A Public Key Crypto System On Real and Complex Numbers Sami Harari - PowerPoint PPT Presentation

A Public Key Crypto System On Real and Complex Numbers Sami Harari ISITV, Universit´ e du Sud Toulon Var BP 56, 83162 La Valette du Var cedex May 8, 2011 Sami Harari A Public Key Crypto System On Real and Complex

Motivation Certain rational interval maps can be used to define a cryptographically difficult problem based on entropy. This is exploited to define a fast and efficient block cipher with a public key, i.e. a Public Key Cryptosystem. The strength of the system will be studied. Parameters requirements will be derived. Implementation details will be presented. Sami Harari A Public Key Crypto System On Real and Complex

Structure of the Presentation 1 The concept of One Way Function for Real Numbers. 2 Trap Door 3 The Interval Maps. 4 The New crypto-system. 5 Complex Field Case Sami Harari A Public Key Crypto System On Real and Complex

One way Functions The concept of ”one way function” (OWF) was introduced in the seventies, linking mathematical considerations of a mapping and its converse with two different computational complexities. The two most studied OWF are : The discrete exponential and the discrete logarithm function on a finite prime field The RSA problem linking a specific couple of exponentiations on a certain finite ring linked to the factorisation problem of factoring two integers. Sami Harari A Public Key Crypto System On Real and Complex

One way Functions 2 These OWF are linked to a computationally hard problem. This means that, in each instance, finding a solution requires testing all ( or nearly all) the elements of a set to see if it is a solution to the problem. The set is of very large cardinality, without any possible strategy of converging to a solution and thus reducing the computational effort. Sami Harari A Public Key Crypto System On Real and Complex

Entropy Based One Way Functions We introduce another kind of computationally hard problem which relies on another paradigm: The Entropy Based Pre-image Search Problem. It is applied using iterated interval maps. Suppose there is a function f (), real valued defined on the unit interval, computable in polynomial time, such that for any y of the unit interval the set A y of preimages of y is a set of (arbitarily) large cardinality, with computation parameters that are tractable. Such a mapping is called an entropy based OWF. Sami Harari A Public Key Crypto System On Real and Complex

Computing the value f ( x ) is done in polynomial time. Finding a pre-image of a given element is done in polynomial time by the well known algorithms. However, with adequate parameters, finding all the pre-images is not possible with available memory resources, in a fixed reasonable interval of time. Sami Harari A Public Key Crypto System On Real and Complex

Quadratic Curve Sami Harari A Public Key Crypto System On Real and Complex

Cubic Curve Sami Harari A Public Key Crypto System On Real and Complex

The Polynomial Interval Maps The new scheme requires interval maps with, for each element, a large set of pre-images. The scheme starts with polynomial interval maps: f ( x ) = α.x 2 mod 1 g ( x ) = β.x 3 mod 1 The coefficients must satisfy α > 2 and β > 2. The variable x belongs to the unit interval I = [0 , 1] Sami Harari A Public Key Crypto System On Real and Complex

Properties of f () The mapping f () has the functional property f ( λ.x ) = λ 2 · f ( x ) mod 1 for all real valued λ < 1. The mappings f () is surjective (onto). Sami Harari A Public Key Crypto System On Real and Complex

Properties of g () The mapping g () has the functional property g ( λ.x ) = λ 3 · g ( x ) mod 1 for all real valued λ < 1. The mappings g () is surjective (onto). This property will serve as a trap door in the PKC. Sami Harari A Public Key Crypto System On Real and Complex

Existence and Computational Problems Associated to Interval Maps To these mappings two algorithmic problems can be deduced. These will be used in a cryptographic context with adequate parameters. Sami Harari A Public Key Crypto System On Real and Complex

Computational Problem for Iterated Polynomial Maps CPPM Given a set of values consisting in the evaluation of two points on the unit interval with the two mapping f () , g () defined in the preceding section, Given ( f ( a ) , g ( a ) , f ( b ) , g ( b )) compute the values ( f ( a.b ) , g ( a.b )). The solution of the problem is the following. The set of pre-images of an element a in the unit interval for f (), E f ( a ) = { x ∈ [0 , 1] s.t. f ( x ) = a } is a set with α points lying in the unit interval. Computation of individual elements is possible in polynomial time. The same can be said for the set E g ( a ) = { x ∈ [0 , 1] s.t. g ( x ) = a } for the mapping g (). In this setting the search for the element a is equivalent to computing the intersection of E g ( a ) and E f ( a ). Sami Harari A Public Key Crypto System On Real and Complex

