Generation of pseudorandom sequence over elliptic curve group and their properties.

*(English)*Zbl 1154.11045The authors propose a pseudorandom sequence generator over elliptic curves in a quite direct way: Pick an element of higher order in an elliptic curve, choose a prime number greater than that order, and, using a linear feedback shift register (LFSR), generate a sequence of random integers in the primitive field of that prime characteristic; then the pseudorandom sequence consists of the multiples of the higher order element by those random integers.

For instance for \(m=8\), \(\mathbb F_{2^8}\) can be realized as \(\mathbb F_{2}[X]/(X^8+X^4+X^3+X^2+1)\) and for a primitive element \(g\in\mathbb F_{2^8}\) consider the elliptic curve \(E\): \([Y^2+XY =X^3+g^4X^2+1]\). The order of \(E\) is \(288 = 2^5 3^2 = 3\cdot 96\). The element \(x=(g^{11},g^{25})\in E\) has order \(n=96 = 2^5 3\). Let \(p=97=n+1\). The polynomial \(C(X)=X^4+X+23\) is irreducible over \(\mathbb F_p\) and it is taken as a “connection polynomial”. A LFSR \((k_j)_j\) is built through the recurrence \(k_j+k_{j-3}+23=0\,\text{mod}\,p\), hence \(k_j=(96\,k_{j-3}+74)\,\text{mod}\,p\), and its period is \(p^4-1 = 88\,529\,280\). Then the random sequence in the elliptic curve \(E\) is \(((x_j,y_j)=k_j(g^{11},g^{25}))_j\).

The authors show that such pseudorandom sequences succeed to pass the FIPS criteria for randomness. They also apply such sequences in well known image encryption procedures [S. Lin, An introduction to error-correcting codes, Prentice-Hall (1970)]. English could be improved (for instance there is a continuous lack of articles), hence reading is difficult in some parts of the paper.

For instance for \(m=8\), \(\mathbb F_{2^8}\) can be realized as \(\mathbb F_{2}[X]/(X^8+X^4+X^3+X^2+1)\) and for a primitive element \(g\in\mathbb F_{2^8}\) consider the elliptic curve \(E\): \([Y^2+XY =X^3+g^4X^2+1]\). The order of \(E\) is \(288 = 2^5 3^2 = 3\cdot 96\). The element \(x=(g^{11},g^{25})\in E\) has order \(n=96 = 2^5 3\). Let \(p=97=n+1\). The polynomial \(C(X)=X^4+X+23\) is irreducible over \(\mathbb F_p\) and it is taken as a “connection polynomial”. A LFSR \((k_j)_j\) is built through the recurrence \(k_j+k_{j-3}+23=0\,\text{mod}\,p\), hence \(k_j=(96\,k_{j-3}+74)\,\text{mod}\,p\), and its period is \(p^4-1 = 88\,529\,280\). Then the random sequence in the elliptic curve \(E\) is \(((x_j,y_j)=k_j(g^{11},g^{25}))_j\).

The authors show that such pseudorandom sequences succeed to pass the FIPS criteria for randomness. They also apply such sequences in well known image encryption procedures [S. Lin, An introduction to error-correcting codes, Prentice-Hall (1970)]. English could be improved (for instance there is a continuous lack of articles), hence reading is difficult in some parts of the paper.

Reviewer: Guillermo Morales-Luna (Mexico)

##### MSC:

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |

11G05 | Elliptic curves over global fields |

94A55 | Shift register sequences and sequences over finite alphabets in information and communication theory |

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\textit{S. V. Sathyanarayana} et al., J. Discrete Math. Sci. Cryptography 10, No. 6, 731--747 (2007; Zbl 1154.11045)

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