qp|m·n(1− 1
p ) are distributed without overlap to
nQ
p|n(1− 1p ) and m
Qp|m(1− 1
p ).
Example
φ(95) = 95 · (1− 15 ) · (1− 1
19 ) = 72 = 4 · 18 = φ(5) · φ(19)
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Examples
Φ(8) = {1, 3, 5, 7}φ(8) = 8 · (1− 1
2 ) = 4
Theorem
If (m, n) = 1, then φ(m · n) = φ(m) · φ(n).
Proof.
Follows from the previous lemma as m and n share no common prime factors. Thusthe terms in the product m · n
Qp|m·n(1− 1
p ) are distributed without overlap to
nQ
p|n(1− 1p ) and m
Qp|m(1− 1
p ).
Example
φ(95) = 95 · (1− 15 ) · (1− 1
19 ) = 72 = 4 · 18 = φ(5) · φ(19)
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
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coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
![Page 31: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/31.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
![Page 32: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/32.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
![Page 33: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/33.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
![Page 34: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/34.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
![Page 35: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/35.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
TheoremPm|n φ(m) = n
Proof.
LetQl
i=1 pkii be the prime factorization of n. Consider the following productQl
i=1(φ(1) + φ(pi ) + φ(p2i ) + · · ·+ φ(pki
i ))
Its easy to see that the i th term in this product is simply pkii . Thus the product is simply
equal to n. If the product is expanded out one term for each divisor of n is produced.
The term corresponding to m =Ql
i=1 pk′ii where 1 ≤ k ′i < ki , is
Qli=1 φ(p
k′ii ). However,
by the previous theorem, this term is simply φ(m).
ExamplePm|27 φ(m) = φ(1) + φ(3) + φ(9) + φ(27) = 1 + 2 + 6 + 18 = 27
Wojciechowski coNP
![Page 36: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/36.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 37: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/37.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 38: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/38.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 39: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/39.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 40: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/40.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 41: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/41.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 42: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/42.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem (Fermat’s Little Theorem)
For all 0 < a < p, ap−1 ≡ 1 mod p, where p is a prime.
Proof.
Lets consider the set a · Φ(p) = {a · i mod p : 0 < i < p}. We have that this set isequal to the set Φ(p) = {i, 0 < i < p}. Suppose otherwise, thus there exist elementsm 6= m′ in Φ(p) such that a ·m ≡ a ·m′ mod p. Thus a · (m −m′) ≡ 0 mod pleading to a contradiction. Now take the products of all the elements in each set, thuswe have that ap−1 · (p − 1)! ≡ (p − 1)! mod p. Thus (ap−1 − 1) · (p − 1)! ≡ 0mod p. Since (p − 1)! 6= 0 mod p we have the desired result.
Corollary
For all a ∈ Φ(n), aφ(n) ≡ 1 mod n
Wojciechowski coNP
![Page 43: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/43.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternative look at primality
Definition (Exponent of a number mod n)
The exponent of a number m ∈ Φ(n) is the smallest integer k > 0 for which mk ≡ 1mod n. It is worth noting that if ml ≡ 1 mod n then k |l . As otherwise l mod k wouldbe the exponent of m.
Example
The exponent of 10 mod 11 is 2 as 102 ≡ 1 mod 11 but 10 6= 1 mod 11.
Definition
Let R(k), for a given prime p, denote the number of residues in Φ(p) which haveexponent k .
Example
Let p = 5 thus R(3) = 0 and R(2) = 1.
Wojciechowski coNP
![Page 44: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/44.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternative look at primality
Definition (Exponent of a number mod n)
The exponent of a number m ∈ Φ(n) is the smallest integer k > 0 for which mk ≡ 1mod n. It is worth noting that if ml ≡ 1 mod n then k |l . As otherwise l mod k wouldbe the exponent of m.
Example
The exponent of 10 mod 11 is 2 as 102 ≡ 1 mod 11 but 10 6= 1 mod 11.
Definition
Let R(k), for a given prime p, denote the number of residues in Φ(p) which haveexponent k .
Example
Let p = 5 thus R(3) = 0 and R(2) = 1.
