refactor: fix minor typos and errors (#34)

Author: AngryMaciek

CHANGELOG.md

@@ -7,6 +7,11 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 
 ## [Unreleased]
 
-- Initial Release
+## [2.0.0] - 2023-02-16
 
-[unreleased]: https://github.com/AngryMaciek/hypercomplex
+### Added
+
+- initial release of the project
+
+[unreleased]: https://github.com/AngryMaciek/hypercomplex/compare/v2.0.0...HEAD
+[2.0.0]: https://github.com/AngryMaciek/hypercomplex/releases/tag/v2.0.0

README.md

@@ -11,7 +11,9 @@
 
 <img src="img/logo.png" alt="drawing" height="30"/>
 
-Header-only *C++* template library for precise operations on hypercomplex numbers from the [Cayley-Dickson algebras](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction).
+Header-only *C++*
+template library for lattice-based cryptosystems in
+high-dimensional algebras.
 
 Full documentation & API are available at [this link](https://angrymaciek.github.io/hypercomplex).  
 Please refer to the [contributing guidelines](CONTRIBUTING.md) if you are interested in contributing to this repository.  

docs/mainpage.dox

@@ -313,7 +313,7 @@
  *
  * \section crypto_sec Cryptographic Application
  *
- * Main feature of _Hypercomplex_ is the implementation of
+ * The main feature of _Hypercomplex_ is an implementation of
  * cryptographic operations as in <a href="https://en.wikipedia.org/wiki/NTRUEncrypt">NTRU cryptosystem</a>
  * but generalized on an arbitrary-high-dimensional algebras generated with the Cayley-Dickson construction.
  * The library provides additional helper class for truncated polynomials
@@ -326,8 +326,8 @@
  * Let \f$R = \mathbb{Z}[x] / (x^N - 1)\f$ be the underlying polynomial quotient ring
  * (\f$R_p, R_q\f$ are modular structures, modulo \f$p,q\f$ respectively)
  * and \f$D\f$ mark the dimension of the algebra we operate in:
- * \f$A = \{\sum^D_{i=1} x_i \cdot e_i | x_i \in R\} \f$, where
- * all \f$e_i\f$ are basis elements (similarly for modular algebras: \f$A_p, A_q\f$).
+ * \f$A = \{x_0 + \sum^{D-1}_{i=1} x_i \cdot e_i | x_i \in R\} \f$, where
+ * all \f$e_i\f$ are imaginary basis elements (similarly for modular algebras: \f$A_p, A_q\f$).
  *
  * Mathematical derivations for these structures and their operations
  * are analogous to those presented for
@@ -391,14 +391,17 @@
  *   Hypercomplex<Polynomial<MaxDeg>, dim> C = DECRYPT(F, E, p, q);
  * \endcode
  *
- * Remarkably, for a cryptosystem based on algebras with 16+ dimensions
+ * Remarkably, for a cryptosystem based on an algebra of \f$D \geq 16\f$ dimensions
  * \f$F\f$ needs to contain at most one \f$ x_i \in R | x_i \neq 0\f$.
  * This is because sedonions (and higher)
  * are not associative, thus the decryption process will only be possible
- * for such specific, reduced subset of private keys.
+ * for a specific, reduced subset of private keys.
  *
  * Cryptographic applications of _Hypercomplex_ have been extensively tested in the
  * test case: _Cryptosystem based on Cayley-Dickson Algebras_ of the
  * <a href="https://github.com/AngryMaciek/hypercomplex/blob/master/.test/unit/test.cpp">following file</a>.
  *
+ * All tests underlying Figure 1 in the publication are available
+ * <a href="https://github.com/AngryMaciek/hypercomplex/blob/master/.test/docs/test.cpp">here</a>.
+ *
  */

paper.md

@@ -110,8 +110,12 @@ The following decryption consist of three steps:
 
 If the decryption was successfull Bob receives $D_3 = M$ (up to coefficients' centered lift in $\mathcal{A_p^\lambda}$).
 Please remember that lattice-based cryptography is always burdened with a chance of decryption failure due to incorrect recovery of polynomial's coefficients.
+Also, for $\lambda \geq 4$ note that $\mathcal{A^\lambda}$ is not alternative
+nor associative thus successful decryption relies on a careful initial choice of $F$
+(e.g. $F: \exists! i\in\{0, \ldots ,2^\lambda-1\}: F_i \neq 0$).
 For a more detailed deriviation of similar
-cryptosystems please see QTRU[@QTRU] and OTRU[@OTRU] paper.
+cryptosystems please see publications
+presenting QTRU[@QTRU] and OTRU[@OTRU].
 
 Three examples of matrix encryption-decryption are presented in the Figure 1.
 All of the data and code required to reproduce these results is available in the code repository.
@@ -121,13 +125,13 @@ All of the data and code required to reproduce these results is available in the
   **(a)** Message (M) composed of seven, relatively-big
   secret numbers is encoded in binary in a [64x7] matrix.
   This is later encrypted (E) and decrypted (D) with a public-key
-  cryptosystem, allowing to transfer the numbers in a secure manner.
+  cryptosystem, allowing to transfer the numbers in a secure manner;
+  $p=2, q=1151$.
   **(b)** Graphical representation of an encrypted/decrypted QR code,
-  encoded in a [32x29] matrix (padded image).
-  **(c)** Encrypted/Decrypted [128x127] meme; credits: [www.nyan.cat](www.nyan.cat).
+  encoded in a [32x29] matrix (padded image); $p=3, q=1723$.
+  **(c)** Encrypted/Decrypted [128x127] meme; $p=17, q=16777213$; credits: [www.nyan.cat](www.nyan.cat).
 ](img/Fig1.png)
 
-