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

2009
*
Lecture Notes in Computer Science
*

Third, we present an infinite family of

doi:10.1007/978-3-642-03356-8_28
fatcat:7eb7b3n2vzdcxox6ragick5tpu
*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*). ...##
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Asymptotically Good Multiplicative LSSS over Galois Rings and Applications to MPC over $$\mathbb {Z}/p^k\mathbb {Z} $$
[chapter]

2020
*
Lecture Notes in Computer Science
*

The standard way to obtain these

doi:10.1007/978-3-030-64840-4_6
fatcat:hq6twvyp65g7pkkm72p47j6ohe
*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. ...##
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Algebraic Geometry Codes: General Theory
[chapter]

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

doi:10.1142/9789812794017_0001
fatcat:3upxrzrbyvc3xizp3cod26lndu
*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*...##
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Fast Large-Scale Honest-Majority MPC for Malicious Adversaries
[chapter]

2018
*
Lecture Notes in Computer Science
*

We present protocol variants for small and large

doi:10.1007/978-3-319-96878-0_2
fatcat:55tehzjfi5h3ba5ot5ckt5fj7a
*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. ...##
###
Secret Sharing with Binary Shares

2018
*
Innovations in Theoretical Computer Science
*

t

doi:10.4230/lipics.itcs.2019.53
dblp:conf/innovations/LinCGSW19
fatcat:o4a4uthdkzemjd7qcjn4ppisc4
*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*. ...##
###
Secure Arithmetic Computation with Constant Computational Overhead
[chapter]

2017
*
Lecture Notes in Computer Science
*

We study the complexity of securely evaluating an arithmetic circuit

doi:10.1007/978-3-319-63688-7_8
fatcat:7cecwsbkwfhbxdvyre3ljh45ou
*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*. ...##
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Secret Sharing with Binary Shares
[article]

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*. ...

##
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On Constructing Homomorphic Encryption Schemes from Coding Theory
[chapter]

2011
*
Lecture Notes in Computer Science
*

This makes code-based schemes particularly interesting as for some codes decryption is simply a

doi:10.1007/978-3-642-25516-8_3
fatcat:y7z3mhyfebbctfbtqetb7pp6oi
*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*. ...##
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Compact VSS and Efficient Homomorphic UC Commitments
[chapter]

2014
*
Lecture Notes in Computer Science
*

Our commitment scheme extends to vectors

doi:10.1007/978-3-662-45608-8_12
fatcat:opv6vmssmzaaxgcb4cncsctpka
*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. ...##
###
Minimising Communication in Honest-Majority MPC by Batchwise Multiplication Verification
[chapter]

2018
*
Lecture Notes in Computer Science
*

In this paper, we present two new and very communicationefficient protocols for maliciously secure multi-party computation

doi:10.1007/978-3-319-93387-0_17
fatcat:ne6gnuy7unfcln3qr2b37kzn7a
*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 ...##
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Rate-1, Linear Time and Additively Homomorphic UC Commitments
[chapter]

2016
*
Lecture Notes in Computer Science
*

We construct the first UC commitment scheme for binary strings

doi:10.1007/978-3-662-53015-3_7
fatcat:sphygde77rgwbltvvuxofav3bm
*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. ...##
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Aspects of Nonabelian Group Based Cryptography: A Survey and Open Problems
[article]

2011
*
arXiv
*
pre-print

Let G be the platform group given by a

arXiv:1103.4093v2
fatcat:7yqcyw2yv5dd3ghgai54bxuzx4
*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 . ...##
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Short Stickelberger Class Relations and Application to Ideal-SVP
[chapter]

2017
*
Lecture Notes in Computer Science
*

The worst-case hardness of finding short vectors in

doi:10.1007/978-3-319-56620-7_12
fatcat:m2b6zy6lmvfehipu4johcentom
*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. ...##
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Towards Sound Fresh Re-keying with Hard (Physical) Learning Problems
[chapter]

2016
*
Lecture Notes in Computer Science
*

In the case of asymmetric algorithms, this is usually obtained by

doi:10.1007/978-3-662-53008-5_10
fatcat:xy63yuhrf5ajpexha2bdjhlkru
*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 ...##
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Correlated Pseudorandom Functions from Variable-Density LPN

2020
*
2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
*

Correlated

doi:10.1109/focs46700.2020.00103
fatcat:eqi522uulnbrtfgl6j2gvsvkra
*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. ...
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