Privacy-Preserving Data Falsification Detection in Smart Grids using Elliptic Curve Cryptography and Homomorphic Encryption

In an advanced metering infrastructure (AMI), the electric utility collects power consumption data from smart meters to improve energy optimization and provides detailed information on power consumption to electric utility customers. However, AMI is vulnerable to data falsification attacks, which organized adversaries can launch. Such attacks can be detected by analyzing customers\u27 fine-grained power consumption data; however, analyzing customers\u27 private data violates the customers\u27 privacy. Although homomorphic encryption-based schemes have been proposed to tackle the problem, the disadvantage is a long execution time. This paper proposes a new privacy-preserving data falsification detection scheme to shorten the execution time. We adopt elliptic curve cryptography (ECC) based on homomorphic encryption (HE) without revealing customer power consumption data. HE is a form of encryption that permits users to perform computations on the encrypted data without decryption. Through ECC, we can achieve light computation. Our experimental evaluation showed that our proposed scheme successfully achieved 18 times faster than the CKKS scheme, a common HE scheme

Faster Final Exponentiation on the KSS18 Curve

The final exponentiation affects the efficiency of pairing computations especially on pairing-friendly curves with high embedding degree. We propose an efficient method for computing the hard part of the final exponentiation on the KSS18 curve at 192-bit security level. Implementations indicate that the computation of the final exponentiation can be 8.74% faster than the previously fastest result

Efficient hash maps to G2 on BLS curves

When a pairing e:G1×G2→GT, on an elliptic curve E defined over a finite field Fq, is exploited for an identity-based protocol, there is often the need to hash binary strings into G1 and G2. Traditionally, if E admits a twist E~ of order d, then G1=E(Fq)∩E[r], where r is a prime integer, and G2=E~(Fqk/d)∩E~[r], where k is the embedding degree of E w.r.t. r. The standard approach for hashing into G2 is to map to a general point P∈E~(Fqk/d) and then multiply it by the cofactor c=#E~(Fqk/d)/r. Usually, the multiplication by c is computationally expensive. In order to speed up such a computation, two different methods—by Scott et al. (International conference on pairing-based cryptography. Springer, Berlin, pp 102–113, 2009) and by Fuentes-Castaneda et al. (International workshop on selected areas in cryptography)—have been proposed. In this paper we consider these two methods for BLS pairing-friendly curves having k∈{12,24,30,42,48}, providing efficiency comparisons. When k=42,48, the application of Fuentes et al. method requires expensive computations which were infeasible for the computational power at our disposal. For these cases, we propose hashing maps that we obtained following Fuentes et al. idea.publishedVersio

Efficient hash maps to G2 on BLS curves

When a pairing e:G1×G2→GT, on an elliptic curve E defined over a finite field Fq, is exploited for an identity-based protocol, there is often the need to hash binary strings into G1 and G2. Traditionally, if E admits a twist E~ of order d, then G1=E(Fq)∩E[r], where r is a prime integer, and G2=E~(Fqk/d)∩E~[r], where k is the embedding degree of E w.r.t. r. The standard approach for hashing into G2 is to map to a general point P∈E~(Fqk/d) and then multiply it by the cofactor c=#E~(Fqk/d)/r. Usually, the multiplication by c is computationally expensive. In order to speed up such a computation, two different methods—by Scott et al. (International conference on pairing-based cryptography. Springer, Berlin, pp 102–113, 2009) and by Fuentes-Castaneda et al. (International workshop on selected areas in cryptography)—have been proposed. In this paper we consider these two methods for BLS pairing-friendly curves having k∈{12,24,30,42,48}, providing efficiency comparisons. When k=42,48, the application of Fuentes et al. method requires expensive computations which were infeasible for the computational power at our disposal. For these cases, we propose hashing maps that we obtained following Fuentes et al. idea

Hashing to elliptic curves of j=0j=0 and quadratic imaginary orders of class number 22

