Index A B ABE, see Attribute-based encryption (ABE) Baby-step giant-step algorithm, 30–31 Additive inverse Bilinear mapping, 20, 26–27, 34–35 addition and subtraction property, 7, 8 Blind signature schemes inverse Chaum’s RSA, 252–253 addition and subtraction property, 7, 8 Okomoto-Schnorr protocol, 254–255 modular arithmetic, 7 Schnorr protocol, 253–254 and multiplicative inverse, 2 BLS short signature/BLS signature scheme, notation, 158–161 260 Asymmetric encryption (ASE), 82 Boneh-Boyen IBE, 152–156 Asymmetric key cryptography, 37, 104, 113, algorithm 244 decryption, 155 Attribute-based encryption (ABE) encryption, 155 access policy, 230–231 extract, 154, 155 basic model basic scheme complexity assumptions, 230 additive notation, 158 categories decrypting, 160–161 CP-ABE, 235–237 encryption, 160 KP-ABE, 233–235 private key extraction, 60 comparisons, 239–240 security, 167–169 difference, 237 setup of parameters, 158–160 encryption, 226 classification of IBE schemes, 156–157 IBE, 228 full scheme literature review, 238–239 decrypting, 167 PKI, 227, 228 encrypting, 166 public key encryption, 226–227 private key extraction, 166 security, 225, 240 setup of parameters, 165–166 working, 231–233 HIBE Attributes decryption, 199 Boneh-Franklin IBE, 138 encryption, 198 FIBE, 206 extraction of the private key, 198 public parameters, 162 setup, 197–198 See also Attribute-based encryption (ABE) limitation, 152 © Springer Nature Switzerland AG 2021 303 K. A. B. Ahmad et al. (eds.), Functional Encryption, EAI/Springer Innovations in Communication and Computing, https://doi.org/10.1007/978-3-030-60890-3 304 Index Boneh-Boyen IBE (cont.) pairing-based (see Pairing-based multiplicative notation cryptography) decrypting, 164 public key, 188–189 encrypting, 163–164 XTR-DH protocol, 50–51 extraction of the private key, 162–163 XTR-DSA signature scheme, 52–53 setup of parameters, 162 XTR-ElGamal encryption scheme, 51 security, 167–169 XTR-Nyberg-Rueppel (NR) signature setup algorithm, 154 scheme, 52 Boneh-Franklin IBE, HIBE XTR version of DH problem and its Boneh-Boyen IBE, 197–199 variants, 49–50 decryption, 196–197 Cryptosystem encryption, 195–196 ElGamal, 247 extraction of the private key, 195 elliptic curve and bilinear mapping-based, lower level setup, 195 34–35 root setup, 194–195 log-based, 34 Boneh-Franklin scheme LUC, 104 bilinear Diffie-Hellman problem, 191 NTRU, 105–111 decryption, 141–143 physical faults, 97 elliptic curve, 156 pitfalls, 190 encryption, 141–143 proposed, 281 examples, 146–148 RSA-based, 34, 60 furtherworks on, 144–146 XTR, 38, 50, 53 HIBE, 194–199 CvA, see Chaum-Van Antwerper (CvA) IBE, 137–139 Cyclotomic polynomial, 40 private key extraction, 140–142 proof, 169 security, 144 D setup of parameters, 139–140, 142 Decryption, 110–111, 125–126, 141–143 algorithm, 155 encoded version, 11 C FE scheme, 2 Chaum-Van Antwerper (CvA), 256–258 HIBE, 196–197, 199 Ciphertext policy-ABE (CP-ABE), 233, hyperelliptic curve, 72–73 235–241 operation, 286–287 Cocks scheme private key, 104 correctness of cocks IBE, 130 public key cryptosystem, 104 decryption, 125–126 Sakai-Kasahara IBE, 177, 180 encryption, 123–125 table, 292, 294, 296 examples, 126–129 Diffie-Hellman key exchange, 35, 86–88, 280, features, 119 283, 289 IBE, 118 Digital signature algorithm (DSA), 24, 34, 49, mathematical concepts, 119–120 52–53, 283 parameters, 121–122 Digital signatures private key extraction, 122–123 algorithms, 244 security, 131–133 hash functions, 245 working, 118–119 hierarchical IBS, 264–267 Conjugate element, 39, 40 other schemes, 267 Convolutional multiplication, 106, 114 security CP-ABE, see Ciphertext policy-ABE models, 244–245 (CP-ABE) services, 243 Cryptography signcryption, 268–270 HECC (see Hyperelliptic curve Digital signature standard (DSS), 250–251, cryptography (HECC)) 260, 272 Index 305 Discrete logarithm (DL) problem, 30, 31, 34, F 38, 60, 83, 95, 138, 245 FE, see Functional encryption (FE) Divisors FIBE, see Fuzzy identity-based encryption explanation 1, 62 (FIBE) explanation 2, 63 Finite field, 24, 61–62, 85–86 explanation 3, 63 definitions and results, 39–40 explanation 4, 63 factoring polynomials, 33 explanation 5, 63 hyperelliptic curves, 68–73 semi-reduced and reduced divisors, 65–67 irreducible polynomials, 33 DL problem, see Discrete logarithm (DL) properties, 25–26 problem Functional encryption (FE) DSA, see Digital signature algorithm (DSA) applications, 6 DSS, see Digital signature standard (DSS) challenges, 5–6 cryptography impact, 98 data security, 1 encrypted version, 2 E fully homomorphic encryption, 4–5 ECC, see Elliptic curve cryptography (ECC) functionality, 3 EC discrete logarithm problem (ECDLP), 83, obfuscation, 4–5 94 scheme, 3–4 ECDLP, see EC discrete logarithm problem symmetric, 2 (ECDLP) Functional field