Portal:NIST

“ Choose the Type T low-complexity normal basis with the smallest T. For each binary field, the parameters are given for the above basis representations.
1.4 Choice of Curves
Two kinds of curves are given:
Pseudo-random curves are those whose coefficients are generated from the output of a seeded cryptographic hash. If the seed value is given along with the coefficients, it can be verified easily that the coefficients were indeed generated by that method. Special curves whose coefficients and underlying field have been selected to optimize the efficiency of the elliptic curve operations. ”

“ In order to generate the primes p and q, a primality test is required.
There are several fast probabilistic algorithms available. The following algorithm is a simplified version of a procedure due to M.O. Rabin, based in part on ideas of Gary L. Miller. [See Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, 1981, Algorithm P, page 379.] If this algorithm is iterated n times, it will produce a false prime with probability no greater than 1/4n. Therefore, n ≥ 50 will give an acceptable probability of error. To test whether an integer is prime:
Step 1. Set i = 1 and n ≥ 50. ”

“ APPENDIX 6.1: IMPLEMENTATION OF MODULAR ARITHMETIC
The prime moduli in the above examples are of a special type (called generalized Mersenne numbers) for which modular multiplication can be carried out more efficiently than in general. This appendix provides the rules for implementing this faster arithmetic, for each of the prime moduli appearing in the examples.
The usual way to multiply two integers (mod m) is to take the integer product and reduce it (mod m) . One therefore has the following problem: given an integer A less than m2, compute
B := A mod m. ”

“ Random Number Generation
FIPS 186-2 includes algorithms for the generation of a user's private key, x, and a user's per message secret number, k. These values must be generated randomly or pseudorandomly and must have values between 0 and the 160-bit prime q (as specified in the standard) . Techniques for generating x and k are provided in Appendix 3 of the standard.
Recently, an unpublished attack on DSA [2] was found that relies on the non-uniformity of the pseudorandom number generators (PRNGs) specified in Appendix 3 of the standard. ”

“ The coefficient b The base point x coordinate Gx The base point y coordinate Gy Pseudo-random curve (Normal Basis representation) :
The 160-bit input seed s to the SHA-1 based algorithm The coefficient b (i.e., the output of the SHA-1 based algorithm) The base point x coordinate Gx The base point y coordinate Gy Integers (such as T, m, and r) are given in decimal form; bit strings and field elements are given in hex. ”

“ Agency heads may act upon a written waiver request containing the information detailed above. Agency heads may also act without a written waiver request when they determine that conditions for meeting the standard cannot be met. Agency heads may approve waivers only by a written decision which explains the basis on which the agency head made the required finding (s) . ”

“ These mandates have given the Secretary of Commerce and NIST important responsibilities for improving the utilization and management of computer and related telecommunications systems in the Federal Government. The NIST, through its Information Technology Laboratory, provides leadership, technical guidance, and coordination of Government efforts in the development of standards and guidelines in these areas. ”

“ Form q from U by setting the most significant bit (the 2159 bit) and the least significant bit to 1. In terms of boolean operations, q = U OR 2159 OR 1. Note that 2159 < q < 2160. Step 4. Use a robust primality testing algorithm to test whether q is prime1. Step 5. If q is not prime, go to step 1. Step 6. Let counter = 0 and offset = 2. Step 7. For k = 0,...,n let Vk = SHA-1 [ (SEED + offset + k) mod 2g ] . 1 A robust primality test is one where the probability of a non-prime number passing the test is at most 2-80. 14 Step 8. Let W be the integer ”