Document Type : Research Article
Authors
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran
Abstract
Oblivious transfer is one of the important tools in cryptography, in which a sender sends a message to a receiver with a probability between 0 and 1, while the sender remains oblivious that the receiver has received the message.
A flavor of $OT$ schemes is chosen $t$-out-of-$k$ oblivious transfer ($OT^t_k$). In an $OT^t_k$ scheme, a sender transfers $k$ messages to a receiver, the receiver can learn only $t$ of them, and the sender remains oblivious to which secrets are extracted by the receiver.
In this paper, we first propose a type of Diffie-Hellman key exchange protocol using the generalized Jacobian of elliptic curves. Next, we introduce simple, secure two-round algorithms for $OT$, $OT^1_2$, $OT^t_k$.
The security of proposed protocols is based on the intractability assumption of solving discrete logarithm problem; furthermore, in our $OT$ schemes, it is not necessary to map the messages to the points on the elliptic curve.
Keywords
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Zheng, Yi Tang, and Mingqiang Wang. Cut-
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based on ddh assumption. Journal of Ambient
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quantum level. Optik, 126(21):3206–3209, 2015.
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transfer protocol based on elgamal encryption for
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Conference on Cryptology and Information Se-
curity in Latin America, pages 40–58. Springer,
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universally composable protocols for oblivious
transfer from the cdh assumption. Cryptology
ePrint Archive, 2017.
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tic curves. In 2006 15th International Conference
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Lempel. A randomized protocol for signing con-
tracts. Communications of the ACM, 28(6):637–
647, 1985.
[3] Gilles Brassard, Claude Cr´epeau, and Jean-
Marc Robert. All-or-nothing disclosure of se-
crets. In Conference on the Theory and Applica-
tion of Cryptographic Techniques, pages 234–238.
Springer, 1987.
[4] Yi Mu, Junqi Zhang, and Vijay Varadharajan.
m out of n oblivious transfer. In Australasian
Conference on Information Security and Privacy,
pages 395–405. Springer, 2002.
[5] Chin-Chen Chang and Jung-San Lee. Robust t-
out-of-n oblivious transfer mechanism based on
crt. Journal of network and computer applications,
32(1):226–235, 2009.
[6] Yalin Chen, Jue-Sam Chou, and Xian-Wu Hou.
A novel k-out-of-n oblivious transfer protocols
based on bilinear pairings. Cryptology ePrint
Archive, 2010.
[7] Der-Chyuan Lou and Hui-Feng Huang. An effi-
cient t-out-of-n oblivious transfer for information
security and privacy protection. International
Journal of Communication Systems, 27(12):3759–
3767, 2014.
[8] Jianchang Lai, Yi Mu, Fuchun Guo, Rongmao
Chen, and Sha Ma. Efficient k-out-of-n oblivious
transfer scheme with the ideal communication
cost. Theoretical Computer Science, 714:15–26,
2018.
[9] Martin Stanek et al. Fast contract signing with
batch oblivious transfer. In IFIP International
Conference on Communications and Multimedia
Security, pages 1–10. Springer, 2005.
[10] Haruna Higo, Keisuke Tanaka, Akihiro Yamada,
and Kenji Yasunaga. A game-theoretic perspec-
tive on oblivious transfer. In Australasian Confer-
ence on Information Security and Privacy, pages
29–42. Springer, 2012.
[11] Han Jiang, Qiuliang Xu, Changyuan Liu, Zhihua
Zheng, Yi Tang, and Mingqiang Wang. Cut-
and-choose bilateral oblivious transfer protocol
based on ddh assumption. Journal of Ambient
Intelligence and Humanized Computing, pages
1–11, 2018.
[12] Carmit Hazay, Peter Scholl, and Eduardo Soria-
Vazquez. Low cost constant round mpc combining
bmr and oblivious transfer. Journal of Cryptology,
33(4):1732–1786, 2020.
[13] Sunil B Mane and Pradeep K Sinha. Oblivi-
ous information retrieval on outsourced database
servers. International Journal of Scientific and
Engineering Research, 5(5), 2014.
[14] Yu-Guang Yang, Si-Jia Sun, Qing-Xiang Pan,
and Peng Xu. Reductions between private in-
formation retrieval and oblivious transfer at the
quantum level. Optik, 126(21):3206–3209, 2015.
[15] Hoda Jannati and Behnam Bahrak. An oblivious
transfer protocol based on elgamal encryption for
preserving location privacy. Wireless Personal
Communications, 97(2):3113–3123, 2017.
[16] Tung Chou and Claudio Orlandi. The simplest
protocol for oblivious transfer. In International
Conference on Cryptology and Information Se-
curity in Latin America, pages 40–58. Springer,
2015.
[17] Eduard Hauck and Julian Loss. Efficient and
universally composable protocols for oblivious
transfer from the cdh assumption. Cryptology
ePrint Archive, 2017.
[18] Martin Hellman. New directions in cryptogra-
phy. IEEE transactions on Information Theory,
22(6):644–654, 1976.
[19] I D´echene. Arithmetic of generalized jacobians,
algorithmic number theory symposium-ants vii.
Lecture Notes in Computer Science, pages 421–
435.
[20] Abhishek Parakh. Oblivious transfer using ellip-
tic curves. In 2006 15th International Conference
on Computing, pages 323–328. IEEE, 2006.