Wojciechowski coNP
![Page 45: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/45.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternative look at primality
Definition (Exponent of a number mod n)
The exponent of a number m ∈ Φ(n) is the smallest integer k > 0 for which mk ≡ 1mod n. It is worth noting that if ml ≡ 1 mod n then k |l . As otherwise l mod k wouldbe the exponent of m.
Example
The exponent of 10 mod 11 is 2 as 102 ≡ 1 mod 11 but 10 6= 1 mod 11.
Definition
Let R(k), for a given prime p, denote the number of residues in Φ(p) which haveexponent k .
Example
Let p = 5 thus R(3) = 0 and R(2) = 1.
Wojciechowski coNP
![Page 46: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/46.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternative look at primality
Definition (Exponent of a number mod n)
The exponent of a number m ∈ Φ(n) is the smallest integer k > 0 for which mk ≡ 1mod n. It is worth noting that if ml ≡ 1 mod n then k |l . As otherwise l mod k wouldbe the exponent of m.
Example
The exponent of 10 mod 11 is 2 as 102 ≡ 1 mod 11 but 10 6= 1 mod 11.
Definition
Let R(k), for a given prime p, denote the number of residues in Φ(p) which haveexponent k .
Example
Let p = 5 thus R(3) = 0 and R(2) = 1.
Wojciechowski coNP
![Page 47: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/47.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternative look at primality
Definition (Exponent of a number mod n)
The exponent of a number m ∈ Φ(n) is the smallest integer k > 0 for which mk ≡ 1mod n. It is worth noting that if ml ≡ 1 mod n then k |l . As otherwise l mod k wouldbe the exponent of m.
Example
The exponent of 10 mod 11 is 2 as 102 ≡ 1 mod 11 but 10 6= 1 mod 11.
Definition
Let R(k), for a given prime p, denote the number of residues in Φ(p) which haveexponent k .
Example
Let p = 5 thus R(3) = 0 and R(2) = 1.
Wojciechowski coNP
![Page 48: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/48.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
Any polynomial of degree k that is not identically zero has at most k distinct rootsmod p.
Proof.
This will be shown by induction on k . If k = 0 this is obvious as the polynomial isconstant. Assume that the theorem holds for al polynomials of degree at most k − 1.Let π(x) = ak xk + . . .+ a1x + a0 be a polynomial of degree k with k + 1 distinct roots,say x1, x2, . . . , xk+1. Now let π′(x) = π(x)− ak ·
Qki=1(x − xi ). Thus π′(x) is a
polynomial of degree at most k − 1, which is not identically 0. Therefore, π′(x) musthave k − 1 distinct roots as per the inductive hypothesis, but x1, . . . , xk are all distinctroots of π′(x) contradicting the hypothesis!
Application to R(k)
Since xk − 1 for k 6∈ φ(p) is a non-zero polynomial it has at most k roots mod p andthus R(k) ≤ k .
Wojciechowski coNP
![Page 49: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/49.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
Any polynomial of degree k that is not identically zero has at most k distinct rootsmod p.
Proof.
This will be shown by induction on k . If k = 0 this is obvious as the polynomial isconstant. Assume that the theorem holds for al polynomials of degree at most k − 1.Let π(x) = ak xk + . . .+ a1x + a0 be a polynomial of degree k with k + 1 distinct roots,say x1, x2, . . . , xk+1. Now let π′(x) = π(x)− ak ·
Qki=1(x − xi ). Thus π′(x) is a
polynomial of degree at most k − 1, which is not identically 0. Therefore, π′(x) musthave k − 1 distinct roots as per the inductive hypothesis, but x1, . . . , xk are all distinctroots of π′(x) contradicting the hypothesis!
Application to R(k)
Since xk − 1 for k 6∈ φ(p) is a non-zero polynomial it has at most k roots mod p andthus R(k) ≤ k .
Wojciechowski coNP
![Page 50: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/50.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
Any polynomial of degree k that is not identically zero has at most k distinct rootsmod p.
Proof.
This will be shown by induction on k . If k = 0 this is obvious as the polynomial isconstant. Assume that the theorem holds for al polynomials of degree at most k − 1.Let π(x) = ak xk + . . .+ a1x + a0 be a polynomial of degree k with k + 1 distinct roots,say x1, x2, . . . , xk+1. Now let π′(x) = π(x)− ak ·
Qki=1(x − xi ). Thus π′(x) is a
polynomial of degree at most k − 1, which is not identically 0. Therefore, π′(x) musthave k − 1 distinct roots as per the inductive hypothesis, but x1, . . . , xk are all distinctroots of π′(x) contradicting the hypothesis!
