As distinct from a classical residue number system, irreducible polynomials over *GF*(2) serve as bases in an NPN.

First of all, an NPN is formed for an *N*-bit block of electronic message [12, 13]. For this purpose, we choose its working bases, i.e. irreducible polynomials

$$\begin{array}{}{\displaystyle {p}_{1}(x),{p}_{2}(x),...,{p}_{S}(x)}\end{array}$$(2)

over *GF*(2) of degrees *m*_{1}, *m*_{2}, …, *m*_{S} respectively [2]. Polynomials (1) subject to their arrangement constitute a certain base system. All bases (1) are to be different including the case when they have the same degree. The working range of the NPN is specified by polynomial (modulus)

$$\begin{array}{}{\displaystyle P(x)={p}_{1}(x)\cdot {p}_{2}(x)\cdot \dots \cdot {p}_{S}(x)}\end{array}$$

Table 1 Examples of the avalanche effect

of degree $\begin{array}{}m=\sum _{i=1}^{S}{m}_{i}.\end{array}$ Therefore, a message of *N*-bit length could be interpreted as a sequence of remainders *α*_{1}(*x*), *α*_{2}(*x*), …, *α*_{S}(*x*) of dividing a polynomial *F*(*x*) by working bases *p*_{1}(*x*) ⋅ *p*_{2}(*x*) ⋅ … ⋅ *p*_{S}(*x*):

$$\begin{array}{}{\displaystyle F(x)=({\alpha}_{1}(x),{\alpha}_{2}(x),\dots ,{\alpha}_{S}(x)),}\end{array}$$(3)

where *F*(*x*) ≡ *α*_{i}(*x*)(mod*p*_{i}(*x*)), *i* = 1, *S*.

In expression (2) remainders *α*_{1}(*x*), *α*_{2}(*x*), …, *α*_{S}(*x*) are chosen so that the first *l*_{1} bits of a message associate to binary coefficients of remainder *α*_{1}(*x*), the next *l*_{2} bits associate to binary coefficients of remainder *α*_{2}(*x*), etc., and the last *l*_{S} bits associate to binary coefficients of *α*_{S}(*x*).

To encrypt a message, it is used a secret key of *N* bits, which is also interpreted as a sequence of remainders resulting from dividing some other polynomial *G*(*x*) by the same working bases of the system:

$$\begin{array}{}{\displaystyle G(x)=({\beta}_{1}(x),{\beta}_{2}(x),\dots ,{\beta}_{S}(x)),}\end{array}$$(4)

where *G*(*x*) ≡ *β*_{i}(*x*)(mod *p*_{i}(*x*)), *i* = 1, *S*.

Hence, some function *H*(*F*(*x*), *G*(*x*)) is considered as a cryptogram:

$$\begin{array}{}{\displaystyle H(x)=({\omega}_{1}(x),{\omega}_{2}(x),\dots ,{\omega}_{S}(x)),}\end{array}$$(5)

where *H*(*x*) ≡ *ω*_{i}(*x*)(mod *p*_{i}(*x*)), *i* = 1, *S*.

In NPNs, a cryptogram is the result of multiplying polynomial *F*(*x*) by *G*(*x*). The members of residue sequence *ω*_{1}(*x*), *ω*_{2}(*x*), …, *ω*_{S}(*x*) are the least remainders on dividing products *α*_{i}(*x*) *β*_{i}(*x*) by respective bases *p*_{i}(*x*):

$$\begin{array}{}{\displaystyle {\alpha}_{i}(x)\cdot {\beta}_{i}(x)\equiv {\omega}_{i}(x)({\textstyle \text{mod}}{p}_{i}(x)),i=\overline{1,S}.}\end{array}$$(6)

The binary form of cryptogram *H*(*x*) is as follows. The binary coefficients of residue *ω*_{1}(*x*) associate to first consecutive *l*_{1} bits of *H*(*x*). The binary coefficients of residue *ω*_{2}(*x*) associate to further consecutive *l*_{2} bits of *H*(*x*), etc. The binary coefficients of the last residue *ω*_{S}(*x*) associate the last consecutive *l*_{S} binary bits of the cryptogram.

When decrypting cryptogram *H*(*x*) with a known key *G*(*x*), for each *β*_{i}(*x*) it is calculated, as follows from (5), a reciprocal (inverse) polynomial $\begin{array}{}{\beta}_{i}^{-1}\end{array}$(*x*) under the following condition:

$$\begin{array}{}{\displaystyle {\beta}_{i}(x)\cdot {\beta}_{i}^{-1}(x)\equiv 1({\textstyle \text{mod}}{p}_{i}(x)),i=\overline{1,S}.}\end{array}$$(7)

The result is polynomial

$$\begin{array}{}{\displaystyle {G}^{-1}(x)=({\beta}_{1}^{-1}(x),{\beta}_{2}^{-1}(x),..,{\beta}_{S}^{-1}(x))}\end{array}$$

inverse to polynomial *G*(*x*). The original message then could be calculated according to (5) and (6) through remainders of the following congruence:

$$\begin{array}{}{\displaystyle {\alpha}_{i}(x)\equiv {\beta}_{i}^{-1}(x){\omega}_{i}(x)({\textstyle \text{mod}}{p}_{i}(x)),i=\overline{1,S}.}\end{array}$$(8)

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