Wish you all the blessing Eid Al-Adha
“Anyone has a mirror ???”
It is interesting to see the property of Little-Dragon-2 (LD2)[1] as the improved version of Little-Dragon[2] developed based on MI (Matsumoto-Imai) scheme as the pioneer in MPKC. One of the known problem in construction Multivariate Public Key Cryptosystem (MPKC) is the possibility to have several equivalent private-keys. The equivalent private-keys in MPKC occur when two unequal private-keys produce the same public-keys (Since the construction of the public-key in MPKC is a composition of 
, where 
is a multivariate polynomial of the degree > 1 (generally is 2) over a prime field 
(
, where 
must be a prime. In case of binary field, 
), 
and 
are two invertible affine transformation of the form 
, where 
is the 
invertible matrix over and 
is a defined-constant on the same field).
The problem of equivalent keys in MPKC can be solved easily by generate permutation polynomial 
over . Such characteristic is produced when the polynomial 
in ![\mathbb{F}_q[x]](http://l.wordpress.com/latex.php?latex=%5Cmathbb%7BF%7D_q%5Bx%5D&bg=FFFFFF&fg=000000&s=0 "\mathbb{F}_q[x]")
is a polynomial function from onto . Thus, there are four condition that make a polynomial is a permutation polynomial over , that are:
- is onto
- is one-to-one
has a solution in , for each 
- has a unique solution in , for each
Then, based on the fact that :
- Every linear polynomial that is a polynomial of the form
with 
over is a permutation polynomial of
- The monomial
is a permutation polynomial of if and only if 
and 
are coprime 
It is proven that polynomial over of the form 
, where 
, is a permutation polynomial if and only if 
and 
are coprime, which 
. The extension of this idea is by having polynomial function of the form 
, where 
is a trace function from 
to 
, in which 
and 
. However, the question is how to choose the value of 
and 
so that the polynomial generated will satisfy the conditions above ?
Choosing and
For the case of public-key cryptosystem, the permutation polynomial only exist where 
is of the form 
. Specifically, for 
, 
, 
and 
, 
is obviously a permutation polynomial, since 
, therefore 
. So the public key is the form of 
where 
is the hidden monomial and must be kept as the private keys, together with and 
as the two invertible linear affine transformation.
However, there is no much improvement on the size of public key and the complexity of the encryption process 
. Moreover, its resistance against many type of algorithm than can solve multivariate polynomial problem (Buchberger, XL, XSL, Linearization, Relinearization, 
and 
, Zhuang-Zhi, etc) as well as the key recovery attack against hidden monomial-based MPKC have not been examined that far.
So as the closing question : anyone interested ? 
[1]. R.P. Singh, A. Saikia, B.K.Sarma., Little Dragon Two: An efficient Multivariate Public Key Cryptosystem. IACR ePrint Archive.
[2]. J.Patarin. Asymmetric cryptography with a hidden monomial. Advances in Cryptology-Crypto ‘96, Springer-Verlag, pp. 4560.
[3] C. Wolf., B. Preenel., Equivalent Keys in Multivariate Quadratic Public Key Systems. IACR ePrint Archive.
Dear folks,
I’ve posted my public key in this blog. Just in case you need to send confidential information or files, I strongly encourage you to encrypt it before send it to my email.
Regards,
Taqabbalallah minna wa minkum
Choose, either pay or pray ...
Here we come everybody, an almost practical related-key attack against reduced round (10 rounds) of AES-256.
AES is the best known and most widely used block cipher. Its three versions (AES-128, AES-192, and AES-256) di ffer in their key sizes (128 bits, 192 bits and 256 bits) and in their number of rounds (10, 12, and 14, respectively). In the case of AES-128, there is no known attack which is faster than the 
complexity of exhaustive search. However, AES-192 and AES-256 were recently shown to be breakable by attacks which require 
and 
time, respectively. While these complexities are much faster than exhaustive search, they are completely non-practical, and do not seem to pose any real threat to the security of AES-based systems. In this paper we describe several attacks which can break with practical complexity variants of AES-256 whose number of rounds are comparable to that of AES-128. One of our attacks uses only two related keys and 
time to recover the complete 256-bit key of a 9-round version of AES-256 (the best previous attack on this variant required 4 related keys and 
time). Another attack can break a 10 round version of AES-256 in 
time, but it uses a stronger type of related subkey attack (the best previous attack on this variant required 64 related keys and 
time). While neither AES-128 nor AES-256 can be directly broken by these attacks, the fact that their hybrid (which combines the smaller number of rounds from AES-128 along with the larger key size from AES-256) can be broken with such a low complexity raises serious concern about the remaining safety margin o ffered by the AES family of cryptosystems.
