FULL PRODUCT VERSION :
java version "1.8.0_45"
Java(TM) SE Runtime Environment (build 1.8.0_45-b14)
Java HotSpot(TM) 64-Bit Server VM (build 25.45-b02, mixed mode)

ADDITIONAL OS VERSION INFORMATION :
Darwin .local 13.2.0 Darwin Kernel Version 13.2.0: Thu Apr 17 23:03:13 PDT 2014; root:xnu-2422.100.13~1/RELEASE_X86_64 x86_64

A DESCRIPTION OF THE PROBLEM :
The results from the constructor for BigInteger that accepts the number of bits and an instance of Random does not produce uniform results. Numbers smaller than a very high limit are never generated.

A "workaround" that produces a better distribution of random numbers is to use
int nBits = random.nextInt(vMax);
BigInteger randomlyChosen = new BigInteger(nBits, random);
but those results also show a bias - the largest of numbers are not generated as often as the nearly uniform distribution of all numbers below them.

STEPS TO FOLLOW TO REPRODUCE THE PROBLEM :
SecureRandom sr = SecureRandom.getInstance("SHA1PRNG");
int nBits = 4096;
for (int i = 0; i < 10000; ++i) {
    // biased towards high numbers. never chooses below a high limit
    BigInteger number = new BigInteger(nBits, sr);
    System.out.println(number.toString());
}

EXPECTED VERSUS ACTUAL BEHAVIOR :
EXPECTED -
A uniform distribution of numbers between 0 to (2^numBits - 1), inclusive.
ACTUAL -
The results are numbers above a very high limit.

REPRODUCIBILITY :
This bug can be reproduced always.

---------- BEGIN SOURCE ----------
import java.security.SecureRandom;
import java.math.BigInteger;

public class T {

    public void test() throws Exception {

        SecureRandom sr = SecureRandom.getInstance("SHA1PRNG");
        int nBits = 4096;
        for (int i = 0; i < 1000; ++i) {
            // biased towards high numbers. never chooses below a high limit
            BigInteger number = new BigInteger(nBits, sr);
            System.out.println(number.toString());
        }
    }
}
---------- END SOURCE ----------

CUSTOMER SUBMITTED WORKAROUND :
A "workaround" that produces a better distribution of random numbers is to use
int nBits = random.nextInt(vMax);
BigInteger randomlyChosen = new BigInteger(nBits, random);
but those results also show a bias - the largest of numbers are not generated as often as the nearly uniform distribution of all numbers below them.