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Elliptic Curve Cryptography (ECC) - Science topic

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I need the difference between Schnorr and ECC in a little bit detail, theoretical and practical. What is the difference in both of the properties like execution time, key sizes, and message sizes?
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In ECDH, when two person wants to share private key, they first select a point G on elliptic curve and after that, each of them pick a random integer a and b, respectively, and multiply with G. After the multiplication each of them shares aG and bG with each other and after that, they multiply again using their keys a(bG) and b(aG), respectively, and creates a shared key between themselves. However, if one person wants to communicate with a group of person (more than 2) using a shared key utilizing ECDH, how he can use this method? because each of the person may choose different integer while establishing the key.
How to employ ECDH in key exchange with a group of people?
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I am trying to implement an experiment to verify the performance of some public key algorithms, in an attempt to evaluate the importance of Elliptic Curve Cryptography (ECC) over other public key crypto-system..
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@Auday sir, can you please mention any one of the public key evaluation tools ?
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I study these terms in
1- Elliptic Curve Cryptography (ECC)
2- Discrete Logarithmic Problem (DLP)
3- Diffie-Hellman Technique
But I don't know the procedure for a protocol that could verifiably protect in the random oracle model against the hardness assumptions of the aforesaid 3 techniques.
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Ok Salah
But tell me some network security experts claim that their protocol will verifiably protect in the random oracle model against the hardness assumptions of the Elliptic Curve Discrete Logarithm Problem and Elliptic Curve Computational Diffie-Hellman problem.
What this mean?
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With the invention of quantum computers, the existing cryptosystems may be broken in the future. This has attracted a new crytosystem known as Quantum Cryptography (QC). What are the advantages of QC over ECC? What are the disadvantages compared to ECC?
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Theoretically, yes. But the required infrastructure is still many years away
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Please, which is more computationally efficient, Modular exponentiation or Elliptic curve scalar multiplication (ECSM)?
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The question should be as you state, but you should have also add at the end "..for achieving the same level of security," I'd assume this is what you meant.
To get a reasonable security in modular exponentiation over an elementary number theoretic group (say, prime order group) one needs to get a security of 2^80 say, an exponent size of about 1000 (or even 2000, more or less, I am not doing exact calculations), given the various methods to extract discrete log in these group representation over the modular subset of the integers. For elliptic curve (if chosen wisely) the best algorithm for d. log. is the generic one, so a size of 160-200 bit scalar is needed so already 5 -- 6 (or even 10) times less bits to perform on, thus making the ECC curve multiplication much faster (even if the basic per bit op is more involved). 
The ECC is therefore faster and consumes less space.
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If A1,...,An be users in a network , and they want to use El-Gamal cryptosystem for their cryptography . I want to know what amount of memory used for the job better for every one of A1 , ... , An. for example n=2^10 . In fact in this example how many initial points is better to use ??
For example If I use about 100 initial point in my program and If keys sizes are 512 bits , 2*100*2^10*512 bits memory need for every users . 2^20*100 bits = 12.5 MB !!!  Is it colophon & reasonable ?
Thank you all .
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You can see this link:
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In imaginary quadratic fields we have:
* ELL(O_K) : = {elliptic curves E/C with End(E) ∼= O_K}/{isomorphism over C}
∼= {lattices L with End(L) ∼= OK}/{homothety}∼=ideal class group CL(K)
* #CL(K)=#ELL(O_K)
this notation at the papper :A SUMMARY OF THE CM THEORY OF ELLIPTIC CURVES
JAYCE GETZ
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As you say, the theory of "complex multiplication" (looking at modular invariants of elliptic curves with CM) solves the problem of determining the maximal abelian extension of an imaginary quadratic field. But as yet no analogous theory is known for real quadratic fields, although class field theory affords  a complete theoretical (but not explicit) description of the abelian extensions of a given number field. You can  take the measure of our ignorance by  listening to the short negative answer given by Serre at the end of his talk on abelian Galois theory at the occasion of Galois' bicentenary (the talk can be found on You Tube, I think).
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I try to search one paper between 2014-2016 talk about encrypt the data in cloud computing using  ECDH + ANN.  Thanks
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not sure for that, but looking for paper talk about that topic
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How are eigen values and eigen vectors connected with elliptic curve cryptography?
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For one, they appear in the study of Frobenius morphism, see e.g.  :
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Dear all,
I'd like to know how many joules a scalar multiplication in the ECDLP  of ECC is needed? 
Best regards
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Everything depends on how you modify your algorithm ,CPU clock cycles to perform it , iteration process capacity of a processor for the running operation.Compare to the previous & the modified one.You'll understand. Better to apply on raw code.You can get the result in per/sec then convert it to j/s.In your needed result must have to know how many works the algorithm will perform in per second. - Thanks.
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It could be thought this question as a different approach if it is compared with research recently.
