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Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Impagliazzo’s Personal View of Average-Case Complexity

summarized Hörður Helgi Helgason

University of Iceland, Faculty of Industrial Engineering, Mechanical Engineering and Computer Science

November 10, 2008

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Levin and Impagliazzo

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Levin and Impagliazzo

Professor Leonid A. Levin, BU CS

◮ Average Case Complete Problems, SIAM J. Comput. (1986)

15(1):285-286

◮ A structural theory of average-case complexity ◮ Many NP-complete problems fast, on average ◮ Distinguishing difficult-on-average problems is beneficial

◮ Save positive efforts ◮ Verify hardness where required, e.g. cryptography

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Levin and Impagliazzo

Professor Leonid A. Levin, BU CS

◮ Average Case Complete Problems, SIAM J. Comput. (1986)

15(1):285-286

◮ A structural theory of average-case complexity ◮ Many NP-complete problems fast, on average ◮ Distinguishing difficult-on-average problems is beneficial

◮ Save positive efforts ◮ Verify hardness where required, e.g. cryptography

Professor Russell Impagliazzo, UCSD CS

◮ A Personal View of Average-Case Complexity, Proceedings

• f the 10th Annual Structure in Complexity Theory

Conference (SCT’95): 134-147

◮ Summarize state of knowledge ◮ Motivate more research

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

What is needed?

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

What is needed?

Conventional completeness results can be relatively meaningless

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

What is needed?

Conventional completeness results can be relatively meaningless

. . . but average-run-time arguments are also unenlightening:

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

What is needed?

Conventional completeness results can be relatively meaningless

. . . but average-run-time arguments are also unenlightening:

A structural theory of distributional complexity is needed

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

What is needed?

Conventional completeness results can be relatively meaningless

. . . but average-run-time arguments are also unenlightening:

A structural theory of distributional complexity is needed

◮ to allow comparisons of the inherent intractability of

distributional problems

◮ to provide meaningful results from arbitrary distributions

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

The five worlds

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

The five worlds

Questions regarding the average case complexity of problems in NP have five possible answers

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

The five worlds

Questions regarding the average case complexity of problems in NP have five possible answers Benefits for AI, VLSI, Cryptography, Comp Sec, Derandomization of algorithms

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

The five worlds

Questions regarding the average case complexity of problems in NP have five possible answers Benefits for AI, VLSI, Cryptography, Comp Sec, Derandomization of algorithms Carl Friedrich Gauß, and his teacher

◮ B¨

uttner constructed an apparently hard problem and posed it to his students, including Gauß, who demonstrated a quick method to solve it, but. . .

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

The five worlds

Questions regarding the average case complexity of problems in NP have five possible answers Benefits for AI, VLSI, Cryptography, Comp Sec, Derandomization of algorithms Carl Friedrich Gauß, and his teacher

◮ B¨

uttner constructed an apparently hard problem and posed it to his students, including Gauß, who demonstrated a quick method to solve it, but. . .

◮ . . . what if B¨

uttner had been a complexity expert, and the main questions abour average-case complexity had been resolved? How would he fare in the five worlds?

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed:

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Revolution in comp sci

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Revolution in comp sci

◮ VLSI would be optimized: Heuristics no more

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Revolution in comp sci

◮ VLSI would be optimized: Heuristics no more ◮ AI: Inductive learning instead of expert systems:

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Revolution in comp sci

◮ VLSI would be optimized: Heuristics no more ◮ AI: Inductive learning instead of expert systems: Feed an

algorithm a training set and it produces the simplest algorithm that produced the same results as an expert system

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Revolution in comp sci

◮ VLSI would be optimized: Heuristics no more ◮ AI: Inductive learning instead of expert systems: Feed an

algorithm a training set and it produces the simplest algorithm that produced the same results as an expert system

Loss of informational-based distinction

◮ No way of telling people, computers apart with information ◮ Any code could be easily broken

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP

B¨ uttner less likely to succeed: To produce easily demonstratable solutions in class, B¨ uttner is confined to NP, now P-solvable

Revolution in comp sci

◮ VLSI would be optimized: Heuristics no more ◮ AI: Inductive learning instead of expert systems: Feed an

algorithm a training set and it produces the simplest algorithm that produced the same results as an expert system

Loss of informational-based distinction

◮ No way of telling people, computers apart with information ◮ Any code could be easily broken

To show we are in Algorithmica we need to demonstrate an efficient algorithm for solving an NP-complete problem

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost:

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Comparison with Algorithmica

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Comparison with Algorithmica

◮ Appears to be similar, since on average finding and solving

NP-problems would be close

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Comparison with Algorithmica

◮ Appears to be similar, since on average finding and solving

NP-problems would be close

◮ But here, the average-case time to solve an NP-problem =

f(average-case time to think it up)

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Comparison with Algorithmica

◮ Appears to be similar, since on average finding and solving

NP-problems would be close

◮ But here, the average-case time to solve an NP-problem =

f(average-case time to think it up): Situation not clear. . .

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Comparison with Algorithmica

◮ Appears to be similar, since on average finding and solving

NP-problems would be close

◮ But here, the average-case time to solve an NP-problem =

f(average-case time to think it up): Situation not clear. . .

