In the projective plane
Its properties are well known (see  for an extensive review) and Hesse curves, which constitute the Hessian pencil, have recently been popular among number theorists in relation to applied elliptic curve cryptography [17, 9]. Hessian parametrization of elliptic curves in particular has been shown to improve resistance to side-channel attacks .
Following , the replacement of the field
Tropical polynomials and tropical curves
A tropical polynomial in n variables
Then to every tropical polynomial f we can associate its hypersurface
The tropical lines are defined by the polynomials
Tropical Hesse curves
The direct tropicalization of the Hessian pencil equation yields the tropical polynomial:
The tropical Hesse curve is the set of points where the maximum is reached more than once. Introducing the inhomogeneous coordinates in the projective plane
2 Group law on the tropical Hesse curve
Tropical intersections and tropical elliptic curves
We use basic facts in tropical intersection theory  to adapt the geometrical presentation of tropical elliptic curve group law from  to the case of the tropical Hesse curve. Let
defining a plane tropical curve. The convex hull of points
Then the balancing condition holds: for any vertex V in the tropical curve
and following  a tropical curve is called smooth if all its vertices are 3-valent and have multiplicity 1, and a tropical elliptic curve is a smooth tropical curve of degree 3 and genus 1 (number of cycles). More generally, the intersection of two segments of a plane tropical curves, with respective weight
Inspection of Figure 2 (right) show that the weight of the Hesse curve’s tentacles are 3, and the weight of the bounded segments are 1 and the balancing condition checks for all three vertices. The degree is 3 and its genus is 1. All vertices are 3-valent, their multiplicity, however, is e.g. for the origin O:
and the tropical Hesse curve is not smooth. The cycle
Geometrical presentation of the group law
On a tropical elliptic curve the group law introduced in , by analogy with the traditional group law on elliptic curves, has a geometrical presentation. We restrict our attention to the cycle C of the tropical elliptic curve, see Figure 3, with
For a general point P, a tropical line intersecting C at P with multiplicity 2 does not necessarily exist. On the tropical Hesse curve, however, intersection of the tropical line rooted in any
Computations of determinants for each case of P lying on the three edges of
when P lies on the upper segment emanating from O, the lower segment emanating from O, the segment opposed to O, respectively. ∎
In [10, 15] Kajiwara and Nobe derive duplication and addition formula for the group law on the tropical Hesse curve using the process of ultradiscretization of the level-three theta functions on the (non-tropical) Hesse cubic curve. Here instead we calculate coordinates of points directly from the geometrical presentation above to obtain short formulas amenable to fast implementations. Let us divide the cycle
In the first step of the addition, the intersection of
Given P and Q on
In order to produce the doubling formulas, we consider the tropical line to be rooted on the point in
Formulas for doubling of P and alternate preimage Q of the double of P.
Note that exactly two points on the tropical Hesse curve have the same image by doubling, with the exception of the origin O. Table 2 displays formulas for the doubling of point P and for Q the other point in the preimage of
The formulas coincide with the ones obtained through ultradiscretization in [10, 14, 15] but involve a smaller number of elementary computations (additions and comparisons). Hence their implementation over either
3 Analysis of the discrete logarithm problem
3.1 Metrics on the tropical Hesse curve
The tropical Hesse curve’s cycle
The explicitation of the homomorphism for tropical elliptic curves in  to the case of the tropical Hesse curve is
where it is enough to specify the images of the vertices. Even though the tropical Hesse curve is not a tropical elliptic curve – it is not smooth –, the proofs in  are still valid for λ as defined above. We can define a signed distance on
3.2 DLP on integral points
Let us now consider the standard cryptographic setting in ECC where Alice and Bob agree on a public elliptic curve, here an integral point
The explicit homomorphism and the signed distance defined in the previous section afford simple analysis of this cryptographic setting on the tropical Hesse curve. Starting with
3.3 Doubling leads to chaos
Although the previous analysis shows that the standard group law is inadequate for cryptographic purposes, the doubling operation shows sensitivity to initial conditions and ergodicity . Although the discretized doubling on integral points of the tropical Hesse curve is a permutation and thus cannot be chaotic, we can extend it to a larger subset of points with coordinates in
The algebraic entropy  of the doubling map, a quantity measuring the complexity of the map defined by
4 A keyed hash function based on the tropical Hessian pencil
4.1 The tropical Hesse chaotic map
For a cryptographic application of the conjunction of both algebraic group properties and chaotic characteristics of the tropical Hessian pencil reviewed in the previous sections, we fix a tropical Hesse curve
which sends a point
We consider the iteration of this chaotic map according to successive bits
4.2 A keyed cipher construction
The construction proposed is iterative and leverages the group structure on the tropical Hesse curve cycle. The message block M and the secret key S have the same bit size n. Note that M and S are mapped each to a point on the upper edge of the public
The cipher round
is iterated r times, with r a public parameter of the cipher, to produce the ciphertext
4.3 Correlation and diffusion properties
As seen in Figure 7, for typical values of the parameters, the cipher construction shows good decorrelation between plaintext bits and ciphertext bits. Bits 0 to 127 of the 128-bit long messages are along the horizontal axis, and the number of times the said message bit is changed after ten rounds over 1,000 runs on random messages is plotted on the vertical axis. The closer the score is to half of the runs, here 500, the closer to random is the correlation between plaintext and ciphertext.
In order to study the diffusion properties, we run the cipher construction on messages that differ on a single bit only while keeping the same secret key. We then compute the Hamming distance, the number of different bits, of the ciphertexts of these single-bit differing plaintexts.
Figure 8 shows the results of running five rounds on 16-bit and 32-bit long messages, and comparing the ciphertext with the sixteen others obtained by changing one bit at each position in the original message. Each round produces an output ciphertext larger than the input message: here ciphertexts are 543-bit long of which an average of 28.67 % are changed when a single bit of the message is switched. Note that the diffusion is larger when the bit changed is higher in the message (most significant bits). By construction the diffusion stays local and with larger key sizes, the ciphertexts are longer, hence the ratio of changed bits decreases: it is 16,4 % for a 32-bit key size yielding 1,055-bit long ciphertexts from 32-bit long plaintext messages.
Properties of elliptic curve geometry naturally carry over to tropical geometry where tropical variety analogues can be defined for notions of degrees, intersections, Jacobians, metrics and group law. Looking at the specific instance of the Hesse cubic curve, a well-studied form of elliptic curve with interesting properties for cryptographic usage, it appears that these properties are lost in the linearization of the group law induced by the tropicalization of the Hessian pencil.
The tropical Hesse group law doubling map
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