A Number-Theoretic Error-Correcting Code

Abstract

In this paper we describe a new error-correcting code (ECC) inspired by the Naccache-Stern cryptosystem. While by far less efficient than Turbo codes, the proposed ECC happens to be more efficient than some established ECCs for certain sets of parameters.

The new ECC adds an appendix to the message. The appendix is the modular product of small primes representing the message bits. The receiver recomputes the product and detects transmission errors using modular division and lattice reduction.

A Toy Example

Let m be the 10-bit message 1100100111. For $t=2$, we let p be the smallest prime number greater than $2 \cdot 29^4$, i.e. $p=707293$. We generate the redundancy:

$$c(m)=2^1 \cdot 3^1\cdot 5^0 \cdot 7^0 \cdot 11^1 \cdot 13^0 \cdot 17^0 \cdot 19^1 \cdot 23^1 \cdot 29^1\,\,\mathrm{{mod}}\,\,707293 $$

$$\Rightarrow c(m)= 836418\,\,\mathrm{{mod}}\,\,707293 = 129125.$$

As we focus on the new error-correcting code we simply omit the Reed-Muller component. The encoded message is

$$\nu (m)=\mathtt{1100100111_2} \Vert \mathtt{129125_{10}}.$$

Let the received encoded message be $\alpha =\mathtt{1100101011_2} \Vert \mathtt{129125_{10}}$. Thus,

$$c(m')=2^1 \cdot 3^1 \cdot 5^0 \cdot 7^0 \cdot 11^1 \cdot 13^0 \cdot 17^1 \cdot 19^0 \cdot 23^1 \cdot 29^1\,\,\mathrm{{mod}}\,\,p$$

$$\Rightarrow c(m')= 748374\,\,\mathrm{{mod}}\,\,707293 = 41081.$$

Dividing by c(m) we get

$$s = \frac{c(m')}{c(m)} = \frac{41081}{129125}\,\,\mathrm{{mod}}\,\,707293 = 632842$$

Applying the rationalize and factor technique we obtain $s = \displaystyle \frac{17}{19}\,\,\mathrm{{mod}}\,\,707293$. It follows that $m' \oplus m = \mathtt{0000001100}$. Flipping the bits retrieved by this calculation, we recover m.