Exponential Lower Bound for 2-Query Locally Decodable Codes via a Quantum Argument

A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in such a way that one can recover any bit x_i from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries need exponential length: m=2^{Omega(n)}. Previously this was known only for linear codes (Goldreich et al. 02). Our proof shows that a 2-query LDC can be decoded with only 1 quantum query, and then proves an exponential lower bound for such 1-query locally quantum-decodable codes. We also show that q quantum queries allow more succinct LDCs than the best known LDCs with q classical queries. Finally, we give new classical lower bounds and quantum upper bounds for the setting of private information retrieval. In particular, we exhibit a quantum 2-server PIR scheme with O(n^{3/10}) qubits of communication, improving upon the O(n^{1/3}) bits of communication of the best known classical 2-server PIR.Comment: 16 pages Latex. 2nd version: title changed, large parts rewritten, some results added or improve

Quantum Cryptography Based Solely on Bell's Theorem

Information-theoretic key agreement is impossible to achieve from scratch and must be based on some - ultimately physical - premise. In 2005, Barrett, Hardy, and Kent showed that unconditional security can be obtained in principle based on the impossibility of faster-than-light signaling; however, their protocol is inefficient and cannot tolerate any noise. While their key-distribution scheme uses quantum entanglement, its security only relies on the impossibility of superluminal signaling, rather than the correctness and completeness of quantum theory. In particular, the resulting security is device independent. Here we introduce a new protocol which is efficient in terms of both classical and quantum communication, and that can tolerate noise in the quantum channel. We prove that it offers device-independent security under the sole assumption that certain non-signaling conditions are satisfied. Our main insight is that the XOR of a number of bits that are partially secret according to the non-signaling conditions turns out to be highly secret. Note that similar statements have been well-known in classical contexts. Earlier results had indicated that amplification of such non-signaling-based privacy is impossible to achieve if the non-signaling condition only holds between events on Alice's and Bob's sides. Here, we show that the situation changes completely if such a separation is given within each of the laboratories.Comment: 32 pages, v2: changed introduction, added reference

Efficient Bit-parallel Multiplication with Subquadratic Space Complexity in Binary Extension Field

Bit-parallel multiplication in GF(2^n) with subquadratic space complexity has been explored in recent years due to its lower area cost compared with traditional parallel multiplications. Based on \u27divide and conquer\u27 technique, several algorithms have been proposed to build subquadratic space complexity multipliers. Among them, Karatsuba algorithm and its generalizations are most often used to construct multiplication architectures with significantly improved efficiency. However, recursively using one type of Karatsuba formula may not result in an optimal structure for many finite fields. It has been shown that improvements on multiplier complexity can be achieved by using a combination of several methods. After completion of a detailed study of existing subquadratic multipliers, this thesis has proposed a new algorithm to find the best combination of selected methods through comprehensive search for constructing polynomial multiplication over GF(2^n). Using this algorithm, ameliorated architectures with shortened critical path or reduced gates cost will be obtained for the given value of n, where n is in the range of [126, 600] reflecting the key size for current cryptographic applications. With different input constraints the proposed algorithm can also yield subquadratic space multiplier architectures optimized for trade-offs between space and time. Optimized multiplication architectures over NIST recommended fields generated from the proposed algorithm are presented and analyzed in detail. Compared with existing works with subquadratic space complexity, the proposed architectures are highly modular and have improved efficiency on space or time complexity. Finally generalization of the proposed algorithm to be suitable for much larger size of fields discussed

Security of quantum key distribution protocols using two-way classical communication or weak coherent pulses

We apply the techniques introduced in [Kraus et. al., Phys. Rev. Lett. 95, 080501, 2005] to prove security of quantum key distribution (QKD) schemes using two-way classical post-processing as well as QKD schemes based on weak coherent pulses instead of single-photon pulses. As a result, we obtain improved bounds on the secret-key rate of these schemes

Decoy state quantum key distribution with two-way classical post-processing

Decoy states have recently been proposed as a useful method for substantially improving the performance of quantum key distribution protocols when a coherent state source is used. Previously, data post-processing schemes based on one-way classical communications were considered for use with decoy states. In this paper, we develop two data post-processing schemes for the decoy-state method using two-way classical communications. Our numerical simulation (using parameters from a specific QKD experiment as an example) results show that our scheme is able to extend the maximal secure distance from 142km (using only one-way classical communications with decoy states) to 181km. The second scheme is able to achieve a 10% greater key generation rate in the whole regime of distances