Converting secret shares between different finite fields (GF(p) and GF(q)) in linear secret sharing schemes (LSSS) is possible, but the feasibility and method depend on the relationship between the fields and whether interaction is allowed. Here's a structured breakdown:

1. Field Relationship and Local Conversion

If GF(q) contains GF(p) as a subfield (i.e., ( q = p^k ) for some intege...

Short answer

• If the two moduli come from the same prime and one field is a subfield of the other (GF(p^m) ⊆ GF(p^k) with m | k), then conversion is trivial and non‑interactive: just re‑interpret the shares in the larger field.

• In all other cases (different primes, no subfield relation, or rings Zp vs Zq), there is no purely local, linear, black‑box way to convert an LSSS sharing over Zp (or ...

Let
– 𝔽ₚ = GF(p) be the field in which the value x is already shared,
– 𝔽_q = GF(q) be the field in which we would like to obtain a sharing of the same integer x (0 ≤ x < p, q) and
– [x]ₚ (resp. [x]₍q₎) denote a sharing of x over 𝔽ₚ (resp. 𝔽_q) with the same access structure (threshold t, Shamir, replicated, …).

There are three essentially different situations.

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Excellent question. This is a fundamental problem in multi-party computation (MPC) when protocols built for arithmetic in one field need to interact with protocols designed for another.

The short answer is yes, it is possible to convert shares of a secret from a field to shares of that same secret in a field . The method and its complexity depend on the relationsh...

The answer depends on the relation between the two fields that you want to work over.
Below we distinguish three practically relevant cases and give a protocol (with perfect information-theoretic security) whenever the task is possible.

Notation
• Fp = ℤp : field of characteristic p
• Fq = ℤq or 𝔽q : field (prime or prime power) in which we want the new sharing
• n : num...

<think> We are dealing with linear secret sharing schemes (LSSS) over finite fields. The question is whether we can convert a secret shared value in one finite field (say, Z_p) to another (Z_q) without compromising security.

Key points from the question:

• The LSSS works over Z_p (p prime or prime power).
• We want to convert the secret and its shares to Z_q (q could be less than or greater than ...

This is an interesting question about converting secret shares between different finite fields. Let me provide a comprehensive answer covering the possibilities and limitations.

The Core Challenge

Converting secret shares between different finite fields and (where and are different primes or prime powers) is a non-trivial problem because:

1. **Non-homomo...

Share Conversion Between Different Finite Fields

Converting shares between arbitrary finite fields and while preserving the properties of a linear secret sharing scheme is indeed challenging, particularly when and are different primes.

The Fundamental Challenge

The main issue is that finite fields of different characteristics have fundamentally differe...

One can actually do no extra interaction at all – as long as one is willing to view the original shares as little‐endian integers and then simply reduce (or embed) into the new field. In fact the argument is almost trivial once you recall that any LSSS is just a linear map from the vector of “secret plus randomness” into each party’s share.

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