[21] Paulo SLM Barreto, Bernardo David, Rafael
Dowsley, Kirill Morozov, and Anderson CA Nasci-
mento. A framework for efficient adaptively se-
cure composable oblivious transfer in the rom.
arXiv preprint arXiv:1710.08256, 2017.
[22] Zengpeng Li, Chunguang Ma, Minghao Zhao,
and Chang Choi. Efficient oblivious transfer con-
struction via multiple bits dual-mode cryptosys-
tem for secure selection in the cloud. Journal of
the Chinese Institute of Engineers, 42(1):97–106,
2019.
[23] Vipul Goyal, Abhishek Jain, Zhengzhong Jin,
and Giulio Malavolta. Statistical zaps and new
oblivious transfer protocols. In Annual Inter-
national Conference on the Theory and Applica-
tions of Cryptographic Techniques, pages 668–699.
Springer, 2020.
[24] Nibedita Kundu, Sumit Kumar Debnath, and
Dheerendra Mishra. 1-out-of-2: post-quantum
oblivious transfer protocols based on multivariate
public key cryptography. S¯adhan¯a, 45(1):1–12,
2020.
[25] Saeid Esmaeilzade, Nasrollah Pakniat, and Ziba
Eslami. A generic construction to build simple
oblivious transfer protocols from homomorphic
encryption schemes. The Journal of Supercom-
puting, 78(1):72–92, 2022.
[26] Joseph H Silverman. The arithmetic of elliptic
curves, volume 106. Springer, 2009.
[27] Lawrence C Washington. Elliptic curves: num-
ber theory and cryptography. Chapman and Hal-
l/CRC, 2008.
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rieties. Annals of Mathematics, pages 505–530,
1954.
[29] H Daghigh and M Bahramian. Generalized jaco-
bian and discrete logarithm problem on elliptic
curves. 2009.
[30] Don Johnson, Alfred Menezes, and Scott Vans-
tone. The elliptic curve digital signature algo-
rithm (ecdsa). International journal of informa-
tion security, 1(1):36–63, 2001.
[31] Cheng-Kang Chu and Wen-Guey Tzeng. Effi-
cient k-out-of-n oblivious transfer schemes with
adaptive and non-adaptive queries. In Interna-
tional Workshop on Public Key Cryptography,
pages 172–183. Springer, 2005.
[32] Jianhong Zhang and Yumin Wang. Two provably
secure k-out-of-n oblivious transfer schemes. Ap-
plied mathematics and computation, 169(2):1211–
1220, 2005.
[33] Neal Koblitz, Alfred Menezes, and Scott Van-
stone. The state of elliptic curve cryptography.
Designs, codes and cryptography, 19(2):173–193,
2000.
Dowsley, Kirill Morozov, and Anderson CA Nasci-
mento. A framework for efficient adaptively se-
cure composable oblivious transfer in the rom.
arXiv preprint arXiv:1710.08256, 2017.
[22] Zengpeng Li, Chunguang Ma, Minghao Zhao,
and Chang Choi. Efficient oblivious transfer con-
struction via multiple bits dual-mode cryptosys-
tem for secure selection in the cloud. Journal of
the Chinese Institute of Engineers, 42(1):97–106,
2019.
[23] Vipul Goyal, Abhishek Jain, Zhengzhong Jin,
and Giulio Malavolta. Statistical zaps and new
oblivious transfer protocols. In Annual Inter-
national Conference on the Theory and Applica-
tions of Cryptographic Techniques, pages 668–699.
Springer, 2020.
[24] Nibedita Kundu, Sumit Kumar Debnath, and
Dheerendra Mishra. 1-out-of-2: post-quantum
oblivious transfer protocols based on multivariate
public key cryptography. S¯adhan¯a, 45(1):1–12,
2020.
[25] Saeid Esmaeilzade, Nasrollah Pakniat, and Ziba
Eslami. A generic construction to build simple
oblivious transfer protocols from homomorphic
encryption schemes. The Journal of Supercom-
puting, 78(1):72–92, 2022.
[26] Joseph H Silverman. The arithmetic of elliptic
curves, volume 106. Springer, 2009.
[27] Lawrence C Washington. Elliptic curves: num-
ber theory and cryptography. Chapman and Hal-
l/CRC, 2008.
[28] Maxwell Rosenlicht. Generalized jacobian va-
rieties. Annals of Mathematics, pages 505–530,
1954.
[29] H Daghigh and M Bahramian. Generalized jaco-
bian and discrete logarithm problem on elliptic
curves. 2009.
[30] Don Johnson, Alfred Menezes, and Scott Vans-
tone. The elliptic curve digital signature algo-
rithm (ecdsa). International journal of informa-
tion security, 1(1):36–63, 2001.
[31] Cheng-Kang Chu and Wen-Guey Tzeng. Effi-
cient k-out-of-n oblivious transfer schemes with
adaptive and non-adaptive queries. In Interna-
tional Workshop on Public Key Cryptography,
pages 172–183. Springer, 2005.
[32] Jianhong Zhang and Yumin Wang. Two provably
secure k-out-of-n oblivious transfer schemes. Ap-
plied mathematics and computation, 169(2):1211–
1220, 2005.
[33] Neal Koblitz, Alfred Menezes, and Scott Van-
stone. The state of elliptic curve cryptography.
Designs, codes and cryptography, 19(2):173–193,
2000.
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