Unformatted text preview: y the signature, Bob computes sk mod n. He also computes a, which is the least integer larger than or equal to two times the number of bits of n divided by 3. If H(m) is less than or equal to sk mod n, and if sk mod n is less than H(m) + 2a, then the signature is considered valid. This algorithm works faster with precomputation. This precomputation can be done at any time and has nothing to do with the message being signed. After picking x, Alice could break step (2) into two partial steps. The first can be precomputed. (2a) Alice computes: u = xk mod n v = 1/(kxk  1) mod p (2b) Alice computes: w = the least integer that is larger than or equal to (H(m) – u)/pq) s = x + (wv mod p)pq For the size of numbers generally used, this precomputation speeds up the signature process by a factor of 10. Almost all the hard work is done in the precomputation stage. A discussion of modular arithmetic operations to speed ESIGN can be found in [1625,1624]. This algorithm can also be extended to work with elliptic curves [1206]. Security of ESIGN
When this algorithm was originally proposed, k was set to 2 [1215]. This was quickly broken by Ernie Brickell and John DeLaurentis [261], who then extended their attack to k = 3. A modified version of this algorithm [1203] was broken by Shamir [1204]. The variant proposed in [1204] was broken in [1553]. ESIGN is the current incarnation of this family of algorithms. Another new attack [963] does not work against ESIGN. The authors currently recommend these values for k: 8, 16, 32, 64, 128, 256, 512, and 1024. They also recommend that p and q each be of at least 192 bits, making n at least 576 bits long. (I think n should be twice that length.) With these parameters, the authors conjecture that ESIGN is as secure as RSA or Rabin. And their analysis shows favorable speed comparison to RSA, ElGamal, and DSA [582]. Patents
ESIGN is patented in the United States [1208], Canada, England, France, Germany, and Italy. Anyone who wishes to license the algorithm should contact Intellectual Property Department, NTT, 1–6 Uchisaiwaicho, 1chome, Chiyadaku, 100 Japan. Previous Table of Contents Next Products  Contact Us  Abo...
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