Application to R(k)
Since xk − 1 for k 6∈ φ(p) is a non-zero polynomial it has at most k roots mod p andthus R(k) ≤ k .
Wojciechowski coNP
![Page 51: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/51.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
Any polynomial of degree k that is not identically zero has at most k distinct rootsmod p.
Proof.
This will be shown by induction on k . If k = 0 this is obvious as the polynomial isconstant. Assume that the theorem holds for al polynomials of degree at most k − 1.Let π(x) = ak xk + . . .+ a1x + a0 be a polynomial of degree k with k + 1 distinct roots,say x1, x2, . . . , xk+1. Now let π′(x) = π(x)− ak ·
Qki=1(x − xi ). Thus π′(x) is a
polynomial of degree at most k − 1, which is not identically 0. Therefore, π′(x) musthave k − 1 distinct roots as per the inductive hypothesis, but x1, . . . , xk are all distinctroots of π′(x) contradicting the hypothesis!
Application to R(k)
Since xk − 1 for k 6∈ φ(p) is a non-zero polynomial it has at most k roots mod p andthus R(k) ≤ k .
Wojciechowski coNP
![Page 52: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/52.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
Any polynomial of degree k that is not identically zero has at most k distinct rootsmod p.
Proof.
This will be shown by induction on k . If k = 0 this is obvious as the polynomial isconstant. Assume that the theorem holds for al polynomials of degree at most k − 1.Let π(x) = ak xk + . . .+ a1x + a0 be a polynomial of degree k with k + 1 distinct roots,say x1, x2, . . . , xk+1. Now let π′(x) = π(x)− ak ·
Qki=1(x − xi ). Thus π′(x) is a
polynomial of degree at most k − 1, which is not identically 0. Therefore, π′(x) musthave k − 1 distinct roots as per the inductive hypothesis, but x1, . . . , xk are all distinctroots of π′(x) contradicting the hypothesis!
Application to R(k)
Since xk − 1 for k 6∈ φ(p) is a non-zero polynomial it has at most k roots mod p andthus R(k) ≤ k .
Wojciechowski coNP
![Page 53: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/53.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
For a given prime p, for all k ∈ Φ(p) we have that R(k) ≤ φ(k).
Proof.
If R(k) = 0 then were done. So we assume that there is an element s with exponent k .Then (1, s, s2, . . . , sk−1) are all distinct. And for all 0 ≤ i < k , (si )k = sik ≡ 1i = 1mod p. Thus these si constitute all k possible roots of xk − 1 mod p. Let sl haveexponent k . If l /∈ Φ(k) then d = (l, k) > 1 and (sl )k/d = s
lkd = (sk )l/d ≡ 1 mod p
leading to a contradiction. Thus if sl has exponent k mod p then l ∈ Φ(k), whichmeans that R(k) ≤ φ(k).
Wojciechowski coNP
![Page 54: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/54.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
For a given prime p, for all k ∈ Φ(p) we have that R(k) ≤ φ(k).
Proof.
If R(k) = 0 then were done. So we assume that there is an element s with exponent k .Then (1, s, s2, . . . , sk−1) are all distinct. And for all 0 ≤ i < k , (si )k = sik ≡ 1i = 1mod p. Thus these si constitute all k possible roots of xk − 1 mod p. Let sl haveexponent k . If l /∈ Φ(k) then d = (l, k) > 1 and (sl )k/d = s
lkd = (sk )l/d ≡ 1 mod p
leading to a contradiction. Thus if sl has exponent k mod p then l ∈ Φ(k), whichmeans that R(k) ≤ φ(k).
Wojciechowski coNP
![Page 55: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/55.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
For a given prime p, for all k ∈ Φ(p) we have that R(k) ≤ φ(k).
Proof.
