Rédei Functions: Conics and Cryptography

• Rédei functions are rational functions defined by decomposing (z+√d)ⁿ into polynomial parts, encapsulating group laws on Pell conics and their generalizations.
• They are characterized by explicit power formulas, recurrence relations, and matrix representations that unify quadratic algebra with conic geometry.
• Rédei functions underpin practical applications in cryptography and rational approximation, serving as the computational core in RSA-like schemes and polynomial Pell equations.

Searching arXiv for recent and foundational papers on Rédei functions and closely related finite-field, conic, Pell, and classification contexts. Rédei functions are rational functions arising from powers in quadratic extensions and from group laws on conics, with classical roots in Pell-type arithmetic and modern manifestations in finite-field permutation theory, cryptography, and exceptional rational function classification. In the classical setting, they are defined from the expansion

$(z+\sqrt d)^n = N_n(d,z)+D_n(d,z)\sqrt d,$

with the associated rational function

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$

while a broader two-parameter generalization replaces the quadratic relation $x^2=d$ by $x^2-hx-d=0$ and yields

$Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$

These functions encode repeated composition under explicit algebraic group laws, especially on Pell-type conics, and over finite fields they also appear as permutation rational functions on $\mathbb P^1(\mathbb F_q)$, often called Rédei permutations (Barbero et al., 2011, Barbero et al., 2012).

1. Classical definition and algebraic origin

Classically, Rédei functions are built from the decomposition of $(z+\sqrt d)^n$ into its $\sqrt d$-free and $\sqrt d$-linear parts: $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$ where

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$0

The associated Rédei rational function is

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$1

This construction is the basic one in Pell theory and is also the specialization $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$2 of the generalized family studied later on conics (Barbero et al., 2011, Barbero et al., 2012).

A matrix realization makes the structure transparent. In the classical case,

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$3

which immediately yields the addition formulas

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$4

In the generalized setting one replaces this matrix by

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$5

and defines

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$6

When $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$7, this recovers the usual Rédei rational functions exactly (Barbero et al., 2012).

The algebraic source is the quadratic algebra

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$8

with multiplication induced by $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$9. In this framework, powers of $x^2=d$0 generate the polynomial sequences $x^2=d$1 and $x^2=d$2, and the quotient $x^2=d$3 encodes the corresponding power in rational form. This mechanism underlies both the classical Pell interpretation and the generalized conic theory (Barbero et al., 2012).

Both the classical and generalized polynomial sequences satisfy second-order linear recurrences. In the classical case,

$x^2=d$4

equivalently

$x^2=d$5

$x^2=d$6

In the generalized case both satisfy

$x^2=d$7

with initial data

$x^2=d$8

These recurrence structures are central in both theoretical manipulations and efficient computation (Barbero et al., 2011, Barbero et al., 2012).

2. Conics, Pell geometry, and group laws

The most direct structural interpretation of Rédei functions is via conic group laws. In the generalized setting, the relevant conic is

$x^2=d$9

Under the identification $x^2-hx-d=0$0 in the quadratic algebra $x^2-hx-d=0$1, the norm

$x^2-hx-d=0$2

shows that the unit norm elements correspond exactly to the points of this conic (Barbero et al., 2012).

The induced product on the conic is

$x^2-hx-d=0$3

The paper states that $x^2-hx-d=0$4 is an abelian group with identity $x^2-hx-d=0$5 and inverse

$x^2-hx-d=0$6

When $x^2-hx-d=0$7, this becomes the Pell hyperbola

$x^2-hx-d=0$8

with group law

$x^2-hx-d=0$9

which is exactly the multiplication law inherited from $Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$0 (Barbero et al., 2012).

A rational parametrization transports this conic group to a one-dimensional parameter set. With

$Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$1

the parametrization map is

$Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$2

and the inverse is

$Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$3

for $Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$4, together with

$Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$5

Transporting the conic law to $Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$6 gives

$Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$7

with $Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$8 when $Q_n(h,d,z)=\frac{N_n(h,d,z)}{D_n(h,d,z)}.$9. The identity is $\mathbb P^1(\mathbb F_q)$0. The generalized Rédei functions are precisely the powers in this parameter group (Barbero et al., 2012).

In the Pell case this reduction is especially explicit. On the extended line $\mathbb P^1(\mathbb F_q)$1, the transported product is

$\mathbb P^1(\mathbb F_q)$2

with identity $\mathbb P^1(\mathbb F_q)$3, inverse $\mathbb P^1(\mathbb F_q)$4, and an isomorphism from $\mathbb P^1(\mathbb F_q)$5 given by

$\mathbb P^1(\mathbb F_q)$6

Within this interpretation, Rédei functions are exactly the iterated powers: $\mathbb P^1(\mathbb F_q)$7 This turns the addition law

$\mathbb P^1(\mathbb F_q)$8

into the ordinary power law of the group $\mathbb P^1(\mathbb F_q)$9 (Barbero et al., 2011).

