Eric Bach is an American

computer scientist A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (al ...

who has made contributions to

computational number theory In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithm ...

. Bach completed his undergraduate studies at the

University of Michigan, Ann Arbor , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

, and got his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

in computer science from the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

, in 1984 under the supervision of

Manuel Blum Manuel Blum (born 26 April 1938) is a Venezuelan-American computer scientist who received the Turing Award in 1995 "In recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and ...

. He is currently a professor at the Computer Science Department,

University of Wisconsin–Madison A university () is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) education and research which awards academic degrees in several Discipline (academia), academic disciplines. Universities ty ...

. Among other work, he gave explicit bounds for the

Chebotarev density theorem Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...

, which imply that if one assumes the

generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...

then

\left(\mathbb/n\mathbb\right)^*

is generated by its elements smaller than 2(log ''n'')². This result shows that the generalized Riemann hypothesis implies tight bounds for the necessary run-time of the deterministic version of the

Miller–Rabin primality test The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen prima ...

. Bach also did some of the first work on pinning down the actual expected run-time of the Pollard rho method where previous work relied on heuristic estimates and empirical data. He is the namesake of

Bach's algorithm Bach's algorithm is a probabilistic polynomial time algorithm for generating random numbers along with their factorizations, named after its discoverer, Eric Bach. It is of interest because no algorithm is known that efficiently factors numbers, ...

for generating random factored numbers.

References

American computer scientists Living people University of Michigan alumni UC Berkeley College of Engineering alumni University of Wisconsin–Madison faculty Year of birth missing (living people) Number theorists {{US-mathematician-stub