If R(k) = 0 then were done. So we assume that there is an element s with exponent k .Then (1, s, s2, . . . , sk−1) are all distinct. And for all 0 ≤ i < k , (si )k = sik ≡ 1i = 1mod p. Thus these si constitute all k possible roots of xk − 1 mod p. Let sl haveexponent k . If l /∈ Φ(k) then d = (l, k) > 1 and (sl )k/d = s
lkd = (sk )l/d ≡ 1 mod p
leading to a contradiction. Thus if sl has exponent k mod p then l ∈ Φ(k), whichmeans that R(k) ≤ φ(k).
Wojciechowski coNP
![Page 56: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/56.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
For a given prime p, for all k ∈ Φ(p) we have that R(k) ≤ φ(k).
Proof.
If R(k) = 0 then were done. So we assume that there is an element s with exponent k .Then (1, s, s2, . . . , sk−1) are all distinct. And for all 0 ≤ i < k , (si )k = sik ≡ 1i = 1mod p. Thus these si constitute all k possible roots of xk − 1 mod p. Let sl haveexponent k . If l /∈ Φ(k) then d = (l, k) > 1 and (sl )k/d = s
lkd = (sk )l/d ≡ 1 mod p
leading to a contradiction. Thus if sl has exponent k mod p then l ∈ Φ(k), whichmeans that R(k) ≤ φ(k).
Wojciechowski coNP
![Page 57: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/57.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
For a given prime p, for all k ∈ Φ(p) we have that R(k) ≤ φ(k).
Proof.
If R(k) = 0 then were done. So we assume that there is an element s with exponent k .Then (1, s, s2, . . . , sk−1) are all distinct. And for all 0 ≤ i < k , (si )k = sik ≡ 1i = 1mod p. Thus these si constitute all k possible roots of xk − 1 mod p. Let sl haveexponent k . If l /∈ Φ(k) then d = (l, k) > 1 and (sl )k/d = s
lkd = (sk )l/d ≡ 1 mod p
leading to a contradiction. Thus if sl has exponent k mod p then l ∈ Φ(k), whichmeans that R(k) ≤ φ(k).
Wojciechowski coNP
![Page 58: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/58.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
For a given prime p, for all k ∈ Φ(p) we have that R(k) ≤ φ(k).
Proof.
If R(k) = 0 then were done. So we assume that there is an element s with exponent k .Then (1, s, s2, . . . , sk−1) are all distinct. And for all 0 ≤ i < k , (si )k = sik ≡ 1i = 1mod p. Thus these si constitute all k possible roots of xk − 1 mod p. Let sl haveexponent k . If l /∈ Φ(k) then d = (l, k) > 1 and (sl )k/d = s
lkd = (sk )l/d ≡ 1 mod p
leading to a contradiction. Thus if sl has exponent k mod p then l ∈ Φ(k), whichmeans that R(k) ≤ φ(k).
Wojciechowski coNP
![Page 59: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/59.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
A number p > 1 is prime if and only if there is a number 1 < r < p such that rp−1 = 1
mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Proof.
p is a prime: As each 0 < i < p has an exponent, that divides p − 1,p − 1 =
Pl|p−1 R(l) ≤
Pl|p−1 φ(l) = p − 1. Thus R(l) = φ(l) for all l|p − 1. Namely
R(p − 1) = φ(p − 1) > 0 and so there is at least one r that has exponent p − 1.p is not a prime: let r ∈ Φ(p) be a number such that rp−1 ≡ 1 mod p, we also havethat rφ(p) ≡ 1 mod p. Let k be the exponent of r mod p. Thus k |p − 1 and k |φ(p).Since p is not a prime k ≤ φ(p) < p − 1. Let q be a prime factor of p−1
k . Thus k | p−1q
and so rp−1
q ≡ 1 mod p
Wojciechowski coNP
![Page 60: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/60.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
A number p > 1 is prime if and only if there is a number 1 < r < p such that rp−1 = 1
mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Proof.
