Computer Security . . . Computer Security . . . Main Idea Newtonian . . . Relativistic Effects Relativistic Bit . . . Can Keep Data Secret: Relativistic Bit . . . Why This Works A Simple Scheme Limitations of This . . . Second Algorithm Vladik Kreinovich Home Page Title Page Department of Computer Science University of Texas at El Paso ◭◭ ◮◮ El Paso, Texas 79968, USA ◭ ◮ vladik@utep.edu Page 1 of 13 Go Back Full Screen Close Quit
Computer Security . . . Computer Security . . . 1. Computer Security without Physics Main Idea • Most existing computer security schemes rely on the Newtonian . . . computational complexity of certain computing tasks. Relativistic Bit . . . Relativistic Bit . . . • For example, RSA relies on the difficulty of factoring Why This Works large integers. Limitations of This . . . • These schemes use sophisticated algorithms. Second Algorithm Home Page • However, these schemes operate within standard com- putational devices. Title Page • These devices are based on classical – non-quantum, ◭◭ ◮◮ non-relativistic – physics. ◭ ◮ Page 2 of 13 Go Back Full Screen Close Quit
Computer Security . . . Computer Security . . . 2. Computer Security and Physics Main Idea • In the 1990s, it was shown that quantum effects can Newtonian . . . be successfully used for secure communications. Relativistic Bit . . . Relativistic Bit . . . • Quantum communications have indeed been used to Why This Works make communications secure. Limitations of This . . . • E.g., supposedly there is a quantum communication Second Algorithm link between the White House and the Pentagon. Home Page • A 2016 Geneva experiment showed that relativistic ef- Title Page fects can also be used to secure communications. ◭◭ ◮◮ ◭ ◮ Page 3 of 13 Go Back Full Screen Close Quit
Computer Security . . . Computer Security . . . 3. Main Idea Main Idea • The corresponding schemes use the fact that: Newtonian . . . Relativistic Bit . . . – according to relativity theory, Relativistic Bit . . . – all communication speeds are limited by the speed Why This Works of light. Limitations of This . . . • These schemes are related to the problem of bit com- Second Algorithm mitment in situations when: Home Page – the two parties do not trust each other Title Page – and there is no third person whom both parties ◭◭ ◮◮ trust. ◭ ◮ • The simplest scheme involves the situation when two Page 4 of 13 companies bid for the same job. Go Back • The smallest bid wins. Full Screen • So, if one party learns about the bid of a competitor, it can offer a slightly smaller amount and win. Close Quit
Computer Security . . . Computer Security . . . 4. Newtonian vs. Relativistic Bidding Main Idea • Thus, if one party submits a bid earlier, the other party Newtonian . . . may learn this bid and win. Relativistic Bit . . . Relativistic Bit . . . • Even if they submit simultaneously, one may submit Why This Works slightly earlier and the other will learn the bid. Limitations of This . . . • Relativistic effects enable to make bidding safe: Second Algorithm Home Page – if both parties submit their bids at the same time but from the different Earth locations, Title Page – then it takes a few milliseconds for each signal to ◭◭ ◮◮ reach the other party, ◭ ◮ – so no one can cheat. Page 5 of 13 • This idea can be extended to cases when we need to Go Back preserve a secret bid for up to 24 hours. Full Screen Close Quit
Computer Security . . . Computer Security . . . 5. Relativistic Bit Commitment: Setting of the Main Idea First Algorithm Newtonian . . . • Suppose that Alice wants to select a bid B and keep it Relativistic Bit . . . secret for time t . Relativistic Bit . . . Why This Works • In the computer, all information is stored as 0s and 1s. Limitations of This . . . • It is thus sufficient to consider each bit d from the bid. Second Algorithm Home Page • Alice does not want Bob to learn the bit until time t . Title Page • Bob wants to make sure that this bit d stays the same. ◭◭ ◮◮ • Alice and Bob do not trust each other. ◭ ◮ • However, Alice has a trusted friend Amy. Page 6 of 13 • At first, all three on them are at the same location. Go Back Full Screen Close Quit
