Cubic Root Extractors of Gaussian Integers and Their Application in Fast Encryption for Time-Constrained Secure Communication

There are settings where encryption must be performed by a sender under a time constraint. This paper describes an encryption/decryption algorithm based on modular arithmetic of complex integers called Gaus-sians. It is shown how cubic extractors operate and how to find all cubic roots of the Gaussian. All validations (proofs) are provided in the Appendix. Detailed numeric illustrations explain how to use the method of digital isotopes to avoid ambiguity in recovery of the original plaintext by the receiver.


Introduction
This paper describes a cryptographic algorithm based on the extraction of cubic roots from complex numbers a + bi with integer components a and b.Such complex integers are called Gaussian integers (Gaussians, for short) [1].Let's denote and , where N is called a norm of (a, b).In modular arithmetic based on Gaussians, if p is a prime and , then for every integer a and b holds an equivalent of the Fermat identity [2]: This means that the cycles in Gaussian modular arithmetic have order , while the cycles in modular arithmetic based on real integers have order . Application of Gaussians for ElGamal cryptosystem is considered in [3]; and the RSA digital signature is described in [4].Public key cryptography based on cubic roots of real integers is provided in [5] and in [6].
Definition1: A Gaussian integer (x, y) is called the cubic root of (a, b) modulo integer n, and defined as Proposition1: If p is a prime, and mod12 1 p        then there exists a cubic root of (a, b) modulo p. Proposition2: If , then for every integer a and b there exists an unique cubic root of (a, b) modulo p.
then there exists a cubic root of (a, b) modulo prime p.

Algorithm-2
If q mod 12 = 5, then a cubic root exists for every (a, b); and each Gaussian has a unique cubic root.The following algorithm computes such a cubic root (1).

Cubic Roots of (1, 0) and Gaussians
In order to find two other roots of (a, b), consider cubic roots of unity: , mod u w x y p are also its cubic roots modulo n.
Proof is provided in the Appendix.

Existence of  3 mod or 3 mod p p
Jacoby symbols [8,9] analyze whether a specified integer is quadratic residue (QR).If p is a Blum prime, then Therefore, if p mod 12 = 11, then 3 is QR.Seven examples are listed in Table 3.

Properties of Gaussian Cubes
Consider Property 1: Property 2:

Cryptographic Protocol
System design (each user's actions): Step 1.3: Selects two large distinct primes p and q, where p mod 12 = 11; ; and q mod 12 = 5; 2 mod 9 1 p  Step 2.3: Computes n = pq; {n is user's public key; p and q are his private keys}; Step 3.3: Finds cubic root (u, w) of (1, 0) modulo p: : mod ; : mod mod ; P q q p Q p p q n  

 
Protocol implementation: Suppose a sender (Sam) wants to securely transmit a plaintext G to receiver (Regina); Sam divides G into an array of blocks

Efficient Encryption of Gaussians
Squaring of a Gaussian requires two multiplications of real integers (MoRI); and multiplication of two Gaussians requires three MoRI [11].Therefore, the cubic power of Gaussian requires five MoRI.Yet, encryption

Asymmetric Tagging of Digital Isotopes
In cryptographic algorithms based on extraction of square roots of real integers [12] or Gaussians [6] there are four pairs of solutions, and only one of them is the original plaintext.To distinguish the original solution from the other three, the authors use methods of tails, which is an analogue of using isotopes to tag various chemical components.
If the digital isotopes repeat r rightmost digits in each component of plaintext (g, h), then the probability of erroneous recovery of the "plaintext" is of order . For instance, if the length of isotope r = 3, then the probability of error is one in one million.
As shown below, a more elaborate strategy must be used to avoid ambiguity in the recovery of the original plaintext.
Definition 2: If there exist Gaussians with distinct components x and y such that then such cubic roots are called Gaussian twins (or CT, for short).Proposition 6: If the square root of 3 modulo prime p exists, then there exists the CT; {see Table4 for examples}.

x y x y x y y x y x x y xy x y y x xy xy x y p i e x y xy p
Let y := Txmodp, then   i.e., which implies that . For instance, if p = 83, then 3  .Yet, if in both components the rightmost digit is "1", it is not clear whether the original plaintext is (0, 1) or (1, 0).For every p mod 12 = 11 there exist CTs that satisfy (25) {examples are provided in Table 4).