p is a prime: As each 0 < i < p has an exponent, that divides p − 1,p − 1 =
Pl|p−1 R(l) ≤
Pl|p−1 φ(l) = p − 1. Thus R(l) = φ(l) for all l|p − 1. Namely
R(p − 1) = φ(p − 1) > 0 and so there is at least one r that has exponent p − 1.p is not a prime: let r ∈ Φ(p) be a number such that rp−1 ≡ 1 mod p, we also havethat rφ(p) ≡ 1 mod p. Let k be the exponent of r mod p. Thus k |p − 1 and k |φ(p).Since p is not a prime k ≤ φ(p) < p − 1. Let q be a prime factor of p−1
k . Thus k | p−1q
and so rp−1
q ≡ 1 mod p
Wojciechowski coNP
![Page 61: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/61.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
A number p > 1 is prime if and only if there is a number 1 < r < p such that rp−1 = 1
mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Proof.
p is a prime: As each 0 < i < p has an exponent, that divides p − 1,p − 1 =
Pl|p−1 R(l) ≤
Pl|p−1 φ(l) = p − 1. Thus R(l) = φ(l) for all l|p − 1. Namely
R(p − 1) = φ(p − 1) > 0 and so there is at least one r that has exponent p − 1.p is not a prime: let r ∈ Φ(p) be a number such that rp−1 ≡ 1 mod p, we also havethat rφ(p) ≡ 1 mod p. Let k be the exponent of r mod p. Thus k |p − 1 and k |φ(p).Since p is not a prime k ≤ φ(p) < p − 1. Let q be a prime factor of p−1
k . Thus k | p−1q
and so rp−1
q ≡ 1 mod p
Wojciechowski coNP
![Page 62: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/62.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
A number p > 1 is prime if and only if there is a number 1 < r < p such that rp−1 = 1
mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Proof.
p is a prime: As each 0 < i < p has an exponent, that divides p − 1,p − 1 =
Pl|p−1 R(l) ≤
Pl|p−1 φ(l) = p − 1. Thus R(l) = φ(l) for all l|p − 1. Namely
R(p − 1) = φ(p − 1) > 0 and so there is at least one r that has exponent p − 1.p is not a prime: let r ∈ Φ(p) be a number such that rp−1 ≡ 1 mod p, we also havethat rφ(p) ≡ 1 mod p. Let k be the exponent of r mod p. Thus k |p − 1 and k |φ(p).Since p is not a prime k ≤ φ(p) < p − 1. Let q be a prime factor of p−1
k . Thus k | p−1q
and so rp−1
q ≡ 1 mod p
Wojciechowski coNP
![Page 63: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/63.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
A number p > 1 is prime if and only if there is a number 1 < r < p such that rp−1 = 1
mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Proof.
p is a prime: As each 0 < i < p has an exponent, that divides p − 1,p − 1 =
Pl|p−1 R(l) ≤
Pl|p−1 φ(l) = p − 1. Thus R(l) = φ(l) for all l|p − 1. Namely
R(p − 1) = φ(p − 1) > 0 and so there is at least one r that has exponent p − 1.p is not a prime: let r ∈ Φ(p) be a number such that rp−1 ≡ 1 mod p, we also havethat rφ(p) ≡ 1 mod p. Let k be the exponent of r mod p. Thus k |p − 1 and k |φ(p).Since p is not a prime k ≤ φ(p) < p − 1. Let q be a prime factor of p−1
k . Thus k | p−1q
and so rp−1
q ≡ 1 mod p
Wojciechowski coNP
![Page 64: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/64.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
An alternate look at primality
Theorem
A number p > 1 is prime if and only if there is a number 1 < r < p such that rp−1 = 1
mod p and rp−1
q 6= 1 mod p for all prime divisors q of p − 1.
Proof.
p is a prime: As each 0 < i < p has an exponent, that divides p − 1,p − 1 =
Pl|p−1 R(l) ≤
Pl|p−1 φ(l) = p − 1. Thus R(l) = φ(l) for all l|p − 1. Namely
R(p − 1) = φ(p − 1) > 0 and so there is at least one r that has exponent p − 1.p is not a prime: let r ∈ Φ(p) be a number such that rp−1 ≡ 1 mod p, we also havethat rφ(p) ≡ 1 mod p. Let k be the exponent of r mod p. Thus k |p − 1 and k |φ(p).Since p is not a prime k ≤ φ(p) < p − 1. Let q be a prime factor of p−1
k . Thus k | p−1q
and so rp−1
q ≡ 1 mod p
Wojciechowski coNP
![Page 65: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/65.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Showing that PRIMES is in NP ∩ coNP
Theorem (Pratt’s Theorem)
PRIMES is in NP ∩ coNP
Proof.