Computational Problem for Polynomial Maps CPPM If another pre-image of f ( a ) say a ′ is used instead of a the inequality g ( a ′ .b ) � = g ( a.b ) will be true and the problem will not be solved. The same can be said with b ′ another pre-image of b for the mapping g (). Sami Harari A Public Key Crypto System On Real and Complex

The Decisional Problem for Polynomial maps DPPM The decision problem associated to the iterated interval maps can be defined as follows for a couple of iterated interval maps f () and g (). Given a set of values for three points of the unit interval a, b, c . ( f ( a ) , g ( a ) , f ( b ) , g ( b ) , f ( c ) , g ( c )) Decide if c = a.b . The decisional problem is solved if an instance of the existence problem are solved. Therefore solving the decision problem is at least as hard as the computational problem. Sami Harari A Public Key Crypto System On Real and Complex

Encoding Data on the Unit Interval Suppose that the words of plaintext are n -bit sequences which must be coded. Pre-compute the sequence a i = 2 − i +1 i = 0 , . . . , n − 1 With the help of the sequence a i , a block of n bits x 0 , . . . , x n − 1 is encoded into the floating point number: n − 1 � y = x i · a i 0 This coding method is efficient if the a i are pre-computed, since the computation of y involves only sums. Sami Harari A Public Key Crypto System On Real and Complex

Decoding Numbers into Bit Sequences To obtain the bit sequence x 0 , . . . , x n − 1 associated to a floating point number y , one uses a knapsack linear decoding algorithm which runs as follows: Input y, output x 0 , . . . , x n − 1 for( i=0,i<n;i++) { if y > a i { x i =1 y=y- a i } else x i =0; } The floating points used to represent the binary data must be coded on at least n bits. However the precision must be 2 · n bits in order that computational noise does not interfere with significant data. Sami Harari A Public Key Crypto System On Real and Complex

Computing a Random pre -image for f () The analytical expression for the modular mapping f () can be quite hard to obtain if α is large. However with the non modular version the algorithm becomes quite simple. Suppose you want to compute a pre-image of y for y = α.x 2 . Choose a random integer r such that 0 < t = y + r < α Compute � x = ( t/α ) the result x is a random pre-image of y Remark One must note that the random integer r can be chosen to be really random since it must never be recomputed. Sami Harari A Public Key Crypto System On Real and Complex

The associated PKC With the usual convention we will suppose that Alice wants to encipher a message to Bob in such a way that only Bob can decipher it. The Secret Key Belonging to Bob The secret key and characteristic quantity of the destination of the message is a real number s in the unit interval. They can be associated is associated to 2 different n bit sequences as shown earlier. The real number s is specific to one user and can be used to encipher many plaintext messages. The images f ( s ) and g ( s ) must be computed and kept secret. The block size n must be specified by Bob. The usual values are 128,256,512,1024. Sami Harari A Public Key Crypto System On Real and Complex

The Public Key of Bob The public key of Bob is made of the following data. 1 The block length n , as well as α and β . 2 ( f ( s ) , g ( s )) which are the images by the mapping f () ang g () of the secret key s . 3 The function f ( x ) = α · x 2 and g ( x ) = β.x 3 . Sami Harari A Public Key Crypto System On Real and Complex

How Alice computes a cryptogram(Enciphering) Let M the plaintext (sequence of bits) and r m the real number associated to a block of n bits, less than 1 associated to the message to be enciphered, by the method described. Alice begins by choosing by computing a (random) pre-image x of r m for the mapping f (). She then computes g ( x.s ) using the trap door property of g (). She also computes g ( x ). Sami Harari A Public Key Crypto System On Real and Complex

Encrypting 2 The cryptogram of r m is a couple of real numbers: 1 The real number c 1 = x.g ( x ) .g ( x.s ) . 2 The real number c 2 = g ( x.s ) Sami Harari A Public Key Crypto System On Real and Complex