Acknowledgement
Supported by : National Research Foundation of Korea (NRF)
References
- M. Abdalla, M. Bellare, and P. Rogaway, The oracle Diffie-Hellman assumptions and an analysis of DHIES, CT-RSA, David Naccache, 143-158, LNCS 2020, 2001.
- M. Abe, E. Kiltz, and T. Okamoto, Chosen ciphertext security with optimal ciphertext overhead, Advances in Cryptology-AsiaCrypt, Josef Pieprzyk, 355-371, LNCS 5350, 2008.
- M. Abe, E. Kiltz, and T. Okamoto, Compact CCA-secure encryption for messages of arbitrary length, PKC, Stanis- law Jarecki and Gene Tsudik, 377-392, LNCS 5443, 2009.
- L. Adleman, The function field sieve, ANTS, Leonard Adleman and Ming-Deh Huang, 108-121, LNCS 877, 1994.
- P. Bateman and R. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367. https://doi.org/10.1090/S0025-5718-1962-0148632-7
-
P. Bateman and R. Stemmler, Waring's problem for algebraic number fields and primes of the form
$(p^r-1)/(p^d-1)$ , Illinois J. Math. 6 (1962), no. 1, 142-156. - K. Barr and K. Asanovic, Energy-aware lossless data compression, ACMTrans. Comput. Syst. 24 (2006), no. 3, 250-291. https://doi.org/10.1145/1151690.1151692
- E. Berlekamp, Factoring polynomials over large finite fields, Math. Comp. 24 (1970), no. 111, 713-735. https://doi.org/10.1090/S0025-5718-1970-0276200-X
- D. Boneh, The decision Diffie-Hellman problem, ANTS, Joe Buhler, 48-63, LNCS 1423, 1998.
- D. Boneh, A. Joux, and P. Q. Nguyen, Why textbook ElGamal and RSA encryption are insecure, Advances in Cryptology-AsiaCrypt, Tatsuaki Okamoto, 30-43, LNCS 1976, 2000.
- R. Brent, An improved Monte Carlo factorization algorithm, BIT 20 (1980), no. 2, 176-184. https://doi.org/10.1007/BF01933190
- D. Cantor and H. Zassenhaus, A new algorithm for factoring polynomials over finite fields, Math. Comp. 36 (1981), no. 154, 587-592. https://doi.org/10.1090/S0025-5718-1981-0606517-5
- G. Castagnos and B. Chevallier-Mames, Towards a DL-based additively homomorphic encryption scheme, ISC, Juan Garay and Arjen Lenstra and Masahiro Mambo and Rene Peralta, 362-375, LNCS 4779, 2007.
- H. Chan, H.-C. Hsiao, A. Perrig, and D. Song, Secure distributed data aggregation, Foundations and Trends in Databases 3 (2011), no. 3, 149-201.
- J. Cooley and J. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297-301. https://doi.org/10.1090/S0025-5718-1965-0178586-1
- I. Damgard and M. Jurik, A generalisation, a simplification and some applications of Paillier's probabilistic public-key system, Public Key Cryptography, Kwangjo Kim, 119-136, LNCS 1992, 2001.
- J. von zur Gathen, Who was who in polynomial factorization, ISSAC, Barry Trager, 2, 2006.
- J. von zur Gathen and D. Panario, Factoring polynomials over finite fields: A survey, J. Symb. Comput. 31 (2001), no. 1-2, 3-17.
- J. von zur Gathen and V. Shoup, Computing Frobenius maps and factoring polynomials, Comput. Complexity 2 (1992), no. 3, 187-224. https://doi.org/10.1007/BF01272074
- J. Fraleigh, A First Course in Abstract Algebra, 7th, Pearson Education, 2003.
- T. El Gamal, A Public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory 31 (1985), no. 4, 469-472. https://doi.org/10.1109/TIT.1985.1057074
- C. Gentry, How to compress Rabin ciphertexts and signatures (and more), Advances in Cryptology-Crypto, Matthew K. Franklin, 179-200, LNCS 3152, 2004.
- O. Goldreich, S. Micali, and A. Wigderson, Proofs that yield nothing but their validity, or All languages in NP have zero-knowledge proof systems, J. Assoc. Comput. Mach. 38 (1991), no. 3, 691-729.
- D. Goldschlag, M. Reed, and Paul Syverson, Onion routing, Commun. ACM 42 (1999), no. 2, 39-41.
- S. Goldwasser and S. Micali, Probabilistic encryption, J. Comput. System Sci. 28 (1984), no. 2, 270-299. https://doi.org/10.1016/0022-0000(84)90070-9
- D. M. Gordon, Discrete logarithms in GF(p) using the number field sieve, SIAM J. Discrete Math. 6 (1993), no. 1, 124-138. https://doi.org/10.1137/0406010
- J. Gower, Prime order primitive subgroups in torus-based cryptography, IACR Cryptology ePrint Archive 2006: 466, 2006.
