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Manindra Agrawal coauthored Primality and Identity Testing via Chinese Remaindering by Manindra Agrawal and Somenath Biswas with Somenath Biswas in 2003. The paper connects primality testing to polynomial identity testing, using randomized algorithms for circuit-represented polynomials over finite fields, the rationals and, for one procedure, integers modulo an arbitrary candidate.^[ref]

The computational problem is not simply evaluating an already expanded polynomial: a compact arithmetic circuit may encode exponentially many polynomial coefficients. A successful test must exploit that representation without materializing the expansion. This distinction connects an algebraic identity to an implementable randomized algorithm.

A randomized decision algorithm has one-sided error when one answer is always correct and only the other answer can be mistaken.

The guarantee concerns each fixed input, with probability taken over the choices of the randomized algorithm. Always state which answer is protected. In polynomial identity testing, call an input positive when its polynomial is identically zero, and let acceptance mean claiming an identity. Then the possibilities are:

Thus a zero polynomial evaluation can leave uncertainty, but a nonzero polynomial evaluation settles the question. An error bound for a fixed nonzero polynomial is not the conditional probability that a particular accepted input is an identity; that latter question would need a distribution on the inputs as well.

In computational complexity theory, a polynomial-time algorithm with this orientation and false acceptance probability at most $1/2$ establishes membership in co-RP. Reversing the question to ask whether the polynomial is nonzero gives the RP orientation: negative instances are always rejected. The random evaluation test for a polynomial identity explains the required field, total degree, sampling and evaluation assumptions.

We give the Agrawal–Biswas primality test, which has one-sided error. Small inputs and perfect powers can first be recognized deterministically. For every remaining integer $n$, put $d=\lceil 4{\log }_{2}n\rceil$ and choose a uniformly random monic polynomialUsing repeated squaring in the quotient ring $(\mathbb{Z}/n\mathbb{Z})[X]/(r)$, test the identityThis takes time polynomial in $\log n$ because every intermediate polynomial has degree below $d=O(\log n)$.

If $n$ is prime, the intermediate binomial coefficients are divisible by $n$, so the identity always holds. Now suppose that $n$ is composite and is not a prime power. Choose a prime divisor $p$ and write $n=p^{a}m$ with $p\nmid m$ and $m>1$. Over ${\mathbb{F}}_p$,because an intermediate coefficient equal to $m$ is nonzero modulo $p$. Henceis a nonzero polynomial of degree below $n$ over ${\mathbb{F}}_p$.

Reduction of random $r$ modulo $p$ is uniform among the $p^d$ monic degree-$d$ polynomials. The polynomial $F$ has at most $n/d$ distinct monic irreducible factors of degree $d$. On the other hand, the number $I_p(d)$ of monic irreducibles of degree $d$ obeys the standard lower boundWhenever $r\mathrm{mod}p$ is one of these irreducibles but does not divide $F$, the tested congruence fails. Thus one trial detects compositeness with probability at leastafter the finitely many small $n$ are handled directly. Repeating $2d$ times makes the probability of missing a composite less than $(1-1/(2d){)}^{2d}<1/2$, while a prime is never rejected. Therefore compositeness is in RP, and primality testing is in

A randomized algorithm makes some choices using random bits.

For a fixed input, the random bits can change the path taken by a randomized algorithm, its output, or its running time. Specify the distribution of those choices and which guarantee is required: one-sided error permits mistakes of only one kind, whereas ZPP requires a correct answer and bounds the expected running time. Polynomial identity testing makes this distinction concrete: choosing different field elements changes whether a nonidentity is detected.

A uniform distribution on a finite set must be implemented faithfully. To choose uniformly from $\{0,1,2,3,4\}$ using fair independent bits, read three bits as a number from zero to seven, discard five through seven, and repeat until a permitted value appears. Each retained value has the same chance. Taking the three-bit number modulo five instead would give zero, one and two twice as many preimages as three and four, invalidating a calculation that assumes equal chances.

Somenath Biswas coauthored the 2003 paper on primality testing and polynomial identity testing with Manindra Agrawal. Their randomized algorithm reduces a candidate integer's primality to a univariate polynomial identity over the integers modulo that candidate.^[ref]

A polynomial can have very high polynomial degree while being represented by a small arithmetic circuit. Expanding every polynomial coefficient can then be impractical. The paper also gives randomized algorithms for multivariate polynomial identity testing with running time polynomial in the circuit representation size and an error parameter. The input representation and probability of error are consequently essential parts of the algorithmic statement.