◮ E.g. VLSI: Sure, an algorithm produced here might work

well on most circuits. But we don’t care about most circuits, just circuits that are minimal instantiations of the provided specifics

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

P = NP worst case, but P = NP on average

B¨ uttner can succeed, but at great cost: Hard instances of a problem can be found, but finding them takes a really long time

Comparison with Algorithmica

◮ Appears to be similar, since on average finding and solving

NP-problems would be close

◮ But here, the average-case time to solve an NP-problem =

f(average-case time to think it up): Situation not clear. . .

◮ E.g. VLSI: Sure, an algorithm produced here might work

well on most circuits. But we don’t care about most circuits, just circuits that are minimal instantiations of the provided specifics

◮ Security: As bad as in Algorithmica: Harder to protect than

to break

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

B¨ uttner can pose hard problems, but can’t solve himself

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

B¨ uttner can pose hard problems, but can’t solve himself

Generating hard instances of NP-problems is easy, but there is no way to generate hard solved instances

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

B¨ uttner can pose hard problems, but can’t solve himself

Generating hard instances of NP-problems is easy, but there is no way to generate hard solved instances

◮ Similar to real life: Progress made slowly, through better

understanding of the relevant situation

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

B¨ uttner can pose hard problems, but can’t solve himself

Generating hard instances of NP-problems is easy, but there is no way to generate hard solved instances

◮ Similar to real life: Progress made slowly, through better

understanding of the relevant situation

◮ Cryptography: No apparent way of making use of the hard

average-case problems

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

B¨ uttner can pose hard problems, but can’t solve himself

Generating hard instances of NP-problems is easy, but there is no way to generate hard solved instances

◮ Similar to real life: Progress made slowly, through better

understanding of the relevant situation

◮ Cryptography: No apparent way of making use of the hard

average-case problems: A problem that no-one knows the answer to cannot be used to distinguish between legitimate and other users

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

Hard average-case problems, but no one-way functions

B¨ uttner can pose hard problems, but can’t solve himself

Generating hard instances of NP-problems is easy, but there is no way to generate hard solved instances

◮ Similar to real life: Progress made slowly, through better

understanding of the relevant situation

◮ Cryptography: No apparent way of making use of the hard

average-case problems: A problem that no-one knows the answer to cannot be used to distinguish between legitimate and other users To show we are in Pessiland we need to give an average-case lower bound for some problem in NP

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

One-way functions can be used to create hard problems, so B¨ uttner can finally pose hard problems to Gauß

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

One-way functions can be used to create hard problems, so B¨ uttner can finally pose hard problems to Gauß

Security implications

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

One-way functions can be used to create hard problems, so B¨ uttner can finally pose hard problems to Gauß

Security implications

◮ Sharing secrets in advance can be used to create secure

• channels. . .

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

One-way functions can be used to create hard problems, so B¨ uttner can finally pose hard problems to Gauß

Security implications

◮ Sharing secrets in advance can be used to create secure

• channels. . .

◮ . . . but no way to initiate private communications over

public networks

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

One-way functions can be used to create hard problems, so B¨ uttner can finally pose hard problems to Gauß

Security implications

◮ Sharing secrets in advance can be used to create secure

• channels. . .

◮ . . . but no way to initiate private communications over

public networks

◮ Many applications not feasible, e.g. anonymous digital

currencies

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

One-way functions, but no PKC

One-way functions can be used to create hard problems, so B¨ uttner can finally pose hard problems to Gauß

Security implications

◮ Sharing secrets in advance can be used to create secure

• channels. . .

◮ . . . but no way to initiate private communications over

public networks

◮ Many applications not feasible, e.g. anonymous digital

currencies To show we are in Minicrypt we would need to show that no efficient algorithm existed for inverting one-way functions, and show how to break any secret-key algorithm

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . .

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . . ◮ No technical way of curbing secure communications

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . . ◮ No technical way of curbing secure communications

. . . but differences remain

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . . ◮ No technical way of curbing secure communications

. . . but differences remain

◮ All known secure PKIs rely on RSA/Rabin/Diffie-Hellman

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . . ◮ No technical way of curbing secure communications

. . . but differences remain

◮ All known secure PKIs rely on RSA/Rabin/Diffie-Hellman ◮ Efficient integer factoring methods would topple this

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . . ◮ No technical way of curbing secure communications

. . . but differences remain

◮ All known secure PKIs rely on RSA/Rabin/Diffie-Hellman ◮ Efficient integer factoring methods would topple this ◮ No theoretical reason why this problem is intractable and

there exist QC models that solve it

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds

Algorithmica Heuristica Pessiland Minicrypt Cryptomania

Definitional issues

PKC is possible

B¨ uttner can selectively stump Gauß, by utilizing secure channels to share solutions with everyone else in class rendering

Security implications: Close to the real world. . .

◮ Implies one-way functions ◮ Apps feasible: electronic voting, anonymous digital cash, . . . ◮ No technical way of curbing secure communications

. . . but differences remain

◮ All known secure PKIs rely on RSA/Rabin/Diffie-Hellman ◮ Efficient integer factoring methods would topple this ◮ No theoretical reason why this problem is intractable and

there exist QC models that solve it To show we are in Cryptomania we need to show that a secret-key protocol is secure, e.g. by proving a strong lower bound on average-case time to factor integers

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Criticism of Levin’s paper

Impagliazzo’s Personal View of Average-Case Complexity H.H.Helgason Abstract Introduction Five possible worlds Definitional issues

Criticism of Levin’s paper

Infinite input distributions versus ensembles of finite input distributions Expected Time versus the Average Case Extensions