Variants of product replacement

Leedham‐Green, C. R.; Murray, Scott H.

doi:10.1090/conm/298/05116

Cited by 6 publications

(5 citation statements)

References 7 publications

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“…The variant Rattle of PRA that we discuss here differs slightly from the original version in [11]; in that, the palette of operations has been increased to make the process symmetric.…”

Section: Rattlementioning

confidence: 99%

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On a variant of the product replacement algorithm

Leedham-Green

2024

Glasgow Math. J.

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We discuss a variant, named ‘Rattle’, of the product replacement algorithm. Rattle is a Markov chain, that returns a random element of a black box group. The limiting distribution of the element returned is the uniform distribution. We prove that, if the generating sequence is long enough, the probability distribution of the element returned converges unexpectedly quickly to the uniform distribution.

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“…The variant Rattle of PRA that we discuss here differs slightly from the original version in [11]; in that, the palette of operations has been increased to make the process symmetric.…”

Section: Rattlementioning

confidence: 99%

“…The variant of PRA that we wish to discuss here is called ‘Rattle’, see [11]. It too is a Markov chain, where a state consists of a team , as in PRA, and an additional group element , the accumulator .…”

Section: Introductionmentioning

confidence: 99%

On a variant of the product replacement algorithm

Leedham-Green

2024

Glasgow Math. J.

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“…To conclude this section, we compare Dixon's technique with two heuristics used in practice: the product replacement algorithm [31] and Prospector [32]. In the product replacement algorithm, the state is an array a of length n elements that generate the group.…”

Section: Protocol 2 Random(g X)mentioning

confidence: 99%

Secure Groups for Threshold Cryptography and Number-Theoretic Multiparty Computation

Schoenmakers,

Segers

2023

Cryptography

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In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed 3max(a,b) in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation.

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“…In practice, by using Product Replacement, Rattle, and recent work by Dixon [5,8,17,18] for groups and normal closure, and by taking random linear combinations of random products of generators for algebras, each of these costs is O (d 3 ), at least after an initialisation phase. However, these methods are not proven to produce independent, uniformly-distributed random elements in general.…”

Section: Complexity Summarymentioning

confidence: 99%

A polynomial-time reduction algorithm for groups of semilinear or subfield class

2009

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We present a Las Vegas algorithm for finding a nontrivial reduction of groups that are irreducible with m generators and either lie in the subfield class of matrix or projective groups or are semilinear or have non-absolutely irreducible derived group. Let R A denote the cost of producing a random element from a matrix algebra A and R H G denote the cost of producing a random element in the normal closure of a group H by a group G. Then the algorithm runsfield operations. We also characterise the absolutely irreducible groups G over arbitrary fields whose derived group consists only of scalars, and prove probabilistic generation results about matrix groups.

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Variants of product replacement

Abstract: We present new variations on the product replacement method for generating random elements in a black box group, together with some basic analysis and conjectures.

Cited by 6 publications

References 7 publications

On a variant of the product replacement algorithm

On a variant of the product replacement algorithm

Secure Groups for Threshold Cryptography and Number-Theoretic Multiparty Computation

A polynomial-time reduction algorithm for groups of semilinear or subfield class

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