- R. Granger and F. Vercauteren, On the discrete logarithm problem on algebraic tori, Advances in Cryptology-Crypto, Victor Shoup, 66-85, LNCS 3621, 2005.
- R. Heath-Brown, Differences between consecutive primes, Seminar on Number Theory, 1979-80 (French) Exp. No. 14, Univ. Bordeaux I, 1980.
- J. Hong, J. W. Kim, J. Kim, K. Park, and J. H. Cheon, Constant-round privacy pre- serving multiset union, Cryptology ePrint Archive 2011:138, 2011.
- T. W. Hungerford, Algebra, Springer, 1980.
- M. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), 164-170.
- G. Itkis and L. Reyzin, Forward-secure signatures with optimal signing and verifying, Advances in Cryptology-Crypto, Joe Kilian, 332-354, LNCS 2139, 2001.
- R. Johnson, D. Molnar, D. Song, and D. Wagner, Homomorphic signature schemes, CT-RSA, Bart Preneel, 244-262, LNCS 2271, 2002.
- A. Joux and R. Lercier, The function field sieve in the medium prime case, Advances in Cryptology-EuroCrypt, Serge Vaudenay, 254-270, LNCS 4004, 2006.
- A. Joux, R. Lercier, N. Smart, and F. Vercauteren, The number field sieve in the medium prime case, Advances in Cryptology-Crypto, Cynthia Dwork, 326-344, LNCS 4117, 2006.
- E. Kaltofen, Polynomial factorization: a success story, ISSAC, J. Rafael Sendra, 3-4, 2003.
- E. Kaltofen and V. Shoup, Subquadratic-time factoring of polynomials over finite fields, Math. Comp. 67 (1998), no. 223, 1179-1197. https://doi.org/10.1090/S0025-5718-98-00944-2
- A. Karatsuba and Y. Ofman, Multiplication of multidigit numbers on automata, Soviet Physics Doklady 7 (1963), 595-596.
- N. Kayal and N. Saxena, On the ring isomorphism & automorphism problems, Electronic Colloquium on Computational Complexity (ECCC), 11 (2004), no. 109.
- M. Kim and H. T. Lee, and J. H. Cheon, Mutual Private Set Intersection with Linear Complexity, WISA, 2011.
- H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), no. 3, 649-673. https://doi.org/10.2307/1971363
- H. W. Lenstra Jr., Finding isomorphisms between finite fields, Math. Comp. 56 (1991), no. 193, 329-347. https://doi.org/10.1090/S0025-5718-1991-1052099-2
- D. Ma, C. Soriente, and G. Tsudik, New adversary and new threats: security in unat- tended sensor networks, IEEE Network 23 (2009), no. 2, 43-48.
- D. Ma and G. Tsudik, Extended abstract: forward-secure sequential aggregate authentication, IEEE Symposium on Security and Privacy, 86-91, 2007.
- H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), no. 2, 221-225. https://doi.org/10.1307/mmj/1029003189
- T. Okamoto and S. Uchiyama, A new public-key cryptosystem as secure as factoring, Advances in Cryptology-EuroCrypt, Kaisa Nyberg, 308-318, LNCS 1403, 1998.
- P. Paillier, Public-key cryptosystems based on composite degree residuosity classes, Ad- vances in Cryptology-EuroCrypt, Jacques Stern, 223-238, LNCS 1592, 1999.
- R. Peralta, On the distribution of quadratic residues and nonresidues modulo a prime number, Math. Comp. 58 (1992), no. 197, 433-440. https://doi.org/10.1090/S0025-5718-1992-1106978-9
- D. H. Phan and D. Pointcheval, Chosen-ciphertext security without redundancy, Ad- vances in Cryptology-AsiaCrypt, Chi-Sung Laih, 1-18, LNCS 2894, 2003.
- J. Pollard, Theorems on factorization and primality testing, Proc. Cambridge Philos. Soc. 76 (1974), 521-528. https://doi.org/10.1017/S0305004100049252
- V. Shoup, A Computational Introduction to Number Theory and Algebra, Cambridge University Press, 2005.
- C. Umans, Fast polynomial factorization and modular composition in small characteristic, STOC'08, 481-490, ACM, New York, 2008.
- A. C. Yao, Protocols for secure computations, FOCS, 160-164, 1982.
- T. Yonemura, Y. Hanatani, T. Isogai, K. Ohkuma, and H. Muratani, Generating parameters for algebraic torus-based cryptosystems, CANS, Swee-Huay Heng and Rebecca Wright and Bok-Min Goi, 156-168, LNCS 6467, 2010.