Algorithm in a nutshell
System design: Let Regina's p = 227 and q = 1109, where , p mod 12 = 11; and q mod 12 = 5; she computes n = pq = 251,743; P and Q: and a cubic root (u w) of (1,0) modulo p: until she detects isotopes Therefore, Regina recovers the original Gaussian block of information; and it is not necessary to compute   3 , x y .

Optimized Recovery of Information
; if the isotopes in Z are detected, then the original information is recovered; otherwise Regina needs to compute four components of two other cubic roots of (a, b): , , , , Yet, to minimize computational burden, instead of computing 2 and 3 M M , she finds and then computes If the isotopes are detected, then she computes , otherwise Regina computes x

Elimination of Ambiguity in Recovery of Original Information
The probability of erroneous recovery can be decreased if, instead of repeating r rightmost digits of g and h, the following procedure is applied: 1) Consider r leftmost digits (prefix r P ) of the first component g in plaintext (g, h) and repeat it as its digital isotope; 2) Consider r rightmost digits (suffix r S ) of the second component h of plaintext (g, h) and repeat it as its digital isotope.Example 2: if (g, h) = (31415926, 27182845) and r = 2, then (3131415926,2718284545).
NB: if n is t-digits long and the number of digits in g is smaller than t, then the To avoid ambiguity, the sender must attach both digital isotopes r and r as suffixes.Below is a simple mnemonic/schematic rule for constructing the digital isotopes:

Algorithm Analysis
The cryptographic algorithm described above is neither a generalization nor a special case of the RSA protocol [13].
First of all, the following identity holds: In the RSA algorithm, if z is the length of group cycle [13], then each user selects a public key e that is co-prime with z.In the proposed algorithm the length of cycle c is equal Therefore in the RSA extension it would have been necessary to compute a multiplicative inverse d of e modulo c.Yet, in the algorithm described above the encryption key e = 3.Hence, the decryption key d cannot be computed as a modular multiplicative inverse, since   gcd 3, 3 z  , which implies that such an inverse does not exist [14].

Communication Speed-Up
Suppose it is necessary to transmit an H digit-long plain-text, where the size of each block must not exceed sixteen digits; in addition, suppose that we want to ensure that the probability of erroneous recovery does not exceed one in one million.There are two options: Option 1 is to select the size of each block equal to ten digits and the size of each tail equal to six digits; Option 2 is to select the size of each block equal to thirteen digits and the size of each tail equal to three digits.
In the 1 st option we will treat each block individually as a real integer; which implies that we need to transmit H/10 real integers.In the 2 nd option we will treat a pair of blocks as a Gaussian; which implies that we need to transmit H/26 Gaussians, i.e., H/13 real integers.Therefore, the first option requires 13/10 = 1.3 times more bandwidth, than the second option.In other words, the bandwidth can be reduced by 30% if Gaussian integers are considered.

Possible Applications and Conclusions
The proposed cryptosystem has significant specifics: the encryption is substantially faster than the decryption.There are certain settings where the sender has limited time to transmit the message: visual images or video, and receiver does not have such restriction.For instance, the sender is a system that urgently needs to transmit information prior to either collision with a target or before it is destroyed by a hostile action [15].Another example is if the sender (say, an interplanetary or interstellar space station) detects an impending collision with an asteroid and is programmed to report about such collision and transmit visual and other details about the asteroid.
In this case it is paramount to ensure the reliability of message delivery [15,16].Yet another example is of a security camera that has detected an imminent explosion and is pre-designed to report the situation (audio, pictures and/or video) [17] prior to its own destruction from the explosion.

Table 1}
It is easy to verify that (5, 2) and

Table 4 . Examples of cubic roots twins (CT) for p = 83.
VSS) and (USS, PVP).Only the former one is authentic.Hence, a receiver (Regina) searches for the cubic root with isotopes in format (PUP, VSS), where P and S are prefix and suffix respectively.In this case (PU, VS) is acceptable as the genuine plaintext.