Computer Security . . . Computer Security . . . 6. Relativistic Bit Commitment: First Algorithm Main Idea • Alice selects (and shares with Amy): Newtonian . . . Relativistic Bit . . . – a bit d and Relativistic Bit . . . – a random integer a from 1 to N . Why This Works • After this, Amy moves to a faraway location, at a dis- Limitations of This . . . tance r > c · t . Second Algorithm • After that, Bob generated a random integer Home Page b ∈ { 1 , . . . , N } , and sends it to Alice. Title Page • Alice replies with r = a + b · d mod N . ◭◭ ◮◮ • So, Bob gets either a or a + b . ◭ ◮ • Then, Amy sends a to Bob. Page 7 of 13 • Once Bob gets a , he compares a with Alice’s answer: Go Back – if r = a , this means that d = 0; Full Screen – if r = a + b , this means that d = 1. Close Quit
Computer Security . . . Computer Security . . . 7. Why This Works Main Idea • Bob cannot find d : Newtonian . . . Relativistic Bit . . . – all he knows is a random number, Relativistic Bit . . . – without knowing a , we cannot tell whether it is a Why This Works or a + b . Limitations of This . . . • Amy cannot cheat: Second Algorithm Home Page – from the moment Alice learns b , Title Page – it takes Alice at least time t to send b to Amy, and at least as long to send the reply to Bob, ◭◭ ◮◮ – so a b -dependent reply cannot get to Bob before ◭ ◮ time t . Page 8 of 13 Go Back Full Screen Close Quit
Computer Security . . . Computer Security . . . 8. Limitations of This Algorithm Main Idea • This algorithm is guaranteed to store a secret bit for Newtonian . . . time t = r/c . Relativistic Bit . . . Relativistic Bit . . . • For Earth locations, this time is limited to milliseconds. Why This Works • To store a secret for a second, Amy needs to move to Limitations of This . . . the Moon. Second Algorithm Home Page • To store a secret for 24 hours, Amy must go beyond Solar systems. Title Page • This is good for the future, but we cannot do it yet. ◭◭ ◮◮ • So, to store a bit for longer than milliseconds, we need ◭ ◮ a different algorithm. Page 9 of 13 Go Back Full Screen Close Quit
Computer Security . . . Computer Security . . . 9. Second Algorithm Main Idea • In the second algorithm, Bob also has a trusted friend Newtonian . . . Brian. Relativistic Bit . . . Relativistic Bit . . . • At first, Alice and Amy select a sequence of random Why This Works numbers a 1 , . . . , a m . Limitations of This . . . • Simultaneously, Bob and Brian select their sequence of Second Algorithm random numbers b 1 , . . . , b m . Home Page • Then, Amy and Brian jointly move away to a distance Title Page r > c · ∆ t . ◭◭ ◮◮ • First, Bob sends b 1 to Alice, she replies with ◭ ◮ r 1 = a 1 + b 1 · d. Page 10 of 13 Go Back • After time ∆ t , Brian sends b 2 to Amy, she replies with Full Screen r 2 = a 2 + b 2 · a 1 . Close Quit
Computer Security . . . Computer Security . . . 10. Second Algorithm (cont-d) Main Idea • Since r > c · ∆ t , neither Amy not Brian have informa- Newtonian . . . tion about the first exchange. Relativistic Bit . . . Relativistic Bit . . . • After time ∆ t , Bob sends b 3 to Alice, she replies with Why This Works r 3 = a 3 + b 3 · a 2 , etc. Limitations of This . . . Second Algorithm • At each cycle m , Bob or Brian send b m , and get Home Page r m = a m + b m · a m − 1 . Title Page ◭◭ ◮◮ • At the end, Amy and/or Alice reveal a m and d . ◭ ◮ • Based on a m and r m = a m + b m · a m − 1 , Bob and Brain can compute b m · a m − 1 . Page 11 of 13 • Since they know b m , they can compute a m − 1 . Go Back • Similarly, from a m − 1 and r m − 1 = a m − 1 + b m − 1 · a m − 2 , Full Screen we can compute b m − 1 · a m − 2 hence a m − 2 , etc. Close Quit
Computer Security . . . Computer Security . . . 11. Second Algorithm: Discussion Main Idea • Eventually, based on r 1 = a 1 + b 1 · d , a 1 , and b 1 , Bob Newtonian . . . and Brian can compute d . Relativistic Bit . . . Relativistic Bit . . . • So, Bob and Brian have: Why This Works – the value d that was officially disclosed by Amy and Limitations of This . . . – the value d that was used originally – that they can Second Algorithm calculate. Home Page • So, Bob and Brian can then check that it is the same Title Page d as before. ◭◭ ◮◮ • If we have m pairs of random numbers, we can keep a ◭ ◮ secret during time m · ∆ t . Page 12 of 13 • The larger m , the longer we can keep a secret. Go Back • In Geneve, Switzerland, the secret was kept for 24 Full Screen hours. Close Quit
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