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It would be great to work together or with other researchers who have desire to work together.
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ECC researchers
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ECC requires shorter key to achieve security as compared to RSA, and that the mathematical operation involved in ECC is scalar multiplication while RSA uses exponentiation.  Because of these, computational complexity involved in ECC is much less as compared to other asymmetric cryptosystems.  
This enables much faster encryption, decryption and signature verification.  Also it saves bandwidh and memory space.  
Hence ECC is best suited for devices which have limited storage and computational power (like smart cards).  Also it is suited for web-browsers, which use a number of encryption sessions.
So your observation about ECC is right.
Regards.
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I have a new method, so the comparison among methods play a crucial role in improving this method. 
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What about Pollard's Rho method and its parallelized?  it may be more efficient and fast.
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I want to determine the number of points on E in terms of æ and ß. For example:
For Koblitz curves one can compute this cardinality using a Lucas sequence but I am working in Elliptic curves over F_{2n} (not necessary Koblitz curves) given by the equation Y2 + XY = X3 + æX2 + ß where æ and ß are elements in F_{2n} and Tr(æ) = 1.
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Yes, there is a relation between the order of the group of points, the size of the base field (check Hasse's condition) and the curve coefficients. You can use a point counting algorithm like Schoof-Elkies-Atkin (SEA) to determine the order. This algorithm is usually already implemented in software like Sage or MAGMA.
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Up to date, multiplication algorithm are considering two-operation multiplication. For multiple-operation multiplication, it is not discussed in the liternature. The existing subquadratic algorithm such as karatsuba algorithm, cook-took algorithm, and winograd FFT, are discussed on two-operation multiplication. multiple-operation multiplications have achieved fast algorithm for computing inversion, exponentiation, and even for point addition, due to these applications is performed by succesive multiplication.
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"multiple-operation" is mistake the word. The word is corrected as "multiple-operand".
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I want to know about process of encryption and decryption of a database by using elliptic curve cryptography.
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Hello sir, the encryption technique you need to use for a database is depend on the data you store inside the database.
If the data is username and passwords, you can use Hash function instead of ECC. But beware of Rainbow table attack(if you chose Hash). This attack can be rectified by using SALT to the hash function.
If you store some important data/value/message that possibly more than 20 characters, you can use multilevel/multistage encryption techniques, that is you can combine two algorithms/ two whole new algorithms. All the best for your work.
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I am talking about the scaler Multiplication Algorithm.
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I will provide you wit a paper to discuss your inquiry
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I want to know the possible attacks over ECC-based cryptographic algorithms and the weakness that allows such attacks.
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Thanks Abdoul, but actually am looking for more advanced attacks such as the ones related to the field attacks, curve generation and so on.
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I need to know if there is such that relation in the case of real quadratic fields.
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There is a notion of (divisor) class group 'Cl(X) ' associated to algebraic varieties X--affine and projective. (I will define this notion below for elliptic curves which are projective.). Now, if A is a Dedekind domain, (e.g. the ring of integers in a number field F (where F could be a quadratic field you are interested in) Cl(Spec(A)) coincides with the ideal class group of A as defined in Algebraic Number Theory. So, this gives you the connection between affine varieties and ideal class group. (Spec A is affine.)
I will now define Cl(X) for an elliptic curve X. . A divisor D of X is a linear combination of points x_i with coefficients k_i in Z(the integers).. The degree of D (deg D) is the sum of the k_i's. The divisors of X form a group Div(X)--the free Abelian group (Z-module) generated by the x_i's . A principal divisor is a divisor D such that the degree of D = 0.The principal divisors (P(X)) form a subgroup of Div (X) and the quotient group Div(X)/P(X) is denoted by Cl(X). Now, there is homomorphism deg: Cl(X) --> Z whose kernel is denoted by Cl^o(X). Cl^o(X) has the following interesting property:
There is a one-one correspondence between the points of x and the elements of the group Cl^o(X). Moreover by using this one-one correspondence, one can transfer the group law from Cl^o(X) to X itself.
Elliptic curves is a very active area of research probably because the curves have various deep ramifications. For example, an elliptic curve is an Abelian variety of dimension 1 i.e an irreducible projective algebraic group; it is also a curve of genus 1. etc.
for further information you may look at the following books:
1) Basic Algebraic Geometry--Varieties in Projective spaces by I. R. Shavarevich.
Springer-Verlag.
2) Algebraic Geometry-- R. Hartshorne. Springer-Verlag.
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Mainly at the doubling of point at Yp=0.
If we consider O(point at infinity) as (0,0) then it must hold P+O=P.
Suppose we take P(3,10) + O(0,0) and perform additive property of ECC then point comes is (3,13) rather than (3,10). What to do in this case?
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Equations do not hold for O related operations. You should add additional “if ” branch codes for these cases.