In SEC 2: Recommended Elliptic Curve Domain Parameters two types of finite fields are utilized - $\mathbb{F}_p$ and $\mathbb{F}_{2^m}$. In the case of sect193r1, $\mathbb{F}_{2^m}$ is the finite field, where $m = 193$ and is defined by $f(x) = x^{193}+x^{15}+1$

In SEC 1: Elliptic Curve Cryptography, the following ASN.1 structures are presented:

FieldTypes FIELD-ID ::= {
    { Prime-p IDENTIFIED BY prime-field } |
    { Characteristic-two IDENTIFIED BY characteristic-two-field }
prime-field OBJECT IDENTIFIER ::= { id-fieldType 1 }
Prime-p ::= INTEGER -- Field of size p.
id-fieldType OBJECT IDENTIFIER ::= { ansi-X9-62 fieldType(1)}
ansi-X9-62 OBJECT IDENTIFIER ::= {
    iso(1) member-body(2) us(840) 10045
characteristic-two-field OBJECT IDENTIFIER ::= { id-fieldType 2 }
Characteristic-two ::= SEQUENCE {
    m INTEGER, -- Field size 2m
    basis CHARACTERISTIC-TWO.&id({BasisTypes}),
    parameters CHARACTERISTIC-TWO.&Type({BasisTypes}{@basis})
    { NULL IDENTIFIED BY gnBasis } |
    { Trinomial IDENTIFIED BY tpBasis } |
    { Pentanomial IDENTIFIED BY ppBasis },
gnBasis OBJECT IDENTIFIER ::= { id-characteristic-two-basis 1 }
tpBasis OBJECT IDENTIFIER ::= { id-characteristic-two-basis 2 }
ppBasis OBJECT IDENTIFIER ::= { id-characteristic-two-basis 3 }
id-characteristic-two-basis OBJECT IDENTIFIER ::= {
    characteristic-two-field basisType(3)
Trinomial ::= INTEGER
Pentanomial ::= SEQUENCE {
    k1 INTEGER, -- k1 > 0
    k2 INTEGER, -- k2 > k1
    k3 INTEGER -- k3 > k2

$x^{193}+x^{15}+1$ is a trinomial but the "basis type" for it only consists of one integer? It seems to me that there are probably close to three integers - the exponents for $x$? eg. $193$, $15$ and $0$? idk - I guess it's just unclear to me how you're supposed to get $x^{193}+x^{15}+1$ from the ASN.1 or, conversely, how you're supposed to denote $x^{193}+x^{15}+1$ in the ASN.1.

Any ideas?


1 Answer 1


For a binary field $\mathbb{F}_{2^m}$, the polynomial necessarily has degree $m$ (otherwise the field would not have cardinal $2^m$), and its least significant coefficient must be $1$, not $0$ (otherwise the polynomial would not be irreducible, and the field would not be a field).

Therefore, for $\mathbb{F}_{2^{193}}$, if the polynomial is a trinomial, then it is already known that is is $x^{193}+x^j+1$ for some integer $1<j<193$. Only that integer $j$ needs to be encoded, which is why you only have one INTEGER. In other words, when trying to encode $x^{193}+x^{15}+1$, only the "15" needs to be encoded; the "193" and the "0" are implicit.

Similarly, for a pentanomial, you only have to encode three exponents, since the two extremes (0 and the field size) are implicit.


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