sieve, 31 ECSTR, see Efficient and compact subgroup Functionalities test trace representation (ECSTR) equality test, 27–28 Efficient and compact subgroup trace inequality test, 28 representation (ECSTR), 38 inner product evaluation, 28 ElGamal scheme, 49, 51, 52, 54 Fuzzy IBE, 157 ElGamal signature method, 73–74 Fuzzy identity-based encryption (FIBE), 206, Elliptic curve cryptography (ECC) 222, 229, 230, 233 data files, 60 formal model, 210 MOV attack, 34 security properties, 210–211 Elliptic curve factorization, 31–32 Encryption flowchart, 298 G functional (see Functional encryption) Greatest common divisor (GCD), 14, 63, 66, hyperelliptic curve, 72 67 simulation, 289–298 Bezout’s algorithm, 15–16 Exponential Euclidean/ Euclid’s algorithm, 14–15 algorithms, 92 extended Euclid’s algorithm, 16–17 Cipher technique, 280 Groups (mathematical), 20 mathematical operations, 281 Group theory See also Quatro-inverse exponential cipher axiom, 21 technique (QUIET) bilinear mapping, 26–27 Extensions of IBE cryptography, 20 FIBE, 206, 209–211 cryptosystem, 33–35 IBCPRE, 220–221 finite field properties, 25–26 IBKA, 207–209 illustrations, 21–22 LEKS, 213–214 properties of group PEKS, 206, 214–216 Abelian group, 22–23 secret session key, 205 finite field, 24 TBEKS, 211–212 Lagrange’s theorem, 23 WIBE, 207, 216–217 Schnorr group, 23–24 WKD-IBE, 218–219 science and conceptual variable-based math, 19 306 Index H Boneh-Boyen (see Boneh-Boyen IBE) HECC, see Hyperelliptic curve cryptography classification (HECC) “commutative blinding”, 157 HECC signature algorithm “exponent inversion”, 157 ElGamal signature method, 73–74 “full domain hash”, 156–157 security, 76 “quadratic residuosity”, 156 signature FIBE (see Fuzzy identity-based encryption generation, 74–75 (FIBE)) verification, 75–76 HIBE (see Hierarchical identity-based HIBE, see Hierarchical identity-based encryption (HIBE)) encryption (HIBE) Sakai-Kasahara (see Sakai-Kasahara IBE) HIBS schemes, 265, 266 See also Boneh-Franklin scheme Hierarchical identity-based encryption (HIBE), Identity-based encryption with wildcard key 192–194 derivation (WKD-IBE) Boneh-Franklin IBE, 194–197 formal model, 218–219 identity-based encryption, 190–192 security requirements, 219 master secret sharing, 199–201 Identity-based key agreement (IBKA), 206, pitfalls, PKC, 190 207 public key cryptography, 188–189 formal model, 208 security, 201–202 security requirements Hyperelliptic curve cryptography (HECC) Oracles, 209 computer algorithm-based systems, 60 divisors, 62–63 ECC, 60 J encryption and decryption, 72–73 Jacobian finite fields hyper ECs, 84 proof, 68–70 hyperelliptic curve Jacobian Hasse-Weil theory, 64 Hasse-Weil theory, 64 instruction, 64 instruction, 64 methodology, 65 methodology, 65 zeta function, 64–65 zeta function, 64–65 quotient group, 74 mathematical terminologies arithmetical closure, 61 cryptography of hyperelliptic curve, 61 finite field, 61–62 K interpretation, 62 Key policy-ABE (KP-ABE), 233–235, 237, Mumford arithmetic, 67–68 238, 241 pair generation, 70–72 Keyword search, 206 semi-reduced and reduced divisors, 65–67 IBCPRE, 220–221 signature algorithm, 73–76 LEKS, 213–214 PEKS, 206, 214–216 WIBE, 207, 216–217 I WKD-IBE, 218–219 IBCPRE, see Identity-based conditional proxy KP-ABE, see Key policy-ABE (KP-ABE) re-encryption (IBCPRE) IBE, see Identity-based encryption (IBE) IBKA, see Identity-based key agreement L (IBKA) Lagrange’s theorem, 23, 34 Identity-based conditional proxy re-encryption Lattice, 104, 112–114 (IBCPRE), 207, 222 LC, see Linear congruence (LC) formal model, 220–221 LEKS, see Linear encryption with keyword security requirements, 221 search (LEKS) Identity-based encryption (IBE), 118, 138–140 Linear congruence (LC), 11–12 Index 307 Linear encryption with keyword search encryption, 109 (LEKS), 206, 222 improvements, 113 formal model, 213 key generation, 107–109 security requirements, 213–214 optimization, 111–112 Low-Hamming-weight product, 112 parameters, 107 public key cryptosystem, 104 security, 112 M symbols and notations, 105–107 Master secret working method, 111 basic scheme—additive notation, 159 PKG, 121 public parameters, 154 O sharing, 199–201 Obfuscation, 2, 4–6 user’s identity, 190 Mathematical tools elliptic curve factorization, 31–32 P factoring polynomials, 33 Pair generation functional field sieve, 31 divisor order, 70–71 functionalities, 27–28 hyperelliptic curve cryptographic irreducible polynomials, 33 arrangement, 71–72 Paley–Wiener hypothesis, 28 Pairing-based cryptography primality test |set 1 Ate, 94 Fermat’s little theorem, 29 drawbacks/vulnerabilities, 94–95 Fermat’s strategy, 29 Eta, 93 Miller-Rabin method, 29 mathematical terms and concepts school method, 29 ASE, 82 Solovay–Strassen primality test, 29–30 Diffie–Hellman algorithm, 86–88 random square