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III.3. OTHER ATTACKS AGAINST ECIES using barcode printer for none control to generate, create none image in none applications. Barcodes FAQs It turns out that t none none his is not a useful thing to do, but for quite a subtle reason. The reason is that the key generation algorithm is a xed algorithm that doesn t depend on any secret values, so there are always attackers that assume something about the key generation algorithm, such as that it always produces one particular key-pair, and attempt attacks based on this assumption. If the key generation algorithm produces a large enough number of di erent key-pairs, then these attackers will not have a signi cant success probability as it is very unlikely that a key-pair that is generated by G will satisfy the assumption that the attacker makes, but this could be a problem for key generation algorithm with small ranges.

It is possibly best to think about this in terms of a simple example. If sk is some xed private key for an asymmetric encryption scheme (G, E, D), then there is an attacker A who will try to decrypt a challenge ciphertext using the function D( , sk) regardless of what the public key is or whether it knows any information about the key generation algorithm. This means that, even if G is unknown, A will still have a high success probability if G outputs the key-pair (pk, sk) with a signi cant probability.

Since we are forced to consider all possible attackers for the scheme (G, E, D), we are forced to conclude that (G, E, D) is a weak encryption algorithm because an attacker exists that will break the scheme with high probability even though we do not know what that attacker is. On the other hand, if G only ever produces (pk, sk) with an insigni cant probability, i.e.

, G has the ability to output lots of di erent keys each with equal probability, then (G, E, D) is secure because the attacker A will not work in all but an insigni cant fraction of the possible cases. This demonstrates another weakness of security proofs. Whilst a security proof tells us about the security of a scheme in general, it doesn t tell us about how the scheme works with one particular key-pair.

It is entirely possible for a scheme to have a good security proof and yet be completely insecure for some set of keys. The best that a security proof can say is that if one generates a key-pair using the key generation algorithm, then the probability that that scheme will be insecure is insigni cant; it doesn t say anything about the suitability of any one speci c key-pair that might actually be in use..


Invalid El liptic Curve Point Attacks. Another assumption that has been made throughout this section is that all the data are valid. In real implementations of elliptic curve cryptosystems, elliptic curve points are usually represented as elements of Fq Fq or Fq by using some form of point compression.

Whenever an attacker has requested the decryption of a ciphertext (U, c, r), we have always implicitly assumed that U is a valid point on the elliptic curve E and an element of the subgroup generated by P . If this is not explicitly checked, then an attacker can exploit this to break the scheme. As an example, consider the following simple attack.

The attacker nds a point U on E with small prime order p1 , chooses any message m M and,. III. PROOFS OF SECURITY FOR ECIES for 0 i < p1 , computes (k1 k2 ) = KD([i]U, l), none for none c(i) = Enc(m, k1 ), r(i) = MAC(c(i) , k2 ). Let C (i) = ([i]U, c(i) , r(i) ) and 0 j < p1 be such that j x (mod p1 ). Since [j]U = [x]U , where x is the secret key, C (j) will decrypt to give the message m.

It is very unlikely that any of the other ciphertexts C (i) (with i = j) will decrypt to give m. Therefore, with only p1 requests to the decryption oracle, the attacker can nd out the value of the private key x (mod p1 ). If an attacker does this for a series of primes p1 , p2 , .

. . , pk such that p1 p2 .

. . pk q, then it can recover the whole private key x using the Chinese Remainder Theorem (and it will only need to make p1 + p2 + .

. . + pk requests to the decryption oracle to do this).

Of course this attack only works if the elliptic curve contains a number of points with small prime orders. However, other variants of this type of attack exist which involve the attacker changing parts of the public parameters, such as the public key Y , the group generator P or even the elliptic curve E itself. These variants may work even if the original attack does not.

Details of methods that can be used to check the validity of elliptic curve points and elliptic curves themselves can be found in Section I.5. III.

3.3. Variable Length Symmetric Keys.

Another implicit assumption that is made in the security proofs is that the keys produced by the key derivation function are of a xed length l. This seems like a trivial point, but it can actually be very important, especially when one combines this with some details about the way in which most key derivation functions work. Most key derivation functions compute KD(U, l) by taking the rst l bits of an in nite pseudo-random sequence produced from the input U .

This means that, for any 0 < l < l, KD(U, l ) is the same as the rst l bits of KD(U, l). This is not a problem for ECIES providing that we use symmetric keys of a xed length l it becomes a problem, however, if we allow l to vary. In particular, some early versions of ECIES allowed the symmetric cipher to be a variable length Vernam cipher (see Section III.

1.3). In this case encryption is given by: Algorithm III.

1: ECIES Encryption (Weak Version) A message m, public key Y and the length of the MAC key l . OUTPUT: A ciphertext (U, c, r). 1.

Choose k R {1, . . .

, q}. 2. U [k]G.

INPUT:. (i) (i) (i) (i).
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