In this article we produce the simplified SWU encoding to some Barreto--Naehrig curves, including BN512, BN638 from the standards ISO/IEC 15946-5 and TCG Algorithm Registry respectively. Moreover, we show (for any $j$-invariant) how to implement the simplified SWU encoding in constant time of one exponentiation in the basic field, namely without quadratic residuosity tests and inversions. Thus in addition to the protection against timing attacks, the new encoding turns out to be much more efficient than the (universal) SWU encoding, which generally requires to perform two quadratic residuosity tests

Efficient Elliptic Curve Diffie-Hellman Computation at the 256-bit Security Level

In this paper we introduce new Montgomery and Edwards form elliptic curve targeted at the 256-bit security level. To this end, we work with three primes, namely $p_1:=2^{506}-45$, $p_2=2^{510}-75$ and $p_3:=2^{521}-1$. While $p_3$ has been considered earlier in the literature, $p_1$ and $p_2$ are new. We define a pair of birationally equivalent Montgomery and Edwards form curves over all the three primes. Efficient 64-bit assembly implementations targeted at Skylake and later generation Intel processors have been made for the shared secret computation phase of the Diffie-Hellman key agreement protocol for the new Montgomery curves. Curve448 of the Transport Layer Security, Version 1.3 is a Montgomery curve which provides security at the 224-bit security level. Compared to the best publicly available 64-bit implementation of Curve448, the new Montgomery curve over $p_1$ leads to a $3\%$-$4\%$ slowdown and the new Montgomery curve over $p_2$ leads to a $4.5\%$-$5\%$ slowdown; on the other hand, 29 and 30.5 extra bits of security respectively are gained. For designers aiming for the 256-bit security level, the new curves over $p_1$ and $p_2$ provide an acceptable trade-off between security and efficiency

Multiplication over Extension Fields for Pairing-based Cryptography: an Hardware Point of View

New Number Field Sieves (NFS) attacks on the discrete logarithm problem have led to increase the key size of pairing-based cryptography and more precisely pairings on most popular curves like BN. To ensure 128-bit security level, recent costs estimations recommand to switch for BLS24 curves. However, using BLS24 curves for pairing requires to have an efficient arithmetic in Fp4. In this paper, we transposed previous work on multiplication over extesnsion fields using Newton\u27s interpolation to construct a new formula for multiplication in Fp4 and propose time x area efficient hardware implementation of this operation. This co-processor is implemented on Kintex-7 Xilinx FPGA. The efficiency of our design in terms of time x area is almost 3 times better than previous specific architecture for multiplication in Fp4. Our architecture is used to estimate the efficiency of hardware implementations of full pairings on BLS12 and BLS24 curves with a 128-bit security level. This co-processeur can be easily modified to anticipate further curve changes

A Revocable Group Signature Scheme with Scalability from Simple Assumptions and Its Application to Identity Management

Group signatures are signatures providing signer anonymity where signers can produce signatures on behalf of the group that they belong to. Although such anonymity is quite attractive considering privacy issues, it is not trivial to check whether a signer has been revoked or not. Thus, how to revoke the rights of signers is one of the major topics in the research on group signatures. In particular, scalability, where the signing and verification costs and the signature size are constant in terms of the number of signers N, and other costs regarding signers are at most logarithmic in N, is quite important. In this paper, we propose a revocable group signature scheme which is currently more efficient compared to previous all scalable schemes. Moreover, our revocable group signature scheme is secure under simple assumptions (in the random oracle model), whereas all scalable schemes are secure under q-type assumptions. We implemented our scheme by employing Barreto-Lynn-Scott curves of embedding degree 12 over a 455-bit prime field (BLS-12-455), and Barreto-Naehrig curves of embedding degree 12 over a 382-bit prime field (BN-12-382), respectively, by using the RELIC library. We showed that the online running times of our signing algorithm were approximately 14 msec (BLS-12-455) and 11 msec (BN-12-382), and those of our verification algorithm were approximately 20 msec (BLS-12-455) and 16 msec (BN-12-382), respectively. Finally, we showed that our scheme is applied to an identity management system proposed by Isshiki et al