## Generation of pseudorandom sequence over elliptic curve group and their properties.(English)Zbl 1154.11045

The 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.

### 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

### Keywords:

pseudorandom generators; elliptic curves
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