| CPC G06F 21/606 (2013.01) [H04L 9/3093 (2013.01); H04L 9/40 (2022.05)] | 8 Claims |
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1. A method of encrypting and decrypting information, referred to as polar lattice cryptography (PL), utilizing a PL key, PLK, comprising two parts: (i) a geometric structure, a graph, comprised of points and connecting lines, between the points, referred to as a polar lattice structure (PLS), onto which, (ii) at least n pairs of markers are written in secret locations on the PLS, where a marker is a designated point on the PLS, and where each pair of markers represents a certain letter of a transmitter-recipient shared plaintext alphabet A comprising n letters a1, a2, . . . an and where each letter of A is associated with at least one pair of markers on the PLS, and where a transmitter transmits a letter ai of A (i=1, 2, . . . n) to a recipient by sending to them a Corresponding Pathway defined as a series of steps marked on the PLS, where a step corresponds to moving from one point on the PLS to a neighboring point on the PLS, where neighborhood between points is well defined, the Corresponding Pathway leads from one marker of a pair of markers representing letter ai (i=1, 2, . . . n) to the other marker of the same pair of markers, and where the Corresponding Pathway does not lead from one marker to the other marker of the same pair where the pair of markers represents any other letter aj for j≠i, each letter of A may be associated with an arbitrary secret number of pairs of markers, marked on arbitrary points on the PLS, and the PLS may be of an arbitrary size and arbitrary configuration, the PL cipher is designed to force the cryptanalyst to use brute-force cryptanalysis, and keep such analysis with a persistent doubt with respect to the identity of the PL key, since it is possible for the PLS to be larger than identified by the cryptanalysis and it is possible for the number of pairs of markers to be larger than identified by the cryptanalysis, the method comprising:
(1) a plaintext alphabet, A, comprising n letters a1, a2, . . . an;
(ii) a polar lattice structure, PLS;
(iii) an arbitrary number of pairs of markers for each of the letters in the plaintext alphabet A;
the PLS is constructed as r concentric rings R1, R2, . . . Rr, where point o is their shared center and where the diameter of ring i is smaller than the diameter of ring (i+1) for i=1, 2 . . . (r−1);
there are l rays or lines, L1, L2, . . . Ll emanating from o and crossing all the rings, each ray j is drawn at direction αj, where αj<αj+1 for j=1, 2, . . . (l−1) and where α1=0, and where L1 is horizontal, drawn to the right of point o;
each line Lj is recognized from a ring gj<r to ring r, it is not recognized or not drawn, from ring 1 to ring gj;
the r rings and the/lines intersect q times, each intersection is designated as a point on the PLS;
a given point x on the PLS has at most four of the following neighbors:
(a) a ‘far’ neighbor which is the point that shares a line with x but is located on the next larger ring,
(b) a ‘close’ neighbor which is the point that shares a line with x but is located on the next smaller ring,
(c) a ‘right’ neighbor which is the point that shares a ring with x but is the next point to x moving in the right direction, the right neighbor is found in the clockwise direction for the upper half of the PLS, 0≤α≤180, and is found in the counterclockwise direction for the lower part of the PLS, 180<α≤360,
(d) a ‘left’ neighbor which is the point that shares a ring with x but is the next point to x moving in the left direction, the left neighbor is found in the counterclockwise direction for the upper half of the PLS, 0≤α≤180, and is found in the clockwise direction for the lower part of the PLS, 180<α≤360;
a traveler on the PLS who is located at point x will reach the far neighbor through a step on the shared ray moving away from o, this step is designated as F;
a traveler on the PLS who is located at point x will reach the close neighbor through a step on the shared ray moving towards o, this step is designated as C;
a traveler on the PLS who is located at point x will reach the right neighbor through a step on the shared ring taken in the right direction, this step is designated as R;
a traveler on the PLS who is located at point x will reach the left neighbor through a step on the shared ring taken in the left direction, this step is designated as L;
the four steps F, C, R, L constitute a steps alphabet As, a trail also called a pathway, on the PLS is a sequence of the As letters, a trail (T), leads a traveler from point x on the PLS to point y on the PLS;
steps R and L are a ‘neutralizing pair’, steps F and C are also a neutralizing pair, a neutralizing pair returns the traveler to the point before the part of the trail marked by the neutralizing pair;
trail Tr is the reverse trail for T comprising the T steps in the reverse order wherein each step is replaced by the reverse of its step in its neutralizing pair, trail T leads a traveler on the PLS from point x to point y, Tr leads a traveler from point y to point x;
a traveler on a point x which is on the outer ring r and on ray Lj when encountering a step F will move to ring gj on the same ray;
a traveler on a point x which is on ray Lj and on ring gj, when encountering a step C will move to ring r on the same ray;
each letter ai∈A has at least one pair of markers, points, a starting point, psi on the PLS, and a finishing point pfi on the PLS, where i=1, 2, . . . n, each pair of points utilized as follows:
regarding letter encryption:
(e.i) to transmit a letter ai∈A, one will draw an arbitrary trail Ti from a certain pair of points psi to pfi,
(e.ii) a transmitter will then check if applying trail Ti to a pair of markers for any other letter, aj in A where i≠j, will lead from the starting point of that pair to the finishing point of that pair somewhere along the trail, this situation is regarded as “collision”,
(e.iii) if collision happens for any aj then the transmitter will draw another arbitrary trail from psi to pfi, return to (e.i),
(e.iv) when a trail T*i is drawn that does not lead from the starting point of any pair of markers representing letter j, aj, to the finishing point of that pair of markers, namely T*i is collision free, then T*i is regarded as the Corresponding Pathway for letter ai, and by transmitting this Corresponding Pathway the transmitter transmits letter ai to an intended recipient,
regarding letter decryption:
letter decryption is implemented by a recipient who shares a polar lattice key (PLK) with the transmitter, the PLK comprising the PLS and the placement on the PLS of all the pairs of markers for all the letters in the plaintext alphabet A,
(d.i) the intended recipient, sharing the key PLK, with the transmitter, will draw the Corresponding Pathway, T*i, for letter ai, from all the starting points of all the pairs of markers representing any of the letters ak∈A, for k=1, 2 . . . n and identify letter ai as the only letter in A for which the Corresponding Pathway, trail T*i leads a traveler from a starting point psi of a pair of markers representing letter ai to pfi, the finishing point for that pair, and thereby conclude that the transmitter transmits letter ai;
regarding message encryption:
any message M written in alphabet A will be transmitted letter by letter according to (e.i) to (e.iv), and the recipient will interpret the series of trails as the letters of M by order, the sequence of trails corresponding to M is the ciphertext CT for M.
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