Part 1: PRIMES is in coNP. Trivially true, since the succinct disqualification forx /∈ PRIMES is simply the factorization of x.eg. 12 = 3 · 4 and 117 = 9 · 13.
Wojciechowski coNP
![Page 66: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/66.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Showing that PRIMES is in NP ∩ coNP
Theorem (Pratt’s Theorem)
PRIMES is in NP ∩ coNP
Proof.
Part 1: PRIMES is in coNP. Trivially true, since the succinct disqualification forx /∈ PRIMES is simply the factorization of x.eg. 12 = 3 · 4 and 117 = 9 · 13.
Wojciechowski coNP
![Page 67: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/67.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Showing that PRIMES is in NP ∩ coNP
Theorem (Pratt’s Theorem)
PRIMES is in NP ∩ coNP
Proof.
Part 1: PRIMES is in coNP. Trivially true, since the succinct disqualification forx /∈ PRIMES is simply the factorization of x.eg. 12 = 3 · 4 and 117 = 9 · 13.
Wojciechowski coNP
![Page 68: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/68.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Showing that PRIMES is in NP ∩ coNP
Theorem (Pratt’s Theorem)
PRIMES is in NP ∩ coNP
Proof.
Part 1: PRIMES is in coNP. Trivially true, since the succinct disqualification forx /∈ PRIMES is simply the factorization of x.eg. 12 = 3 · 4 and 117 = 9 · 13.
Wojciechowski coNP
![Page 69: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/69.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 70: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/70.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 71: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/71.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 72: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/72.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 73: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/73.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 74: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/74.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 75: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/75.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 76: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/76.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 77: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/77.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Part 2: PRIMES is in NP. First we will try to construct a certificate for any x ∈ PRIMES.Once a reasonable certificate is found we will show that it is succinct.
Possible Certificates, C(p), for p ∈ PRIMES
1 C(p) = r such that rp−1 = 1 mod p.Insufficient as 20 is a ”valid” certificate for 21 /∈ PRIMES.
2 C(p) = (r , p1, p2, . . . , pk ) where rp−1 = 1 mod p and rp−1
pi 6= 1 mod p for1 ≤ i ≤ k and p1 · · · · · pk = p − 1.Insufficient as (10, 2, 45) is a ”valid” certificate for 91 /∈ PRIMES.Need some way to ensure that p1, . . . , pk are primes without having to check.
3 C(p) = (r ; p1,C(p1), p2,C(p2), . . . , pk ,C(pk )) where C(1)=(1), rp−1 = 1 mod p,
and rp−1
pi 6= 1 mod p for 1 ≤ i ≤ k and p1 · · · · · pk = p − 1.eg. C(67) = (2; 2, (1), 3, (2; 2, (1)), 11, (8; 2, (1), 5, (3; 2, (1)))).
Wojciechowski coNP
![Page 78: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/78.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 79: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/79.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 80: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/80.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 81: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/81.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 82: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/82.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 83: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/83.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 84: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/84.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 85: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/85.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 86: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/86.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 87: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/87.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
First we will show that the certificate is succinct. We will show that for all primes p thecertificate has length at most 4 · log2(p). If p = 2 or p = 3 this is trivial. For any p > 3,p − 1 will have k < log(p) prime divisors q1 = 2, q2, . . . , qk . Thus C(p) will contain 2kseparators, the number r , 2 and its certificate (1), the qi s (at most 2 log p bits), and theC(qi )s.By the inductive hypothesis, we have that |C(qi )| ≤ 4 log2 qi . Thus|C(p)| ≤ 4 log p + 5 + 4
Pki=2 log2 qi
The logarithms of the q1s add up to log p−12 < log p − 1, so the sum of their squares is
at most (log p − 1)2. Thus|C(p)| ≤ 4 log2 p + 9− 4 log p, which is less than 4 log2 p when p ≥ 5.
