A method is provided for performing elliptic curve cryptography that reduces the number of required computations to produce, for example, a key pair. The number of computations is reduced by changing how a random nonce used in the computations is selected. In an embodiment, a look-up table is generated having pre-computed scalar values and elliptic curve points. Every time a new pseudo-random value is created for use in the ECDSA, a combination of the look-up table values is used to create multiple intermediate values. One of the multiple intermediate values is randomly chosen as a replacement value for one of the existing table entries. Each time the look-up table is used, multiple entries in the look-up table are updated to new look-up table values as described. In this manner, new randomness is provided in every step to generate the next pseudo-random nonce as a combination of multiple internally stored temporary look-up table values. Alternately, another mathematical group may be used.