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Asymptotically Good Ideal Linear Secret Sharing with Strong Multiplication over Any Fixed Finite Field [chapter]

Ignacio Cascudo, Hao Chen, Ronald Cramer, Chaoping Xing
2009 Lecture Notes in Computer Science  
Third, we present an infinite family of ideal schemes with t-strong multiplication that does not rely on algebraic geometry and that works over every finite field Fq.  ...  Second, we show that for every finite field Fq, there exists an infinite family of LSSS over Fq that is asymptotically good in the following sense: the schemes are "ideal," i.e., each share consists of  ...  Secret Sharing In this section we give precise definitions of (linear) secret sharing (with strong multiplication).  ... 
doi:10.1007/978-3-642-03356-8_28 fatcat:7eb7b3n2vzdcxox6ragick5tpu

Asymptotically Good Multiplicative LSSS over Galois Rings and Applications to MPC over $$\mathbb {Z}/p^k\mathbb {Z} $$ [chapter]

Mark Abspoel, Ronald Cramer, Ivan Damgård, Daniel Escudero, Matthieu Rambaud, Chaoping Xing, Chen Yuan
2020 Lecture Notes in Computer Science  
The standard way to obtain these over fields is with a family of linear codes C, such that C, C ⊥ and C 2 are asymptotically good (strongly multiplicative).  ...  Self-orthogonal codes are multiplicative, therefore we can use existing results of asymptotically good self-dual codes over fields to obtain arithmetic secret sharing over Galois rings.  ...  It is well-known that any linear code over a field with good parameters yields a good linear secret-sharing scheme [25] , and it is straightforward to show this also holds over Galois rings.  ... 
doi:10.1007/978-3-030-64840-4_6 fatcat:hq6twvyp65g7pkkm72p47j6ohe

Algebraic Geometry Codes: General Theory [chapter]

Iwan M. Duursma
2008 Series on Coding Theory and Cryptology  
Special attention is given to recent results on two-point codes from Hermitian curves and to applications for secret sharing.  ...  Roos bound for the minimum distance [22] , Linear secret sharing schemes [12] , Weight distributions and codes over extension fields [21] , [76] , Dual BCH codes [20] , [32] , [69] , Codes from  ...  books [5] , [36] , [44] , [49] , [54] , [62] , [68] , [71] , [72] , [75] , [77] , [79] , as well as the survey chapters [10] , [42] , [45] , [47] , discuss algebraic geometry codes, each with  ... 
doi:10.1142/9789812794017_0001 fatcat:3upxrzrbyvc3xizp3cod26lndu

Fast Large-Scale Honest-Majority MPC for Malicious Adversaries [chapter]

Koji Chida, Daniel Genkin, Koki Hamada, Dai Ikarashi, Ryo Kikuchi, Yehuda Lindell, Ariel Nof
2018 Lecture Notes in Computer Science  
We present protocol variants for small and large fields, and show how to efficiently instantiate them based on replicated secret sharing and Shamir sharing.  ...  Protocols for semi-honest adversaries are often far more efficient, but in many cases the security guarantees are not strong enough.  ...  Let σ be a statistical security parameter, let F be a finite field, and let f be a n-party functionality over F.  ... 
doi:10.1007/978-3-319-96878-0_2 fatcat:55tehzjfi5h3ba5ot5ckt5fj7a

Secret Sharing with Binary Shares

Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, Huaxiong Wang, Michael Wagner
2018 Innovations in Theoretical Computer Science  
t shares does not reveal any information about the secret and, (ii) any choice of t + 1 shares fully reveals the secret.  ...  For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any τ fraction of observed shares, and reconstruction from any ρ fraction of shares, for any choices  ...  Then, Shamir's scheme treats the secret as an element of the finite field F q , where q = 2 , padded with t uniformly random and independent elements from the same field.  ... 
doi:10.4230/lipics.itcs.2019.53 dblp:conf/innovations/LinCGSW19 fatcat:o4a4uthdkzemjd7qcjn4ppisc4

Secure Arithmetic Computation with Constant Computational Overhead [chapter]

Benny Applebaum, Ivan Damgård, Yuval Ishai, Michael Nielsen, Lior Zichron
2017 Lecture Notes in Computer Science  
We study the complexity of securely evaluating an arithmetic circuit over a finite field F in the setting of secure two-party computation with semi-honest adversaries.  ...  First, we present a general way to combine any linear code that has a fast encoder and a cryptographic ("LPNstyle") pseudorandomness property with another linear code that supports fast encoding and erasuredecoding  ...  It is natural to assume that, for every m = poly(k), a random m × k matrix is pseudorandom over any finite field.  ... 
doi:10.1007/978-3-319-63688-7_8 fatcat:7cecwsbkwfhbxdvyre3ljh45ou

Secret Sharing with Binary Shares [article]

Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, Huaxiong Wang
2018 arXiv   pre-print
any t shares does not reveal any information about the secret and, (ii) any choice of t+1 shares fully reveals the secret.  ...  For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any τ fraction of observed shares, and reconstruction from any ρ fraction of shares, for any choices  ...  Then, Shamir's scheme treats the secret as an element of the finite field F q , where q = 2 ℓ , padded with t uniformly random and independent elements from the same field.  ... 
arXiv:1808.02974v3 fatcat:3ii4upekbjdsbmksrdna6qyfvm

On Constructing Homomorphic Encryption Schemes from Coding Theory [chapter]

