Answer Posted / boss

Let E be a product cipher that maps N-bit blocks to N-bit blocks.
Let E_K(X) be the encryption of X under key K. Then, for any fixed K,
the map sending X to E_K(X) is a permutation of the set of N-bit
blocks. Denote this permutation by P_K. The set of all N-bit
permutations is called the symmetric group and is written S_{2^N}.
The collection of all these permutations P_K, where K ranges over all
possible keys, is denoted E(S_{2^N}). If E were a random mapping from
plaintexts to ciphertexts then we would expect E(S_{2^N}) to generate
a large subset of S_{2^N}.

Coppersmith and Grossman [COP74] have shown that a very simple
product cipher can generate the alternating group A_{2^N} given a
sufficient number of rounds. (The alternating group is half of the
symmetric group: it consists of all ``even'' permutations, i.e., all
permutations which can be written as an even number of swaps.)
Even and Goldreich [EVE83] were able to extend these results to show
that Feistel ciphers can generate A_{2^N}, given a sufficient number
of rounds.

The security of multiple encipherment also depends on the
group-theoretic properties of a cipher. Multiple encipherment is an
extension over single encipherment if for keys K1, K2 there does
not exist a third key K3 such that

E_K2(E_K1(X)) == E_(K3)(X) (**)

which indicates that encrypting twice with two independent keys
K1, K2 is equal to a single encryption under the third key K3. If
for every K1, K2 there exists a K3 such that eq. (**) is true then
we say that E is a group.

This question of whether DES is a group under this definition was
extensively studied by Sherman, Kaliski, and Rivest [SHE88]. In their
paper they give strong evidence for the hypothesis that DES is not a
group. In fact DES is not a group [CAM93].

What exactly is DES?

1449

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What are IP Tunnels?

1656

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How was NSA involved in the design of DES?

1340

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What is a product cipher?

1980

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hi,pls help me for the preparation of interview for iob's it manager post?

1875

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Is ECB cipher mode faster than CBC ?

1775

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Does IAIK support RC4 ? Is the code written the same way as for RC2 ?

1419

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Name different symmetric ciphers?

1931

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Can symmetric block ciphers be used for message authentication?

1413

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what is ECB?

1765

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What can be proven about the security of a product cipher?

1453

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what is CBC?

1828

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Can DES be used to protect classified information?

1381

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shall we use a journalling filesystem on top of /dev/loop?

1648

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Is DES available in software?

1320

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