A New Construction of Multisender Authentication Codes from Pseudosymplectic Geometry over Finite Fields

Multisender authentication codes allow a group of senders to construct an authenticated message for one receiver such that the receiver can verify authenticity of the received message. In this paper, we construct one multisender authentication code from pseudosymplectic geometry over finite fields. The parameters and the probabilities of deceptions of this code are also computed.


Introduction
Multisender authentication code was firstly constructed by Gilbert et al. in [1] in 1974.Multisender authentication system refers to a group of senders that cooperatively send a message to the receiver, and then the receiver should be able to ascertain that the message is authentic.About this case, many scholars had also much researches and had made great contributions to multisender authentication codes [2][3][4][5][6].
In the actual computer network communications, multisender authentication codes include sequential model and simultaneous model.Sequential model is that each sender uses its own encoding message to the receiver, and the receiver receives the message and verifies whether the message is legal or not.Simultaneous model is that all senders use their own encoding rules to encode a source state, and each sender sends the encoded message to the synthesizer, respectively, and then the synthesizer forms an authenticated message and verifies whether the message is legal or not.In this paper, we will adopt the second model.
In a simultaneous model, there are four participants: a group of senders  = { 1 ,  2 , . . .,   }, the keys distribution center, he is responsible for the key distribution to senders and receiver, including solving the disputes between them, a receiver , a synthesizer, he only runs the trusted synthesis algorithm.The code works as follows: each sender and receiver has their own cartesian authentication code, respectively.Let (,   ,   ;   ) ( = 1, 2, . . ., ) be the senders' cartesian authentication code, (,   , ; ) be the receiver's cartesian authentication code, ℎ :  1 ×  2 × ⋅ ⋅ ⋅ ×   →  the synthesis algorithm.  :  →   is a subkey generation algorithm, where  is the key set of the key distribution center.When authenticating a message, the senders and the receiver should comply with the protocol.The key distribution center randomly selects an encoding rule  ∈  and sends   =   () to the th sender   ( = 1, 2, . . ., ) secretly, and then he calculates   by  according to an effective algorithm and secretly sends   to the receiver ; if the senders would like to send a source state  to the receiver ,   computes   =   (,   ) ( = 1, 2, . . ., ) and sends   = (,   ) ( = 1, 2, . . ., ) to the synthesizer through an open channel; the synthesizer receives the message   = (,   ) ( = 1, 2, . . ., ) and calculates  = ℎ( 1 ,  2 , . . .,   ) by the synthesis algorithm ℎ and then sends message  = (, ) to the receiver , he checks the authenticity by verifying whether  = (,   ) or not.If the equality holds, the message is authentic and is accepted.Otherwise, the message is rejected.
We assume that the key distribution center is credible, though he know the senders' and receiver's encoding rules, he will not participate in any communication activities.When transmitters and receiver are disputing, the key distribution center settles it.At the same time, we assume that the system follows Kerckhoff's principle in which except for the actual used keys, the other information of the whole system is public.
In a multisender authentication system, we assume that the whole senders are cooperating to form a valid message; that is, all senders as a whole and receiver are reliable.But there are some malicious senders which they together cheat the receiver, the part of senders and receiver are not credible, they can take impersonation attack and substitution attack.In the whole system, we assume that { 1 ,  2 , . . .,   } are senders,  is a receiver,   is the encoding rules set of the sender   , and   is the decoding rules set of receiver .If the source state space  and the key space   of receiver  are according to a uniform distribution, then the probability distribution of message space  and tag space  is determined by the probability distribution of  and   .Consider  = { ) . ( In this paper, we give a construction about multisender authentication code from pseudosymplectic geometry over finite fields.

Pseudosymplectic Geometry
Let   be the finite field with  elements, where  is a power of 2,  = 2] + , and  = 1, 2.
Let  (2]+)  be the (2] + )-dimensional row vector space over   . 2]+ (  ) has an action on  (2]+)  defined as follows: The vector space  (2]+)  together with this group action is called the pseudosymplectic space over the finite field   of characteristic 2.

𝑞
. Denote by  ⊥ the set of vectors which are orthogonal to every vector of ; that is, Obviously, .More properties of pseudosymplectic geometry over finite fields can be found in [7].
In [2], Desmedt et al. gave two constructions for MRAcodes based on polynomials and finite geometries, respectively.There are other constructions of multisender authentication codes which are given in [3][4][5][6].The construction of authentication codes is of combinational design in its nature.We know that the geometry of classical groups over finite fields, including symplectic geometry, pseudosymplectic geometry, unitary geometry, and orthogonal geometry, can provide a better combination of structure and can be easy to count.In this paper, we construct one multisender authentication code from pseudosymplectic geometry over finite fields.The parameters and the probabilities of deceptions of this code are also computed.We realize the generalization and application of the similar idea and method of article [8] from symplectic geometry to pseudosymplectic geometry over finite fields.

Construction
Let F  be a finite field with  elements and   (1 ≤  ≤ 2] + 2) the row vector in F (2]+2)  whose th coordinate is 1 and all other coordinates are 0. Assume that 2 <  + and  ⊂  ⊂  ⊥ }; the set of th sender's encoding rules
From Lemmas 1 and 2, we know that such construction of multisender authentication codes is reasonable, and there are  senders in this system.Next, we compute the parameters of this code and the maximum probability of success in impersonation attack and substitution attack by group of senders.

Lemma 4. (1) For any
(2) The number of the th sender's tag is Proof.(1) Considering the transitivity properties of the same subspaces under the pseudosymplectic groups, we may take   as follows: twocolumngrid If    ⊂   , then we assume that from    ⊥   , we know that  7 = 1, where  9 ,  11 arbitrarily, and therefore the number of We know that every   contains only one source state   ∩ ⊥ and the number of   containing    .Therefore, we have
(2) Considering the transitivity properties of the same subspaces under the pseudosymplectic groups, we may choose  as follows: If   ⊂ , then where  5 and  7 arbitrarily.Therefore, the number of   which contained  is  (−+1) . ( Without loss of generality, we assume that  = { If   ⊃   , then   has the following form: where   6 ,   7 arbitrarily.Therefore, the number of   containing   is  (]−+1)(−) .Lemma 7.For any  ∈  and   = {  1 ,   2 , . . .,    } ∈   , the number of   which contained in  and containing   is  (−+1)(−) .Proof.For any  ∈ , we assume  to be as follows: If   ⊂ , then   has the following form: Since   ⊂   ⊂ , then we assume   to be as follows: From the definition of , we may take   ,  = 1, 2, as follows: Let From the above mentioned, we know that where every row of (0  7 0  9 0) (39) is the linear combination of the base of ( 0  7 0 0 0 0 0 0 1 0 ) .