Frederik Armknecht, Daniel Augot, Ludovic Perret, Ahmad-Reza Sadeghi
2011 Lecture Notes in Computer Science  
This makes code-based schemes particularly interesting as for some codes decryption is simply a linear operation over the underlying field.  ...  First, they are not restricted to linear homomorphism but allow for evaluating multivariate polynomials up to a fixed (but arbitrary) degree µ on encrypted field elements.  ...  As opposed to other constructions, our scheme works over finite fields.  ... 
doi:10.1007/978-3-642-25516-8_3 fatcat:y7z3mhyfebbctfbtqetb7pp6oi

Compact VSS and Efficient Homomorphic UC Commitments [chapter]

Ivan Damgård, Bernardo David, Irene Giacomelli, Jesper Buus Nielsen
2014 Lecture Notes in Computer Science  
Our commitment scheme extends to vectors over any finite field and is additively homomorphic.  ...  We present a new compact verifiable secret sharing scheme, based on this we present the first construction of a homomorphic UC commitment scheme that requires only cheap symmetric cryptography, except  ...  We thank Yuval Ishai for pointing out interesting applications of our results and Ignacio Cascudo for clarifying key facts about algebraic geometric secret sharing schemes.  ... 
doi:10.1007/978-3-662-45608-8_12 fatcat:opv6vmssmzaaxgcb4cncsctpka

Minimising Communication in Honest-Majority MPC by Batchwise Multiplication Verification [chapter]

Peter Sebastian Nordholt, Meilof Veeningen
2018 Lecture Notes in Computer Science  
In this paper, we present two new and very communicationefficient protocols for maliciously secure multi-party computation over fields in the honest-majority setting with abort.  ...  Using the so far overlooked tool of batchwise multiplication verification, we speed up their technique for checking correctness of multiplications (with some other improvements), reducing communication  ...  The protocol for computing an arithmetic circuit over a finite field from [LN17] with the batchwise multiplication check from Fig. 1 computes any n-party functionality f with computational security in  ... 
doi:10.1007/978-3-319-93387-0_17 fatcat:ne6gnuy7unfcln3qr2b37kzn7a

Rate-1, Linear Time and Additively Homomorphic UC Commitments [chapter]

Ignacio Cascudo, Ivan Damgård, Bernardo David, Nico Döttling, Jesper Buus Nielsen
2016 Lecture Notes in Computer Science  
We construct the first UC commitment scheme for binary strings with the optimal properties of rate approaching 1 and linear time complexity (in the amortised sense, using a small number of seed OTs).  ...  the first almost universal hash function with small seed that can be computed in linear time, and we introduce a new primitive called interactive proximity testing that can be used to verify whether a  ...  Fix a finite field F of constant size.  ... 
doi:10.1007/978-3-662-53015-3_7 fatcat:sphygde77rgwbltvvuxofav3bm

Aspects of Nonabelian Group Based Cryptography: A Survey and Open Problems [article]

Benjamin Fine, Maggie Habeeb, Delaram Kahrobaei, Gerhard Rosenberger
2011 arXiv   pre-print
Let G be the platform group given by a finite prsentation and with the assumptions on normal forms as described above. Alice and Bob want to communicate a shared secret.  ...  Specifically if G is a finite group, such as the cyclic multiplicative group of Z p where p is a prime, and h = g k for some k then the discrete log of h to the base g is any integer t with h = g t .  ... 
arXiv:1103.4093v2 fatcat:7yqcyw2yv5dd3ghgai54bxuzx4

Short Stickelberger Class Relations and Application to Ideal-SVP [chapter]

Ronald Cramer, Léo Ducas, Benjamin Wesolowski
2017 Lecture Notes in Computer Science  
The worst-case hardness of finding short vectors in ideals of cyclotomic number fields (Ideal-SVP) is a central matter in lattice based cryptography.  ...  Combined with the previous results, this solves Ideal-SVP in the worst case in quantum polynomial time for an approximation factor of exp(Õ( √ n)).  ...  If c can be made as small as 1/2, then the asymptotic tradeoffs for Ideal-SVP are as good as the tradeoffs for Principal-Ideal-SVP.  ... 
doi:10.1007/978-3-319-56620-7_12 fatcat:m2b6zy6lmvfehipu4johcentom

Towards Sound Fresh Re-keying with Hard (Physical) Learning Problems [chapter]

Stefan Dziembowski, Sebastian Faust, Gottfried Herold, Anthony Journault, Daniel Masny, François-Xavier Standaert
2016 Lecture Notes in Computer Science  
In the case of asymmetric algorithms, this is usually obtained by secret sharing (aka masking) the key, which is made easy by their algebraic properties.  ...  Most leakage-resilient cryptographic constructions aim at limiting the information adversaries can obtain about secret keys.  ...  A similar technique to our reduction from LPL to LPN was used in [11] , who also analyze physical noise used as a countermeasure to leakage in the context of finite field multiplication and attack this  ... 
doi:10.1007/978-3-662-53008-5_10 fatcat:xy63yuhrf5ajpexha2bdjhlkru

Correlated Pseudorandom Functions from Variable-Density LPN

Elette Boyle, Geoffroy Couteau, Niv Gilboa, Yuval Ishai, Lisa Kohl, Peter Scholl
2020 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)  
Correlated secret randomness is a useful resource for many cryptographic applications.  ...  Parity with Noise assumption (VDLPN).  ...  In Fig. 6 , we give a simple construction of a PCF for VOLE, from any function secret sharing scheme for scalar multiples of a WPRF family.  ... 
doi:10.1109/focs46700.2020.00103 fatcat:eqi522uulnbrtfgl6j2gvsvkra
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