D. J. Bernstein x[i],x[i+1] = ( Challenges in Prelude: What is the … · 2017. 10. 3. · 1...

1

Challenges in

quantum algorithms for

integer factorization

D. J. Bernstein

University of Illinois at Chicago

Prelude: What is the fastest

algorithm to sort an array?

def blindsort(x):

while not issorted(x):

permuterandomly(x)

2

def bubblesort(x):

for j in range(len(x)):

for i in reversed(range(j)):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

bubblesort takes poly time.

Θ(n2) comparisons.

Huge speedup over blindsort!

Is this the end of the story?

1

Challenges in

D. J. Bernstein

def blindsort(x):

permuterandomly(x)

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

No, still not optimal.

1

Challenges in

D. J. Bernstein

def blindsort(x):

permuterandomly(x)

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

Analogous: What is the fastest

algorithm to factor integers?

Shor’s algorithm takes poly time.

Huge speedup over NFS!

b2(log b)1+o(1) qubit operations

to factor b-bit integer,

using standard subroutines

for fast integer arithmetic.

1

Challenges in

D. J. Bernstein

def blindsort(x):

permuterandomly(x)

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

1

Challenges in

D. J. Bernstein

def blindsort(x):

permuterandomly(x)

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

“Shor’s algorithm: the bubble sort

of integer factorization.”

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

4

A simple exercise to illustrate

suboptimality of Shor’s algorithm:

Find a prime divisor of¨103009ı

˝.

3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208

2

def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

4

˝.

3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208

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def bubblesort(x):

x[i],x[i+1] = (

min(x[i],x[i+1]),

max(x[i],x[i+1])

)

Θ(n2) comparisons.

3

4

˝.

3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208

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˝.

3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208

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˝.

3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208

5

Important variations in the

factorization problem:

• Maybe need one factor.• Maybe need all factors.• Maybe factors are small.• Maybe factors are large.• Maybe there are many inputs.• Maybe inputs in superposition.

Important variations in metrics

(even assuming perfect devices):

• Qubits.• Area (“A”, including wire area).• Qubit operations (“gates”).• Depth.• Time (“T”: latency).

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Short-term RSA security

1995 Kitaev, 1996 Vedral–

Barenco–Ekert, 1996 Beckman–

Chari–Devabhaktuni–Preskill,

1998 Zalka, 1999 Mosca–Ekert,

2000 Parker–Plenio, 2001 Seifert,

2002 Kitaev–Shen–Vyalyi, 2003

Beauregard, 2006 Takahashi–

Kunihiro, 2010 Ahmadi–Chiang,

2014 Svore–Hastings–Freedman,

2015 Grosshans–Lawson–Morain–

Smith, 2016 Häner–Roetteler–

Svore, 2017 Eker̊a–Håstad, 2017

Johnston: try to squeeze constant

factors out of Shor’s algorithm.

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2003 Beauregard: 2b + 3 qubits.

: : : 2016 Häner–Roetteler–Svore:

2b + 2 qubits; 64b3(lg b + O(1))

Toffoli gates; similar number of

CNOT gates; depth O(b3).

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2b + 2 qubits; 64b3(lg b + O(1))

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2b + 2 qubits; 64b3(lg b + O(1))

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2b + 2 qubits; 64b3(lg b + O(1))

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2b + 2 qubits; 64b3(lg b + O(1))

Conventional wisdom:

cannot avoid 2b qubits

for controlled mulmod.

e.g. 4096 qubits for b = 2048,

very common RSA key size.

So 2048-bit factorization

needs 4096 qubits?

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2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

No: NFS uses 0 qubits.

6

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

NFS takes Lp+o(1) operations

with p =3p

92 + 26√

13=3 > 1:9,

log L = (log 2b)1=3(log log 2b)2=3.

Analysis for b = 2048 (not easy!):

very roughly 2112 operations.

6

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

6

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

2017 Bernstein–Biasse–Mosca:

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

(and many non-quantum bits).

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

Open: Analyze for b = 2048.

Fewer than 4096 qubits?

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

Counting operations is an

oversimplified cost model: ignores

communication costs, parallelism.

See, e.g., 1981 Brent–Kung AT

theorem for realistic chip model.

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

7

2b + 2 qubits; 64b3(lg b + O(1))

needs 4096 qubits?

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

NFS suffers somewhat from

communication costs inside

big linear-algebra subroutine.

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

using b2=3+o(1) qubits.

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

Actually have many inputs.

Lower cost for some output?

Lower cost for many outputs?

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

8

with p =3p

92 + 26√

13=3 > 1:9,

Lq+o(1) operations

with q = 3p

8=3 ≈ 1:387,using b2=3+o(1) qubits

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

1993 Coppersmith:

L1:638:::+o(1) operations

after precomp(b) involving

L2:006:::+o(1) operations.

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

1993 Coppersmith:

2014 Bernstein–Lange:

AT = L2:204:::+o(1)

to factor L0:5+o(1) inputs;

L1:704:::+o(1) per input.

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

1993 Coppersmith:

AT = L2:204:::+o(1)

Open: Any quantum speedups

for factoring many integers?

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

Long-term RSA security

Long history of advances

in integer factorization.

Long history of RSA users

switching to larger key sizes,

not far beyond broken sizes.

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

9

2001 Bernstein:

AT = Lp′+o(1) with p′ ≈ 1:976.

AT = Lq′+o(1) with q′ ≈ 1:456

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

“Expert” cryptographers:

“Obviously they won’t react to

Shor’s algorithm this way! They’ll

switch to codes, lattices, etc. long

before quantum computers break

RSA-2048! We don’t need to

analyze the security of RSA-4096,

RSA-8192, RSA-16384, etc.!”

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

RSA-8192, RSA-16384, etc.!”

12

We consider possible impact of

quantum computers. Shouldn’t

we also consider possible impact

of users wanting to stick to RSA?

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

RSA-8192, RSA-16384, etc.!”

12

10

1993 Coppersmith:

AT = L2:204:::+o(1)

11

RSA-8192, RSA-16384, etc.!”

12

11

RSA-8192, RSA-16384, etc.!”

12

11

RSA-8192, RSA-16384, etc.!”

12

2017 Bernstein–Heninger–Lou–

Valenta “Post-quantum RSA”

(pqRSA): Generated 1-terabyte

RSA key; 2000000 core-hours.

Shor’s algorithm: >2100 gates.

11

RSA-8192, RSA-16384, etc.!”

12

Bernstein–Fried–Heninger–Lou–

Valenta: Draft NIST submission

proposing 1-gigabyte RSA keys.

Much faster to generate.

11

RSA-8192, RSA-16384, etc.!”

12

13

The secret primes are small:

4096 bits in terabyte key;

1024 bits in gigabyte key.

Important time-saver in

keygen, signing, decryption.

Is this a weakness?

ECM finds any prime

11

RSA-8192, RSA-16384, etc.!”

12

13

Is this a weakness?

ECM finds any prime

11

RSA-8192, RSA-16384, etc.!”

12

13

Is this a weakness?

ECM finds any prime

Date post:	08-Feb-2021
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