量勵⼦孑密码学 Contents: Quantum Cryptography 1. Fake Quantum Theory. 2. Simple Quantum Protocols. 3. More Fake Quantum Theory: Fake Tensor Products, Entanglement. 4. A Glance at Genuine Quantum Theory. 5. Quantum Computing, Shor’s Algorithm and the Threat to RSA. 1. 假量勵⼦孑理痢论。 2. 简单的量勵⼦孑协议。 3. 更車多假量勵⼦孑理痢论：假张量勵积，纠缠。 4. 真正的量勵⼦孑理痢论⼀丁瞥。 5. 量勵⼦孑计算， Shor 算法和对 RSA 的威胁。 Bennett and Brassard’s QKD apparatus R. Banach, Computer Science, University of Manchester: Quantum Cryptography 1 of 32 R. Banach, Computer Science, University of Manchester: Quantum Cryptography 2 of 32 1. Fake Quantum Theory. Two State System (TSS) Basics There are two bases, called R and D . Normal (classical) physical systems are described using the usual kind of applied R consists of |0 R 〉 and |1 R 〉 mathematics — the descriptions are expressed using variables which take values Each basis consists of two states: in the real (or maybe complex) numbers, and these variables are constrained by D consists of |0 D 〉 and |1 D 〉 algebraic or differential equations of a normal kind. The R and D bases give different and incompatible views of the same system. The quantum world is very different. the |0 R 〉 or the |1 R 〉 state in the R basis Thus, a TSS can be in EITHER Developing the full machinery would take time and effort. We can get away with a the |0 D 〉 or the |1 D 〉 state in the D basis OR much simpler ‘fake’ version of quantum theory that is enough to get the main points across. We will have a quick look at the ‘real’ version of quantum theory at the end, When the TSS is in one or other state of the R basis, just to show how the fake elements correspond to the more honest picture. then NOTHING CAN BE SAID ABOUT ITS STATE IN THE D BASIS . When the TSS is in one or other state of the D basis, then NOTHING CAN BE SAID ABOUT ITS STATE IN THE R BASIS . The price to pay for fakery is that we will be restricted to only saying the simplest things about the simplest kinds of quantum system. Still, it will be enough. Quantum physics provides PERFECT CONCEALMENT. We are restricted to (so called) two state quantum systems (TSS). The above facts express the relationship between classical information (i.e. everyday 0’s and 1’s) and the quantum world. R. Banach, Computer Science, University of Manchester: Quantum Cryptography 3 of 32 R. Banach, Computer Science, University of Manchester: Quantum Cryptography 4 of 32
Two State System (TSS) Measurement Measurement and Preparation The way of getting classical information out of a quantum TSS is measurement. Therefore: If you don’t know which basis the TSS state belongs to, there is no way of reliably extracting what classical information it might contain. • You can measure in either the R or the D basis. • If the TSS is in the |0 R 〉 or the |1 R 〉 state in the R basis, and you measure in the How do you ever find anything out about a TSS? You measure it! R basis, the result is reliable: |0 R 〉 yields ‘0’ and |1 R 〉 yields ‘1’, and the TSS stays Once you have measured the TSS you know something about it. If you repeat the in the state it was. same measurement, you reliably get the same answer. • If the TSS is in the |0 D 〉 or the |1 D 〉 state in the D basis, and you measure in the This gives a method of preparing a TSS in a desired state: D basis, the result is reliable: |0 D 〉 yields ‘0’ and |1 D 〉 yields ‘1’, and the TSS stays 1. Measure the TSS in the basis to which the desired state belongs. in the state it was. • If the TSS is in the D basis, and you measure in the R basis, the result is random: 2. If the answer comes out right, that’s it; else repeat 1. ( either with a fresh TSS, EITHER the TSS state becomes |0 R 〉 and yields ‘0’ OR becomes |1 R 〉 and yields ‘1’; or with the same TSS having measured in the other basis first to randomise). and ‘0’ and ‘1’ are equally likely. Having two incompatible bases gives: two incompatible ways of storing ‘0’ in a TSS • If the TSS is in the R basis, and you measure in the D basis, the result is random: (i.e. as either |0 R 〉 or |0 D 〉 ), and two incompatible ways of storing ‘1’ in a TSS (i.e. as EITHER the TSS state becomes |0 D 〉 and yields ‘0’ OR becomes |1 D 〉 and yields ‘1’; either |1 R 〉 or |1 D 〉 ). Useful for cryptography! and ‘0’ and ‘1’ are equally likely. R. Banach, Computer Science, University of Manchester: Quantum Cryptography 5 of 32 R. Banach, Computer Science, University of Manchester: Quantum Cryptography 6 of 32 2. Simple Quantum Protocols. Weisner’s Quantum Money Weisner’s Quantum Money (WQM) depends on using several TSS as a MAC. The PERFECT CONCEALMENT capapbility that the availability of two incompatible bases for TSS gives, yields a useful cryptographic primitive that can be exploited in Create a banknote containing: various ways. — the value desired, — a normal, classical, serial number, We look at a couple of simple protocols. — a series of TSS, each in a random state, such that only the issuing bank knows • Weisner’s Quantum Money. the state of each TSS (and in particular • Bennett and Brassard’s BB84 Quantum Key Distribution protocol. only the issuing bank knows the basis that each state belongs to). • Bennett’s B92 Quantum Key Distribution protocol. The perfect concealment of the quantum world provides authentication. R. Banach, Computer Science, University of Manchester: Quantum Cryptography 7 of 32 R. Banach, Computer Science, University of Manchester: Quantum Cryptography 8 of 32
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