A determinant identity ties these constructions back to the conic. Since

$(z+\sqrt d)^n$0

one gets

$(z+\sqrt d)^n$1

Hence, when

$(z+\sqrt d)^n$2

the pair $(z+\sqrt d)^n$3 lies on the conic $(z+\sqrt d)^n$4. This exhibits the polynomial pair itself as a generator of conic powers and thereby links recurrence, conic geometry, and rational parametrization in a single formalism (Barbero et al., 2012).

3. Composition, powers, and approximation theory

A fundamental property of classical Rédei functions is the multiplicative index law

$(z+\sqrt d)^n$5

which implies that Rédei functions commute under composition. In the group-law interpretation this is immediate: once $(z+\sqrt d)^n$6 is viewed as the $(z+\sqrt d)^n$7-th power of $(z+\sqrt d)^n$8 in the $(z+\sqrt d)^n$9-group, composition corresponds to multiplication of exponents (Barbero et al., 2011).

The generalized functions satisfy the analogous role on the parameter group of the conic. The paper on generalized Rédei functions emphasizes that they “play exactly the same structural role as in the Pell hyperbola”: they encode repeated composition under the group law and provide explicit formulas for powers of points over the conic (Barbero et al., 2012).

The same formalism also yields rational approximations to quadratic irrationalities. In the Pell setting, the paper explicitly emphasizes that $\sqrt d$0 converges to $\sqrt d$1, describing Rédei functions as “rational approximations of $\sqrt d$2, for any parameter $\sqrt d$3” (Barbero et al., 2011). The generalized conic paper extends this approximation viewpoint and states that it obtains “a new result for the approximation of quadratic irrationalities” (Barbero et al., 2012).

A polynomial analogue appears in the study of the polynomial Pell equation

$\sqrt d$4

Using Rédei polynomials

$\sqrt d$5

the identity

$\sqrt d$6

produces explicit polynomial solutions once one chooses $\sqrt d$7 and $\sqrt d$8, so that $\sqrt d$9. This yields all integer polynomial solutions for

$\sqrt d$0

in the cases $\sqrt d$1 described in the paper (Murru, 2019).

This suggests a unifying interpretation: Rédei functions and polynomials are compression devices for powers in quadratic algebras. In arithmetic settings, the compression is rational and suited to Pell-type groups; in polynomial settings, the same binomial decomposition yields complete families of Pell-type solutions (Barbero et al., 2011, Murru, 2019).

4. Finite-field Rédei functions and Rédei permutations

Over finite fields, Rédei functions become rational self-maps of the projective line

$\sqrt d$2

For $\sqrt d$3, writing

$\sqrt d$4

one defines

$\sqrt d$5

A Rédei function that induces a bijection on $\sqrt d$6 is called a Rédei permutation (Capaverde et al., 2020, Capaverde et al., 2021).

A standard criterion is

$\sqrt d$7

where $\sqrt d$8 is the quadratic character. Since $\sqrt d$9 is even for odd $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$0, no even $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$1 yields a Rédei permutation (Capaverde et al., 2020, Capaverde et al., 2021).

The cycle structure is controlled entirely by arithmetic modulo divisors of $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$2. If $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$3 is a permutation, then it consists of

$(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$4

disjoint $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$5-cycles for each divisor $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$6, together with $(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$7 fixed points. The total number of fixed points is

$(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$8

Moreover,

$(z + \sqrt{d})^n = N_n(d, z) + D_n(d, z)\sqrt{d},$9

These formulas place the dynamics of Rédei permutations squarely in the arithmetic of multiplicative orders (Capaverde et al., 2020).

A complete characterization of when two Rédei permutations have the same cycle structure is given in terms of the condition

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$00

assuming the parameters have the same quadratic character. The classification is then refined prime by prime using multiplicative orders and $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$01-adic valuation constraints (Capaverde et al., 2021).

Special attention has been given to Rédei permutations whose nontrivial cycles all have the same length. For cycle decompositions consisting only of $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$02-cycles and $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$03-cycles, with $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$04 prime, the paper gives an exact admissibility criterion for divisors $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$05 and an existence theorem: such a Rédei permutation exists if and only if $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$06 or $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$07 has a prime factor of the form $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$08 or is divisible by $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$09 (Capaverde et al., 2020).

A different finite-field presentation, especially useful structurally, expresses Rédei functions by Möbius conjugation. For odd $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$10, if $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$11 is irreducible over $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$12 and $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$13 is a root, define

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$14

Then $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$15 maps $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$16 bijectively onto $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$17, the group of $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$18-st roots of unity in $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$19. As a result, the functional graph of $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$20 on $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$21 is isomorphic to that of $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$22 on $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$23, and

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$24

in that nonsquare setting (Ding et al., 2021).

This Möbius-conjugation viewpoint also yields an addition law: $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$25 For odd $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$26, where $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$27, this simplifies to

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$28

This is the finite-field counterpart of the Pell-style addition law seen over $Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$29 (Ding et al., 2021).

5. Cryptographic and algorithmic roles

Rédei functions have been used as the computational core of RSA-like schemes based on conics, especially the Pell hyperbola. The central idea is to replace ordinary exponentiation by powering in a non-standard parameter group

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$30

or, in the generalized conic setting,

$Q_n(d,z)=\frac{N_n(d,z)}{D_n(d,z)},$31

Since [ z^{\