Answer to thorny question could unlock internet security

Is it easier to cheque that a solution to a occupation is close than it is to lick the problem?

The question—known arsenic the "NP versus P" problem—is the deepest cardinal occupation successful computer science and cryptography, lying astatine the bosom of whether immoderate net information tin ever beryllium genuinely private.

In the improbable lawsuit that P = NP, each encryption schemes and methods of keeping our information connected the net backstage would beryllium insecure. But adjacent if P is not adjacent to NP, and adjacent if idiosyncratic manages to beryllium this, we inactive don't cognize however to get an encryption strategy that is genuinely secure.

Rafael Pass, prof of machine subject astatine Cornell Tech and astatine the Cornell Ann S. Bowers College of Computing and Information Science, and co-author Yanyi Liu, a doctoral pupil successful the tract of machine science, person offered a solution—sort of.

Their enactment is elaborate successful "On the Possibility of Basing Cryptography connected EXP ≠ BPP," which won the Best Paper grant astatine CRYPTO '21 and volition beryllium presented astatine the league Aug. 17.

The question posed successful the rubric of the insubstantial deals with the thought of randomness, a thorny machine subject and mathematics question. The EXP versus BPP problem—while not arsenic celebrated arsenic "NP versus P"– is different longstanding unfastened problem, and origin for adjacent much embarrassment successful the field, according to Pass.

"The question fundamentally is, tin randomness exponentially velocity up computations?" Pass said. "That's intelligibly believed to beryllium impossible. We wouldn't deliberation that conscionable tossing immoderate random coins volition let america to velocity up our computations exponentially. That would beryllium benignant of crazy, but radical inactive person not been capable to beryllium that."

If computations tin beryllium exponentially sped up utilizing randomness past each encryption schemes tin beryllium broken. The alleged "brute-force" attacks, successful which each imaginable keys are enumerated, could present beryllium efficiently implemented.

Pass and Liu tackle the question of whether simply assuming that EXP is not adjacent to BPP—that computations cannot beryllium exponentially sped up utilizing randomness—suffices to get unbreakable encryption schemes. Toward this, Pass and Liu revisit a transportation betwixt encryption schemes and time-bounded Kolmogorov Complexity that they established past year.

The time-bounded Kolmogorov Complexity of a drawstring (x) is the magnitude of the shortest programme that tin output x successful a acceptable magnitude of time. But the caller enactment considers a antithetic conception of Kolmogorov complexity: computing the "Levin-Kolmogorov Complexity" of a drawstring (x). The problem: Given x, find the "most efficient" programme that prints x, wherever "efficiency" is the sum of the magnitude of the programme and the logarithm of the moving clip of the program.

Their insubstantial shows that unbreakable encryptions are imaginable if and lone if determination does not beryllium an businesslike algorithm that tin compute the Levin-Kolmogorov Complexity for astir strings, without making excessively galore mistakes.

"So to get an unbreakable encryption," Pass said, "we conscionable request to amusement that nary efficient algorithm tin lick this peculiar problem."

While they are not capable to beryllium that nary specified algorithm exists, they amusement that assuming EXP is not adjacent to BPP, determination does not beryllium an businesslike "errorless" algorithm (an algorithm that either produces the close reply oregon says "I don't know") for determining the Levin-Kolmogorov Complexity of a ample fraction of random strings.

"It doesn't person to lick it for each the strings—it tin springiness up sometimes," Pass said. "But erstwhile it gives an answer, it ever needs to beryllium the close one."

In different words, algorithms that whitethorn err bash large connected tests wherever you are rewarded based connected the fig of questions you get right, whereas errorless algorithms besides bash good connected tests wherever you are penalized for questions you get wrong.

Their results reason that the Levin-Kolmogorov Complexity occupation is cardinal for knowing some the EXP versus BPP problem, and the occupation of whether unbreakable encryption schemes exist.

"This occupation holds the cardinal to immoderate of the astir important questions successful machine science," Pass said. "This circumstantial occupation is cardinal and we truly request to recognize the spread betwixt errorless algorithms and algorithms that whitethorn err."

The authors amusement that if the spread tin beryllium closed—a gigantic "if" successful machine science—then you person not lone proven that unbreakable cryptography exists if EXP does not adjacent BPP, but successful information you person besides proven that NP is not adjacent to P.

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More information: Yanyi Liu and Rafael Pass, On the Possibility of Basing Cryptography connected EXP 6 ≠ BPP. eprint.iacr.org/2021/535.pdf