Which statements are true

$$ Cyclic groups play a critical role in asymmetric cryptosystems.

$$ The historically main application for hash functions is in digital signatures, as they can shorten messages to be signed.

$$ The main difference between digital signatures and MACs are that the latter use same private keys and therefore primarily address symmetric cryptography $($however with public$-$key cryptography possibly used to exchange keys with conditional security, or with quantum key.

$$ Itis sensible to combine symmetric cryptography with asymmetric cryptography.

Which of the statements on cryptography are correct

$$ Security of hash functions can be related to the birthday paradox.

$$ Elgamal scheme's security is conditioned by the DLP problem being a hard computational problem $($which requires usage of the appropriately large prime numbers$).$

$$ DSA and Elgamal belong to the integer factorization algorithms family.

$$ If one can solve the discrete log problem one can break the Diffie$-$Hellman protocol.

Which below statements on cryptography are correct

$$ The order of an element of a groups the smallest positive integer such that this element to the power of that integer is equal to the identity element of the group.

$$ The Elgamal encryption scheme generally includes the Diffie$-$Hellman key exchange to establish the secret session key, which is used for encryption by multiplication of the message to be encrypted with the session key modulo the public prime parameter from the Diffie$-$Hellman protocol $($the decryption requires multiplication of the encrypted message by the inverse of the session key modulo the same public prime parameter of the Diffie$-$Hellman scheme$),$ although the Elgamal scheme reorders these steps.

$$ Secure hash functions are required to take arbitrary input lengths and produce fixed and short output lengths in a computationally efficient way, with a property of one$-$wayness operation $($also referred to as preimage resistance$),$ that could be explained with outputs of very similar inputs bing very different, as well as with properties of weak collision resistance and collision resistance.

$$ Acyclic group can have only one primitive element.

Which of the following statements on cryptography are correct

$$ The two most important classes of public$-$key cryptography include factorization problems and discrete logarithm problems.

$$ In the Key Distribution Center protocols class the KDC does not have to be a trusted authority.

$$ Non$-$repudiation in cryptography ensures that the author of the message will not be able to dispute his authorship.

$$ Diffie$-$Hellman protocol has $3$ public parameters.

Which of the statements below are correct

$$ The problem with the AES encryption standard is that it is slow.

$$ Objectives of a security system may be referred to as security services and may include confidentiality, authenticity, integrity and non$-$repudiation.

$$ The MITM attack works against all cryptosystems which do not solve the authentication problem.

$$ Factoring is the only way to break the RSA cipher.

Which of following statements are correct

$$ In the Diffie$-$Hellman protocol Bob's public key to the power of Alice' private key modulo public parameter p $($a large prime$)$ will have the same value as the Alice's public key to the power of Bob' private key modulo public parameter p$,$ and this value remains unknown to all other parties.

$$ Collision attacks are easier to prevent than second preimage attacks.

$$ Multiplicative groups of prime fields are less frequently applied to practical cryptosystems than multiplicative groups of extension fields.

$$ The most secure hash functions do not have any collisions.

Which of these statements are correct

$$ RSA cryptosystem belongs to the integer factoring public$-$key cryptography algorithms family.

$$ Using the prime number of $160$ bits binary encoding would result with circa $2$ to the power $80$ steps n the hest known attack, i$.$e$.$ well beyond current computational power.

$$ A sufficient level of security with RSA or DLP based cryptosystems can he achieved with keys of $1024-2048$ bts$,$ while for the elliptic curves cryptosystems the sufficient length of the key is $80$ bts$.$

$$ Symmetric cryptography security level of $256$ bits $($for AES and $3$DES$)$ corresponds to $512$ bits key for ECDH and ECDSA asymmetric cryptosystems, and to well above $10000$ bits for DH$,$ DSA, Elgamal and RSA asymmetric cryptosystems.