Thomas C. Hales (University of Pittsburgh)

(This article will be published in the Notices of the American Mathematical Society.)

Use once. Die once. — activist saying about insecure communication

This article gives a brief mathematical description of the NIST standard for cryptographically secure pseudo-random number generation by elliptic curves, the back door to the algorithm discovered by Ferguson and Shumow, and finally the design of the back door based on the Diffie-Hellman key exchange algorithm.

NIST (the National Institute for Standards and Technology) of the U.S. Department of Commerce derives its mandate from the U.S. Constitution, through the congressional power to "fix the standard of weights and measures." In brief, NIST establishes the basic standards of science and commerce. Whatever NIST says about cryptography becomes implemented in cryptographic applications throughout U.S. government agencies. Its influence leads to the widespread use of its standards in industry and the broad adoption of its standards internationally.

Through the Snowden disclosures, the NIST standard for pseudo-random number generation has fallen into disrepute. Here I describe the back door to the NIST standard for pseudo-random number generation in elementary and mathematically precise terms. The NIST standard offers three methods for pseudo-random number generation [NIST]. My remarks are limited to the third of the three methods, which is based on elliptic curves.

Random number generators can either be truly random (obtaining their values from randomness in the physical world such as a quantum mechanical process) or pseudo-random (obtaining their values from a deterministic algorithm, yet displaying a semblance of randomness). The significance of random number generation within the theory of algorithms can be gauged by Knuth's multivolume book, The Art of Computer Programming. It devotes a massive 193 pages (half of volume two) to the subject! A subclass of pseudo-random number generators are cryptographically secure, intended for use in cryptographic applications such as key generation, one-way hash functions, signature schemes, private key cryptosystems, and zero knowledge interactive proofs [Luby].

Elliptic curves as pseudo-random number generators

The NIST standard gives a list of explicit mathematical data (E,p,n,f,P,Q) to be used for pseudo-random number generation [NIST]. Here E is an elliptic curve defined over a finite field \mathbb{F}_p of prime order p. The group E(\mathbb{F}_p) has order n, which is prime for all of the curves that occur in the NIST standard. The elements of the group E(\mathbb{F}_p) consist of the set of points on an affine curve, together with a point at infinity which serves as the identity element of the group. The affine curve is defined by an equation y^2 = f(x) for some explicit cubic polynomial f in \mathbb{F}_p[x]. Finally, P and Q are given points on the affine curve.

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