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7

Prime theory is of great interest to me! It is currently used in many cryptosystems to protect data (in making public keys, for example). There are always a few obscure researchers studying how to make prime factorization easier (or stronger I suppose). There are so many branches of math that we use in cryptography (matrices, primes, ellipses, modular ...


4

Shamir's (m,n) secret sharing scheme has a secret $s_0$ which is represented as an element of a finite field $\mathbb F_q$ of $q$ elements. There are also $m-1$ other "randomly chosen" elements $s_1, s_2, \ldots, s_{m-1}$ that the designer uses. The scheme creates a polynomial $$S(x) = s_0 + s_1x + \cdots + s_{m-1}x^{m-1}$$ and evaluates $S(x)$ at $n$ ...


4

Generating prime integers and computing things modulo integers is easy with a programming language which features support for big integers. I usually use Java: it includes java.math.BigInteger, with which one can: generate prime integers of any length; do all basic operations, including modular reduction (mod() method); compute modular inverses ...


4

I would like to add my two cents (mostly related to asymmetric cryptography): Number theoretic primitives (RSA/DH/EC/Pairing based crypto) [Mainstream] Coding theory based crypto systems (McEliece) [research] Lattice based systems [research] Other models of information theory, e.g. wiretap model [research] Combinatorics (knapsack problems) [research]


4

"Algebraic Geometry Codes: Basic Notions" by Tsfasman, Vladut, and Nogin is a textbook that is available as a PDF. Discussion of Hermitian curves begins on page 167. I haven't read it and no very little about coding theory. It was the reference a friend provided in his dissertation, which included constructing universal hash functions from Hermitian curves. ...


4

Abstract mathematics has played an important role in the development of cryptography. From Analytical number theory, tools like factorization and computing logarithms in a finite field. Enough is said and known about these techniques! Combinatorial problems, like knapsack and subset-sum has been used in cryptosystem. You can find a very nice connection ...


3

Not sure if hash trees miss some of your requirements, but many of requirements you have could be satisfied with hash trees. Note: The scheme described below is essentially "Merkle Hash Tree-based Storage Enforcing Scheme (MHT-SE)[Golle et al. 2002]". So my question is, if we relax the requirement of being able to perform an unbounded number of ...


3

I do not know of any approaches in context of proofs or retrievability (PoRs)/provable data possession (PDP) that use homomorphic encryption. However, many of those schemes employ homomorphic (linear) authenticators/tags for the metadata such that the proofs delivered by the server can be of constant size, i.e., by aggregating single tags. Now to some ...


3

Yes, we need symmetric cryptosystems, for many reasons; to give three of these: We need a hash function to make most asymmetric cryptosystems secure (e.g. we simply do not have a secure signature system based on RSA without a hash), and current hash functions are (or are built from) symmetric cryptosystems. All asymmetric encryption cryptosystems are bound ...


3

Surprised nobody has mentioned this. Abstract algebra is a big player in the design of AES, specifically AES uses finite field arithmetic over a specific field. This article introduces the field in question. The field in question is also called a Galois Field, from Galois theory which neatly solves questions about higher order polynomials as well as linking ...


2

I am not sure on the implementation status of hyperelliptic curves. Two other significant uses of mathematical techniques: Bilinear pairings on appropriate elliptic curves - mainstream (Voltage) Ideal lattices - mainstream (NTRU) and in research (Gentry's fully homomorphic encryption) Another area of interest is based on coding theory: Learning ...


2

If you use a high-level mathematical language (Mathematica, Maple, etc.), generating this data is very easy. I use Mathematica personally, but some are free and apparently very good. In a pinch, you can use Wolfram Alpha to do a lot: Generate random prime: RandomPrime[{2^1023,2^1024}] Random integer: RandomInteger[{2^1023,2^1024}] Inverse mod p: ...


2

In general, an $(n,k,k')$ erasure code must have the property that any $k'$ out of $n$ symbols must be sufficient to recover the original $k$-symbol message. An $(n,k')$ threshold secret sharing scheme, however, requires an additional property: knowing less than $k'$ shares out of $n$ must not be sufficient to recover any information about ...


1

Scroll to the end for tl;dr. Regarding your contrived example: Alice doesn't have Bob's keypair, but sends a message in such a way that only Bob can read it, eg. puts it in a dead-drop. So she takes out her pen and writes Hey Bob, could you sign and send me $X$ along with your public key? Here's mine: $P_A$ Signed, Alice Bob has no way of knowing if ...


1

First of all, the choice of word "Erasure code" is not appropriate. RS code can be used as an erasure code, but when you are talking about erasure code, you implicitly work in the area where some bits of the data has been deleted (note the difference from ECC where you know that the data has been corrupted). An erasure code is more efficient in terms of ...



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