Wojciechowski coNP
![Page 88: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/88.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 89: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/89.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 90: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/90.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 91: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/91.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 92: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/92.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 93: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/93.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 94: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/94.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 95: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/95.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 96: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/96.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 97: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/97.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
Now it needs to be shown that C(p) is verifiable in polynomial time. This hinges of thecomputation of rp−1 mod p. If repeated multiplication by r is done then this processclearly takes exponential time. However, repeated squaring can be used.Let l = dlog(p)e. First r , r2, r4, . . . , r2l
mod p are computed. Each of these stepstakes O(l2) time. Then multiply the appropriate exponents of r to obtain rp−1 mod p.As there are O(l) multiplications this entire process takes O(l3) time.
Example
p = 7, r = 5thus p − 1 = 6 and l = 3.so r2 = 25 ≡ 4, r4 = 16 ≡ 2, r8 = 4 mod 7 thus rp−1 = r4 · r2 = 8 ≡ 1 mod 7.
Wojciechowski coNP
![Page 98: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/98.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
However, it is not enough that rp−1 mod p takes O(l3) time. We need to show that theentire verification process of C(p) runs in polynomial time. To do this we need to
compute rp−1 mod p, rp−1
qi mod p for each of the O(l) qi s, q1, q2, . . . , qk , and eachof the C(qi )s. This entire process takes O(l4) time.
Wojciechowski coNP
![Page 99: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/99.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
However, it is not enough that rp−1 mod p takes O(l3) time. We need to show that theentire verification process of C(p) runs in polynomial time. To do this we need to
compute rp−1 mod p, rp−1
qi mod p for each of the O(l) qi s, q1, q2, . . . , qk , and eachof the C(qi )s. This entire process takes O(l4) time.
Wojciechowski coNP
![Page 100: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/100.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
However, it is not enough that rp−1 mod p takes O(l3) time. We need to show that theentire verification process of C(p) runs in polynomial time. To do this we need to
compute rp−1 mod p, rp−1
qi mod p for each of the O(l) qi s, q1, q2, . . . , qk , and eachof the C(qi )s. This entire process takes O(l4) time.
Wojciechowski coNP
![Page 101: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/101.jpg)
coNPNP∩ coNP
NP, coNP, and P
Properties of NP∩ coNPProblems in NP∩ coNP
Proof.
However, it is not enough that rp−1 mod p takes O(l3) time. We need to show that theentire verification process of C(p) runs in polynomial time. To do this we need to
compute rp−1 mod p, rp−1
qi mod p for each of the O(l) qi s, q1, q2, . . . , qk , and eachof the C(qi )s. This entire process takes O(l4) time.
Wojciechowski coNP
![Page 102: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/102.jpg)
coNPNP∩ coNP
NP, coNP, and PHierarchy
Outline
1 Description of coNP and examples of problemsWhat is coNPExamples of problems in coNP
2 The NP ∩ coNP complexity classProperties of NP ∩ coNPProblems in NP ∩ coNP
3 NP, coNP, and PThe P, NP, coNP Hierarchy
Wojciechowski coNP
![Page 103: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/103.jpg)
coNPNP∩ coNP
NP, coNP, and PHierarchy
Inclusion Relationships
Relation to P
Just as P ⊆ NP, we have that P = coP ⊆ coNP. Thus P ⊆ NP ∩ coNP.It is also unknown if P = NP ∩ coNP.
Wojciechowski coNP
![Page 104: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/104.jpg)
coNPNP∩ coNP
NP, coNP, and PHierarchy
Inclusion Relationships
Relation to P
Just as P ⊆ NP, we have that P = coP ⊆ coNP. Thus P ⊆ NP ∩ coNP.It is also unknown if P = NP ∩ coNP.
Wojciechowski coNP
![Page 105: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/105.jpg)
coNPNP∩ coNP
NP, coNP, and PHierarchy
Inclusion Relationships
Relation to P
Just as P ⊆ NP, we have that P = coP ⊆ coNP. Thus P ⊆ NP ∩ coNP.It is also unknown if P = NP ∩ coNP.
Wojciechowski coNP
![Page 106: The complexity class coNP - West Virginia University](https://reader035.vdocuments.mx/reader035/viewer/2022071602/613d5512736caf36b75c1305/html5/thumbnails/106.jpg)
coNPNP∩ coNP
NP, coNP, and PHierarchy
The Complexity Picture
P
NP ∩ coNP
coNPNP
NP-complete coNP-complete
Wojciechowski coNP